Anisotropic x16 LOD (line direction, u)

Percentage Accurate: 76.0% → 76.2%

Time: 8.5s

Alternatives: 8

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_3 := t\_2 \cdot t\_2 + t\_0 \cdot t\_0\\ t_4 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_5 := t\_1 \cdot t\_1 + t\_4 \cdot t\_4\\ t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t\_3, t\_5\right)}}\\ \mathbf{if}\;t\_3 \geq t\_5:\\ \;\;\;\;t\_6 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_6 \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor w) dX.u))
        (t_3 (+ (* t_2 t_2) (* t_0 t_0)))
        (t_4 (* (floor h) dY.v))
        (t_5 (+ (* t_1 t_1) (* t_4 t_4)))
        (t_6 (/ 1.0 (sqrt (fmax t_3 t_5)))))
   (if (>= t_3 t_5) (* t_6 t_2) (* t_6 t_1))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = (t_2 * t_2) + (t_0 * t_0);
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = (t_1 * t_1) + (t_4 * t_4);
	float t_6 = 1.0f / sqrtf(fmaxf(t_3, t_5));
	float tmp;
	if (t_3 >= t_5) {
		tmp = t_6 * t_2;
	} else {
		tmp = t_6 * t_1;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(fmax(t_3, t_5)))
	tmp = Float32(0.0)
	if (t_3 >= t_5)
		tmp = Float32(t_6 * t_2);
	else
		tmp = Float32(t_6 * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(w) * dX_46_u;
	t_3 = (t_2 * t_2) + (t_0 * t_0);
	t_4 = floor(h) * dY_46_v;
	t_5 = (t_1 * t_1) + (t_4 * t_4);
	t_6 = single(1.0) / sqrt(max(t_3, t_5));
	tmp = single(0.0);
	if (t_3 >= t_5)
		tmp = t_6 * t_2;
	else
		tmp = t_6 * t_1;
	end
	tmp_2 = tmp;
end

Initial Program: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_3 := t\_2 \cdot t\_2 + t\_0 \cdot t\_0\\ t_4 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_5 := t\_1 \cdot t\_1 + t\_4 \cdot t\_4\\ t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t\_3, t\_5\right)}}\\ \mathbf{if}\;t\_3 \geq t\_5:\\ \;\;\;\;t\_6 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_6 \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor w) dX.u))
        (t_3 (+ (* t_2 t_2) (* t_0 t_0)))
        (t_4 (* (floor h) dY.v))
        (t_5 (+ (* t_1 t_1) (* t_4 t_4)))
        (t_6 (/ 1.0 (sqrt (fmax t_3 t_5)))))
   (if (>= t_3 t_5) (* t_6 t_2) (* t_6 t_1))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = (t_2 * t_2) + (t_0 * t_0);
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = (t_1 * t_1) + (t_4 * t_4);
	float t_6 = 1.0f / sqrtf(fmaxf(t_3, t_5));
	float tmp;
	if (t_3 >= t_5) {
		tmp = t_6 * t_2;
	} else {
		tmp = t_6 * t_1;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(fmax(t_3, t_5)))
	tmp = Float32(0.0)
	if (t_3 >= t_5)
		tmp = Float32(t_6 * t_2);
	else
		tmp = Float32(t_6 * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(w) * dX_46_u;
	t_3 = (t_2 * t_2) + (t_0 * t_0);
	t_4 = floor(h) * dY_46_v;
	t_5 = (t_1 * t_1) + (t_4 * t_4);
	t_6 = single(1.0) / sqrt(max(t_3, t_5));
	tmp = single(0.0);
	if (t_3 >= t_5)
		tmp = t_6 * t_2;
	else
		tmp = t_6 * t_1;
	end
	tmp_2 = tmp;
end

Alternative 1: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\\ t_1 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_2 := \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\\ t_3 := \mathsf{fma}\left(t\_0, \left\lfloor h\right\rfloor , t\_2 \cdot \left\lfloor w\right\rfloor \right)\\ t_4 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_5 := \mathsf{fma}\left(t\_1 \cdot \left\lfloor h\right\rfloor , dY.v, t\_4 \cdot t\_4\right)\\ \mathbf{if}\;\mathsf{fma}\left(t\_2, \left\lfloor w\right\rfloor , t\_0 \cdot \left\lfloor h\right\rfloor \right) \geq t\_5:\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_4 \cdot dY.u, \left\lfloor w\right\rfloor , \left(t\_1 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), t\_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(t\_5, t\_3\right)}}\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (* dX.v (floor h)) dX.v))
        (t_1 (* dY.v (floor h)))
        (t_2 (* (* dX.u (floor w)) dX.u))
        (t_3 (fma t_0 (floor h) (* t_2 (floor w))))
        (t_4 (* dY.u (floor w)))
        (t_5 (fma (* t_1 (floor h)) dY.v (* t_4 t_4))))
   (if (>= (fma t_2 (floor w) (* t_0 (floor h))) t_5)
     (/
      (* (- dX.u) (floor w))
      (-
       (sqrt
        (fmax (fma (* t_4 dY.u) (floor w) (* (* t_1 dY.v) (floor h))) t_3))))
     (/ (* (- dY.u) (floor w)) (- (sqrt (fmax t_5 t_3)))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = (dX_46_v * floorf(h)) * dX_46_v;
	float t_1 = dY_46_v * floorf(h);
	float t_2 = (dX_46_u * floorf(w)) * dX_46_u;
	float t_3 = fmaf(t_0, floorf(h), (t_2 * floorf(w)));
	float t_4 = dY_46_u * floorf(w);
	float t_5 = fmaf((t_1 * floorf(h)), dY_46_v, (t_4 * t_4));
	float tmp;
	if (fmaf(t_2, floorf(w), (t_0 * floorf(h))) >= t_5) {
		tmp = (-dX_46_u * floorf(w)) / -sqrtf(fmaxf(fmaf((t_4 * dY_46_u), floorf(w), ((t_1 * dY_46_v) * floorf(h))), t_3));
	} else {
		tmp = (-dY_46_u * floorf(w)) / -sqrtf(fmaxf(t_5, t_3));
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(Float32(dX_46_v * floor(h)) * dX_46_v)
	t_1 = Float32(dY_46_v * floor(h))
	t_2 = Float32(Float32(dX_46_u * floor(w)) * dX_46_u)
	t_3 = fma(t_0, floor(h), Float32(t_2 * floor(w)))
	t_4 = Float32(dY_46_u * floor(w))
	t_5 = fma(Float32(t_1 * floor(h)), dY_46_v, Float32(t_4 * t_4))
	tmp = Float32(0.0)
	if (fma(t_2, floor(w), Float32(t_0 * floor(h))) >= t_5)
		tmp = Float32(Float32(Float32(-dX_46_u) * floor(w)) / Float32(-sqrt(fmax(fma(Float32(t_4 * dY_46_u), floor(w), Float32(Float32(t_1 * dY_46_v) * floor(h))), t_3))));
	else
		tmp = Float32(Float32(Float32(-dY_46_u) * floor(w)) / Float32(-sqrt(fmax(t_5, t_3))));
	end
	return tmp
end

Derivation

1. Initial program 76.0%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

3. Applied rewrites 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\color{blue}{\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

5. Applied rewrites 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

7. Applied rewrites 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}:\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]
8. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \color{blue}{\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }:\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \color{blue}{\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right)} \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   3. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \left(\color{blue}{\left(dY.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   4. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \color{blue}{\left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)} \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   5. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \left(dY.v \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   6. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \color{blue}{dY.v \cdot \left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)} + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   7. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v} + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   8. lower-fma.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \color{blue}{\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}:\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   9. lower-*.f32 76.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor }, dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   10. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dY.v\right)} \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   11. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\color{blue}{\left(dY.v \cdot \left\lfloor h\right\rfloor \right)} \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   12. lift-*.f32 76.2

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\color{blue}{\left(dY.v \cdot \left\lfloor h\right\rfloor \right)} \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   13. rem-square-sqrt N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \color{blue}{\sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }}\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   14. sqrt-unprod N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \color{blue}{\sqrt{\left(\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left(\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   15. sqrt-prod N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \color{blue}{\sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }}\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   16. lower-unsound-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \color{blue}{\sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }}\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

9. Applied rewrites 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]
10. Step-by-step derivation

    1. lift-fma.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\color{blue}{dY.u}\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    2. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    3. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    4. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    5. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    6. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(dY.v \cdot \left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    7. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    8. lower-fma.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\color{blue}{dY.u}\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    9. lower-*.f32 76.2

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    10. lift-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    11. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    12. lift-*.f32 76.2

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    13. rem-square-sqrt N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }\right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    14. sqrt-unprod N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \sqrt{\left(\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left(\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}\right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    15. sqrt-prod N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }\right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

    16. lower-unsound-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }\right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

11. Applied rewrites 76.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\color{blue}{dY.u}\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]
12. Add Preprocessing

Alternative 2: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\\ t_1 := \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\\ t_2 := \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\\ t_3 := \mathsf{fma}\left(t\_0, \left\lfloor h\right\rfloor , t\_2 \cdot \left\lfloor w\right\rfloor \right)\\ t_4 := \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\\ t_5 := \mathsf{fma}\left(t\_1, \left\lfloor h\right\rfloor , t\_4 \cdot \left\lfloor w\right\rfloor \right)\\ \mathbf{if}\;\mathsf{fma}\left(t\_2, \left\lfloor w\right\rfloor , t\_0 \cdot \left\lfloor h\right\rfloor \right) \geq t\_5:\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_4, \left\lfloor w\right\rfloor , t\_1 \cdot \left\lfloor h\right\rfloor \right), t\_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(t\_5, t\_3\right)}}\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (* dX.v (floor h)) dX.v))
        (t_1 (* (* dY.v (floor h)) dY.v))
        (t_2 (* (* dX.u (floor w)) dX.u))
        (t_3 (fma t_0 (floor h) (* t_2 (floor w))))
        (t_4 (* (* dY.u (floor w)) dY.u))
        (t_5 (fma t_1 (floor h) (* t_4 (floor w)))))
   (if (>= (fma t_2 (floor w) (* t_0 (floor h))) t_5)
     (/
      (* (- dX.u) (floor w))
      (- (sqrt (fmax (fma t_4 (floor w) (* t_1 (floor h))) t_3))))
     (/ (* (- dY.u) (floor w)) (- (sqrt (fmax t_5 t_3)))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = (dX_46_v * floorf(h)) * dX_46_v;
	float t_1 = (dY_46_v * floorf(h)) * dY_46_v;
	float t_2 = (dX_46_u * floorf(w)) * dX_46_u;
	float t_3 = fmaf(t_0, floorf(h), (t_2 * floorf(w)));
	float t_4 = (dY_46_u * floorf(w)) * dY_46_u;
	float t_5 = fmaf(t_1, floorf(h), (t_4 * floorf(w)));
	float tmp;
	if (fmaf(t_2, floorf(w), (t_0 * floorf(h))) >= t_5) {
		tmp = (-dX_46_u * floorf(w)) / -sqrtf(fmaxf(fmaf(t_4, floorf(w), (t_1 * floorf(h))), t_3));
	} else {
		tmp = (-dY_46_u * floorf(w)) / -sqrtf(fmaxf(t_5, t_3));
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(Float32(dX_46_v * floor(h)) * dX_46_v)
	t_1 = Float32(Float32(dY_46_v * floor(h)) * dY_46_v)
	t_2 = Float32(Float32(dX_46_u * floor(w)) * dX_46_u)
	t_3 = fma(t_0, floor(h), Float32(t_2 * floor(w)))
	t_4 = Float32(Float32(dY_46_u * floor(w)) * dY_46_u)
	t_5 = fma(t_1, floor(h), Float32(t_4 * floor(w)))
	tmp = Float32(0.0)
	if (fma(t_2, floor(w), Float32(t_0 * floor(h))) >= t_5)
		tmp = Float32(Float32(Float32(-dX_46_u) * floor(w)) / Float32(-sqrt(fmax(fma(t_4, floor(w), Float32(t_1 * floor(h))), t_3))));
	else
		tmp = Float32(Float32(Float32(-dY_46_u) * floor(w)) / Float32(-sqrt(fmax(t_5, t_3))));
	end
	return tmp
end

Derivation

1. Initial program 76.0%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

3. Applied rewrites 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\color{blue}{\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

5. Applied rewrites 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

7. Applied rewrites 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}:\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-dY.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]
8. Add Preprocessing

Alternative 3: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\\ t_1 := \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\\ t_2 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_3 := \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\\ t_4 := \mathsf{fma}\left(t\_0, \left\lfloor h\right\rfloor , t\_3 \cdot \left\lfloor w\right\rfloor \right)\\ t_5 := t\_2 \cdot dY.u\\ t_6 := \mathsf{fma}\left(t\_1, \left\lfloor h\right\rfloor , t\_5 \cdot \left\lfloor w\right\rfloor \right)\\ \mathbf{if}\;\mathsf{fma}\left(t\_3, \left\lfloor w\right\rfloor , t\_0 \cdot \left\lfloor h\right\rfloor \right) \geq t\_6:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_5, \left\lfloor w\right\rfloor , t\_1 \cdot \left\lfloor h\right\rfloor \right), t\_4\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{max}\left(t\_6, t\_4\right)}}\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (* dX.v (floor h)) dX.v))
        (t_1 (* (* dY.v (floor h)) dY.v))
        (t_2 (* dY.u (floor w)))
        (t_3 (* (* dX.u (floor w)) dX.u))
        (t_4 (fma t_0 (floor h) (* t_3 (floor w))))
        (t_5 (* t_2 dY.u))
        (t_6 (fma t_1 (floor h) (* t_5 (floor w)))))
   (if (>= (fma t_3 (floor w) (* t_0 (floor h))) t_6)
     (*
      (/ dX.u (sqrt (fmax (fma t_5 (floor w) (* t_1 (floor h))) t_4)))
      (floor w))
     (/ t_2 (sqrt (fmax t_6 t_4))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = (dX_46_v * floorf(h)) * dX_46_v;
	float t_1 = (dY_46_v * floorf(h)) * dY_46_v;
	float t_2 = dY_46_u * floorf(w);
	float t_3 = (dX_46_u * floorf(w)) * dX_46_u;
	float t_4 = fmaf(t_0, floorf(h), (t_3 * floorf(w)));
	float t_5 = t_2 * dY_46_u;
	float t_6 = fmaf(t_1, floorf(h), (t_5 * floorf(w)));
	float tmp;
	if (fmaf(t_3, floorf(w), (t_0 * floorf(h))) >= t_6) {
		tmp = (dX_46_u / sqrtf(fmaxf(fmaf(t_5, floorf(w), (t_1 * floorf(h))), t_4))) * floorf(w);
	} else {
		tmp = t_2 / sqrtf(fmaxf(t_6, t_4));
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(Float32(dX_46_v * floor(h)) * dX_46_v)
	t_1 = Float32(Float32(dY_46_v * floor(h)) * dY_46_v)
	t_2 = Float32(dY_46_u * floor(w))
	t_3 = Float32(Float32(dX_46_u * floor(w)) * dX_46_u)
	t_4 = fma(t_0, floor(h), Float32(t_3 * floor(w)))
	t_5 = Float32(t_2 * dY_46_u)
	t_6 = fma(t_1, floor(h), Float32(t_5 * floor(w)))
	tmp = Float32(0.0)
	if (fma(t_3, floor(w), Float32(t_0 * floor(h))) >= t_6)
		tmp = Float32(Float32(dX_46_u / sqrt(fmax(fma(t_5, floor(w), Float32(t_1 * floor(h))), t_4))) * floor(w));
	else
		tmp = Float32(t_2 / sqrt(fmax(t_6, t_4)));
	end
	return tmp
end

Derivation

1. Initial program 76.0%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

3. Applied rewrites 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\color{blue}{\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot \left\lfloor w\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

5. Applied rewrites 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

7. Applied rewrites 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}:\\ \;\;\;\;\frac{dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot \left\lfloor w\right\rfloor \\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]
8. Add Preprocessing

Alternative 4: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\\ t_1 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_2 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_3 := \frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_2 \cdot \left\lfloor h\right\rfloor , dY.v, t\_1 \cdot t\_1\right), \mathsf{fma}\left(t\_0, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}}\\ \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , t\_0 \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(t\_1 \cdot dY.u, \left\lfloor w\right\rfloor , \left(t\_2 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;t\_3 \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot dY.u\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (* dX.u (floor w)) dX.u))
        (t_1 (* dY.u (floor w)))
        (t_2 (* dY.v (floor h)))
        (t_3
         (/
          (floor w)
          (sqrt
           (fmax
            (fma (* t_2 (floor h)) dY.v (* t_1 t_1))
            (fma t_0 (floor w) (* (* (* dX.v (floor h)) dX.v) (floor h))))))))
   (if (>=
        (fma (* (* dX.v dX.v) (floor h)) (floor h) (* t_0 (floor w)))
        (fma (* t_1 dY.u) (floor w) (* (* t_2 dY.v) (floor h))))
     (* t_3 dX.u)
     (* t_3 dY.u))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = (dX_46_u * floorf(w)) * dX_46_u;
	float t_1 = dY_46_u * floorf(w);
	float t_2 = dY_46_v * floorf(h);
	float t_3 = floorf(w) / sqrtf(fmaxf(fmaf((t_2 * floorf(h)), dY_46_v, (t_1 * t_1)), fmaf(t_0, floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))));
	float tmp;
	if (fmaf(((dX_46_v * dX_46_v) * floorf(h)), floorf(h), (t_0 * floorf(w))) >= fmaf((t_1 * dY_46_u), floorf(w), ((t_2 * dY_46_v) * floorf(h)))) {
		tmp = t_3 * dX_46_u;
	} else {
		tmp = t_3 * dY_46_u;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(Float32(dX_46_u * floor(w)) * dX_46_u)
	t_1 = Float32(dY_46_u * floor(w))
	t_2 = Float32(dY_46_v * floor(h))
	t_3 = Float32(floor(w) / sqrt(fmax(fma(Float32(t_2 * floor(h)), dY_46_v, Float32(t_1 * t_1)), fma(t_0, floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h))))))
	tmp = Float32(0.0)
	if (fma(Float32(Float32(dX_46_v * dX_46_v) * floor(h)), floor(h), Float32(t_0 * floor(w))) >= fma(Float32(t_1 * dY_46_u), floor(w), Float32(Float32(t_2 * dY_46_v) * floor(h))))
		tmp = Float32(t_3 * dX_46_u);
	else
		tmp = Float32(t_3 * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 76.0%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

3. Applied rewrites 75.9%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ } \end{array}} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(dX.v \cdot \color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

5. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

7. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right), \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

9. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

11. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\color{blue}{\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

13. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
14. Step-by-step derivation

    1. lift-fma.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }, \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    2. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right)} \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    3. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\color{blue}{\left(dY.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    4. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\color{blue}{\left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)} \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    5. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(dY.v \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    6. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\color{blue}{dY.v \cdot \left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)} + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    7. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\color{blue}{\left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v} + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    8. lower-fma.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    9. lower-*.f32 75.9

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor }, dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    10. lift-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot dY.v\right)} \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    11. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(dY.v \cdot \left\lfloor h\right\rfloor \right)} \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    12. lift-*.f32 75.9

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(dY.v \cdot \left\lfloor h\right\rfloor \right)} \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    13. rem-square-sqrt N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \color{blue}{\sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }}\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    14. sqrt-unprod N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \color{blue}{\sqrt{\left(\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left(\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    15. sqrt-prod N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \color{blue}{\sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }}\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    16. lower-unsound-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \color{blue}{\sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }}\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

15. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}, \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
16. Step-by-step derivation

    1. lift-fma.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    2. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    3. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    4. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    5. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    6. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(dY.v \cdot \left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    7. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v + \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    8. lower-fma.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    9. lower-*.f32 75.9

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    10. lift-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    11. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    12. lift-*.f32 75.9

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    13. rem-square-sqrt N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    14. sqrt-unprod N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \sqrt{\left(\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left(\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    15. sqrt-prod N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    16. lower-unsound-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

17. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor , dY.v, \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
18. Add Preprocessing

Alternative 5: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_2 := \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\\ t_3 := \frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_2, \left\lfloor h\right\rfloor , t\_0 \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, t\_1 \cdot t\_1\right)\right)}}\\ \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(t\_0, \left\lfloor w\right\rfloor , t\_2 \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;t\_3 \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot dY.u\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (* dY.u (floor w)) dY.u))
        (t_1 (* (floor h) dX.v))
        (t_2 (* (* dY.v (floor h)) dY.v))
        (t_3
         (/
          (floor w)
          (sqrt
           (fmax
            (fma t_2 (floor h) (* t_0 (floor w)))
            (fma (* (* (floor w) (floor w)) dX.u) dX.u (* t_1 t_1)))))))
   (if (>=
        (fma
         (* (* dX.v dX.v) (floor h))
         (floor h)
         (* (* (* dX.u (floor w)) dX.u) (floor w)))
        (fma t_0 (floor w) (* t_2 (floor h))))
     (* t_3 dX.u)
     (* t_3 dY.u))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = (dY_46_u * floorf(w)) * dY_46_u;
	float t_1 = floorf(h) * dX_46_v;
	float t_2 = (dY_46_v * floorf(h)) * dY_46_v;
	float t_3 = floorf(w) / sqrtf(fmaxf(fmaf(t_2, floorf(h), (t_0 * floorf(w))), fmaf(((floorf(w) * floorf(w)) * dX_46_u), dX_46_u, (t_1 * t_1))));
	float tmp;
	if (fmaf(((dX_46_v * dX_46_v) * floorf(h)), floorf(h), (((dX_46_u * floorf(w)) * dX_46_u) * floorf(w))) >= fmaf(t_0, floorf(w), (t_2 * floorf(h)))) {
		tmp = t_3 * dX_46_u;
	} else {
		tmp = t_3 * dY_46_u;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(Float32(dY_46_u * floor(w)) * dY_46_u)
	t_1 = Float32(floor(h) * dX_46_v)
	t_2 = Float32(Float32(dY_46_v * floor(h)) * dY_46_v)
	t_3 = Float32(floor(w) / sqrt(fmax(fma(t_2, floor(h), Float32(t_0 * floor(w))), fma(Float32(Float32(floor(w) * floor(w)) * dX_46_u), dX_46_u, Float32(t_1 * t_1)))))
	tmp = Float32(0.0)
	if (fma(Float32(Float32(dX_46_v * dX_46_v) * floor(h)), floor(h), Float32(Float32(Float32(dX_46_u * floor(w)) * dX_46_u) * floor(w))) >= fma(t_0, floor(w), Float32(t_2 * floor(h))))
		tmp = Float32(t_3 * dX_46_u);
	else
		tmp = Float32(t_3 * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 76.0%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

3. Applied rewrites 75.9%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ } \end{array}} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(dX.v \cdot \color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

5. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

7. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right), \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

9. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

11. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\color{blue}{\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

13. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
14. Step-by-step derivation

    1. lift-fma.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    2. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left\lfloor w\right\rfloor \cdot \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right)} + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    3. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left\lfloor w\right\rfloor \cdot \color{blue}{\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right)} + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    4. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left\lfloor w\right\rfloor \cdot \left(\color{blue}{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)} \cdot dX.u\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    5. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left\lfloor w\right\rfloor \cdot \color{blue}{\left(dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)} + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    6. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left\lfloor w\right\rfloor \cdot \left(dX.u \cdot \color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    7. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)} + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    8. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    9. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    10. lift-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)} + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    11. swap-sqr N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)} + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    12. associate-*r* N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot dX.u} + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    13. lower-fma.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

15. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)}\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
16. Step-by-step derivation

    1. lift-fma.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    2. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left\lfloor w\right\rfloor \cdot \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    3. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left\lfloor w\right\rfloor \cdot \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    4. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left\lfloor w\right\rfloor \cdot \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    5. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left\lfloor w\right\rfloor \cdot \left(dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    6. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left\lfloor w\right\rfloor \cdot \left(dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    7. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    8. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    9. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    10. lift-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    11. swap-sqr N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    12. associate-*r* N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot dX.u + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    13. lower-fma.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

17. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right)\right)}} \cdot dY.u\\ \end{array} \]
18. Add Preprocessing

Alternative 6: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\\ t_1 := \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_3 := \frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0, \left\lfloor h\right\rfloor , t\_1 \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, t\_2 \cdot t\_2\right)\right)}}\\ \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(t\_1, \left\lfloor w\right\rfloor , t\_0 \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;t\_3 \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot dY.u\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (* dY.v (floor h)) dY.v))
        (t_1 (* (* dY.u (floor w)) dY.u))
        (t_2 (* (floor w) dX.u))
        (t_3
         (/
          (floor w)
          (sqrt
           (fmax
            (fma t_0 (floor h) (* t_1 (floor w)))
            (fma (* (floor h) (floor h)) (* dX.v dX.v) (* t_2 t_2)))))))
   (if (>=
        (fma
         (* (* dX.v dX.v) (floor h))
         (floor h)
         (* (* (* dX.u (floor w)) dX.u) (floor w)))
        (fma t_1 (floor w) (* t_0 (floor h))))
     (* t_3 dX.u)
     (* t_3 dY.u))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = (dY_46_v * floorf(h)) * dY_46_v;
	float t_1 = (dY_46_u * floorf(w)) * dY_46_u;
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = floorf(w) / sqrtf(fmaxf(fmaf(t_0, floorf(h), (t_1 * floorf(w))), fmaf((floorf(h) * floorf(h)), (dX_46_v * dX_46_v), (t_2 * t_2))));
	float tmp;
	if (fmaf(((dX_46_v * dX_46_v) * floorf(h)), floorf(h), (((dX_46_u * floorf(w)) * dX_46_u) * floorf(w))) >= fmaf(t_1, floorf(w), (t_0 * floorf(h)))) {
		tmp = t_3 * dX_46_u;
	} else {
		tmp = t_3 * dY_46_u;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(Float32(dY_46_v * floor(h)) * dY_46_v)
	t_1 = Float32(Float32(dY_46_u * floor(w)) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(floor(w) / sqrt(fmax(fma(t_0, floor(h), Float32(t_1 * floor(w))), fma(Float32(floor(h) * floor(h)), Float32(dX_46_v * dX_46_v), Float32(t_2 * t_2)))))
	tmp = Float32(0.0)
	if (fma(Float32(Float32(dX_46_v * dX_46_v) * floor(h)), floor(h), Float32(Float32(Float32(dX_46_u * floor(w)) * dX_46_u) * floor(w))) >= fma(t_1, floor(w), Float32(t_0 * floor(h))))
		tmp = Float32(t_3 * dX_46_u);
	else
		tmp = Float32(t_3 * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 76.0%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

3. Applied rewrites 75.9%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ } \end{array}} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(dX.v \cdot \color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

5. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

7. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right), \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

9. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

11. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\color{blue}{\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

13. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
14. Step-by-step derivation

    1. lift-fma.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    2. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor } + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    3. +-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor }\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    4. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor } + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    5. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right)} \cdot \left\lfloor h\right\rfloor + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    6. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right)} + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    7. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    8. unswap-sqr N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)} + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    9. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(dX.v \cdot dX.v\right)} \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    10. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)} + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    11. lower-fma.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    12. lower-*.f32 75.9

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor }, dX.v \cdot dX.v, \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    13. lift-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \color{blue}{\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor }\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    14. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \color{blue}{\left\lfloor w\right\rfloor \cdot \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right)}\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    15. lift-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left\lfloor w\right\rfloor \cdot \color{blue}{\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right)}\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    16. lift-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left\lfloor w\right\rfloor \cdot \left(\color{blue}{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)} \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    17. associate-*l* N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left\lfloor w\right\rfloor \cdot \color{blue}{\left(dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)}\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    18. lift-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left\lfloor w\right\rfloor \cdot \left(dX.u \cdot \color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

15. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \color{blue}{\mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)}\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
16. Step-by-step derivation

    1. lift-fma.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    2. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    3. +-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    4. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    5. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    6. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    7. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    8. unswap-sqr N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(dX.v \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    9. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(dX.v \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    10. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) + \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]

    11. lower-fma.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    12. lower-*.f32 75.9

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    13. lift-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    14. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left\lfloor w\right\rfloor \cdot \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right)\right)\right)}} \cdot dY.u\\ \end{array} \]

    15. lift-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left\lfloor w\right\rfloor \cdot \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right)\right)\right)}} \cdot dY.u\\ \end{array} \]

    16. lift-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left\lfloor w\right\rfloor \cdot \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right)\right)\right)}} \cdot dY.u\\ \end{array} \]

    17. associate-*l* N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left\lfloor w\right\rfloor \cdot \left(dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)\right)}} \cdot dY.u\\ \end{array} \]

    18. lift-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left\lfloor w\right\rfloor \cdot \left(dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)\right)}} \cdot dY.u\\ \end{array} \]

17. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dX.v \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right)\right)}} \cdot dY.u\\ \end{array} \]
18. Add Preprocessing

Alternative 7: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\\ t_1 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_2 := \frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , t\_1 \cdot t\_1\right), \mathsf{fma}\left(t\_0, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}}\\ \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , t\_0 \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(t\_1 \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;t\_2 \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot dY.u\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (* dX.u (floor w)) dX.u))
        (t_1 (* dY.u (floor w)))
        (t_2
         (/
          (floor w)
          (sqrt
           (fmax
            (fma (* (* dY.v dY.v) (floor h)) (floor h) (* t_1 t_1))
            (fma t_0 (floor w) (* (* (* dX.v (floor h)) dX.v) (floor h))))))))
   (if (>=
        (fma (* (* dX.v dX.v) (floor h)) (floor h) (* t_0 (floor w)))
        (fma (* t_1 dY.u) (floor w) (* (* (* dY.v (floor h)) dY.v) (floor h))))
     (* t_2 dX.u)
     (* t_2 dY.u))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = (dX_46_u * floorf(w)) * dX_46_u;
	float t_1 = dY_46_u * floorf(w);
	float t_2 = floorf(w) / sqrtf(fmaxf(fmaf(((dY_46_v * dY_46_v) * floorf(h)), floorf(h), (t_1 * t_1)), fmaf(t_0, floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))));
	float tmp;
	if (fmaf(((dX_46_v * dX_46_v) * floorf(h)), floorf(h), (t_0 * floorf(w))) >= fmaf((t_1 * dY_46_u), floorf(w), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)))) {
		tmp = t_2 * dX_46_u;
	} else {
		tmp = t_2 * dY_46_u;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(Float32(dX_46_u * floor(w)) * dX_46_u)
	t_1 = Float32(dY_46_u * floor(w))
	t_2 = Float32(floor(w) / sqrt(fmax(fma(Float32(Float32(dY_46_v * dY_46_v) * floor(h)), floor(h), Float32(t_1 * t_1)), fma(t_0, floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h))))))
	tmp = Float32(0.0)
	if (fma(Float32(Float32(dX_46_v * dX_46_v) * floor(h)), floor(h), Float32(t_0 * floor(w))) >= fma(Float32(t_1 * dY_46_u), floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h))))
		tmp = Float32(t_2 * dX_46_u);
	else
		tmp = Float32(t_2 * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 76.0%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

3. Applied rewrites 75.9%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ } \end{array}} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(dX.v \cdot \color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

5. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

7. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right), \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

9. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

11. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\color{blue}{\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

13. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
14. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v}, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    2. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(dY.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    3. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)}, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    4. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \color{blue}{\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    5. associate-*r* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    6. lower-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    7. lower-*.f32 75.9

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(dY.v \cdot dY.v\right)} \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    8. rem-square-sqrt N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \color{blue}{\sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }}\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    9. sqrt-unprod N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \color{blue}{\sqrt{\left(\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left(\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    10. sqrt-prod N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \color{blue}{\sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }}\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    11. lower-unsound-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \color{blue}{\sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }}\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

15. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)}, \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
16. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    2. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    3. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right), \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    4. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.v \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right), \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    5. associate-*r* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    6. lower-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    7. lower-*.f32 75.9

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    8. rem-square-sqrt N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    9. sqrt-unprod N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \sqrt{\left(\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left(\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    10. sqrt-prod N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

    11. lower-unsound-*.f32 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor } \cdot \sqrt{\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor }\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

17. Applied rewrites 75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
18. Add Preprocessing

Alternative 8: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\\ t_1 := \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\\ t_2 := \frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(t\_0, t\_1\right)}}\\ \mathbf{if}\;t\_1 \geq t\_0:\\ \;\;\;\;t\_2 \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot dY.u\\ \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0
         (fma
          (* (* dY.u (floor w)) dY.u)
          (floor w)
          (* (* (* dY.v (floor h)) dY.v) (floor h))))
        (t_1
         (fma
          (* (* dX.v dX.v) (floor h))
          (floor h)
          (* (* (* dX.u (floor w)) dX.u) (floor w))))
        (t_2 (/ (floor w) (sqrt (fmax t_0 t_1)))))
   (if (>= t_1 t_0) (* t_2 dX.u) (* t_2 dY.u))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = fmaf(((dY_46_u * floorf(w)) * dY_46_u), floorf(w), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)));
	float t_1 = fmaf(((dX_46_v * dX_46_v) * floorf(h)), floorf(h), (((dX_46_u * floorf(w)) * dX_46_u) * floorf(w)));
	float t_2 = floorf(w) / sqrtf(fmaxf(t_0, t_1));
	float tmp;
	if (t_1 >= t_0) {
		tmp = t_2 * dX_46_u;
	} else {
		tmp = t_2 * dY_46_u;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = fma(Float32(Float32(dY_46_u * floor(w)) * dY_46_u), floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))
	t_1 = fma(Float32(Float32(dX_46_v * dX_46_v) * floor(h)), floor(h), Float32(Float32(Float32(dX_46_u * floor(w)) * dX_46_u) * floor(w)))
	t_2 = Float32(floor(w) / sqrt(fmax(t_0, t_1)))
	tmp = Float32(0.0)
	if (t_1 >= t_0)
		tmp = Float32(t_2 * dX_46_u);
	else
		tmp = Float32(t_2 * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 76.0%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

3. Applied rewrites 75.9%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ } \end{array}} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(dX.v \cdot \color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

5. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

7. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(dX.v \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right), \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   5. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

   7. lower-*.f32 75.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]

9. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right), \mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dY.u\\ \end{array} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025170 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (line direction, u)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (== maxAniso 16.0))
  (if (>= (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor w) dX.u)) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor w) dY.u))))