Anisotropic x16 LOD (line direction, v)

Percentage Accurate: 76.5% → 76.5%

Time: 24.3s

Alternatives: 5

Speedup: N/A×

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

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

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_0) (* 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 = 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_0;
	} else {
		tmp = 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(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_0);
	else
		tmp = Float32(t_6 * t_4);
	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_0;
	else
		tmp = t_6 * t_4;
	end
	tmp_2 = tmp;
end

Initial Program: 76.5% accurate, 1.0× speedup

Math

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

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_0) (* 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 = 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_0;
	} else {
		tmp = 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(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_0);
	else
		tmp = Float32(t_6 * t_4);
	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_0;
	else
		tmp = t_6 * t_4;
	end
	tmp_2 = tmp;
end

Alternative 1: 76.5% accurate, N/A× speedup

Math

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

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* -1.0 (* (floor w) dY.u)))
        (t_2 (* -1.0 (* (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_0) (* 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 = floorf(h) * dX_46_v;
	float t_1 = -1.0f * (floorf(w) * dY_46_u);
	float t_2 = -1.0f * (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_0;
	} else {
		tmp = 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(floor(h) * dX_46_v)
	t_1 = Float32(Float32(-1.0) * Float32(floor(w) * dY_46_u))
	t_2 = Float32(Float32(-1.0) * 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_0);
	else
		tmp = Float32(t_6 * t_4);
	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 = single(-1.0) * (floor(w) * dY_46_u);
	t_2 = single(-1.0) * (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_0;
	else
		tmp = t_6 * t_4;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 75.3%

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

3. Final simplification 75.3%

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

Alternative 2: 76.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_2 := -1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ t_3 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_4 := -1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ t_5 := t\_4 \cdot t\_4 + t\_0 \cdot t\_0\\ t_6 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_7 := t\_2 \cdot t\_2 + t\_6 \cdot t\_6\\ \mathbf{if}\;t\_5 \geq t\_7:\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(t\_1 \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot t\_3\right)\right) \cdot dX.u, \left(t\_1 \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot t\_3\right)\right) \cdot dY.u\right)\right)}^{-0.5}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(t\_5, t\_7\right)}} \cdot t\_6\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (pow (floor h) 2.0))
        (t_2 (* -1.0 (* (floor w) dY.u)))
        (t_3 (pow (floor w) 2.0))
        (t_4 (* -1.0 (* (floor w) dX.u)))
        (t_5 (+ (* t_4 t_4) (* t_0 t_0)))
        (t_6 (* (floor h) dY.v))
        (t_7 (+ (* t_2 t_2) (* t_6 t_6))))
   (if (>= t_5 t_7)
     (*
      (*
       (floor h)
       (pow
        (fmax
         (- (* (* t_1 dX.v) dX.v) (* (* -1.0 (* dX.u t_3)) dX.u))
         (- (* (* t_1 dY.v) dY.v) (* (* -1.0 (* dY.u t_3)) dY.u)))
        -0.5))
      dX.v)
     (* (/ 1.0 (sqrt (fmax t_5 t_7))) t_6))))

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 = powf(floorf(h), 2.0f);
	float t_2 = -1.0f * (floorf(w) * dY_46_u);
	float t_3 = powf(floorf(w), 2.0f);
	float t_4 = -1.0f * (floorf(w) * dX_46_u);
	float t_5 = (t_4 * t_4) + (t_0 * t_0);
	float t_6 = floorf(h) * dY_46_v;
	float t_7 = (t_2 * t_2) + (t_6 * t_6);
	float tmp;
	if (t_5 >= t_7) {
		tmp = (floorf(h) * powf(fmaxf((((t_1 * dX_46_v) * dX_46_v) - ((-1.0f * (dX_46_u * t_3)) * dX_46_u)), (((t_1 * dY_46_v) * dY_46_v) - ((-1.0f * (dY_46_u * t_3)) * dY_46_u))), -0.5f)) * dX_46_v;
	} else {
		tmp = (1.0f / sqrtf(fmaxf(t_5, t_7))) * t_6;
	}
	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 = floor(h) ^ Float32(2.0)
	t_2 = Float32(Float32(-1.0) * Float32(floor(w) * dY_46_u))
	t_3 = floor(w) ^ Float32(2.0)
	t_4 = Float32(Float32(-1.0) * Float32(floor(w) * dX_46_u))
	t_5 = Float32(Float32(t_4 * t_4) + Float32(t_0 * t_0))
	t_6 = Float32(floor(h) * dY_46_v)
	t_7 = Float32(Float32(t_2 * t_2) + Float32(t_6 * t_6))
	tmp = Float32(0.0)
	if (t_5 >= t_7)
		tmp = Float32(Float32(floor(h) * (fmax(Float32(Float32(Float32(t_1 * dX_46_v) * dX_46_v) - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_3)) * dX_46_u)), Float32(Float32(Float32(t_1 * dY_46_v) * dY_46_v) - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_3)) * dY_46_u))) ^ Float32(-0.5))) * dX_46_v);
	else
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(t_5, t_7))) * t_6);
	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(h) ^ single(2.0);
	t_2 = single(-1.0) * (floor(w) * dY_46_u);
	t_3 = floor(w) ^ single(2.0);
	t_4 = single(-1.0) * (floor(w) * dX_46_u);
	t_5 = (t_4 * t_4) + (t_0 * t_0);
	t_6 = floor(h) * dY_46_v;
	t_7 = (t_2 * t_2) + (t_6 * t_6);
	tmp = single(0.0);
	if (t_5 >= t_7)
		tmp = (floor(h) * (max((((t_1 * dX_46_v) * dX_46_v) - ((single(-1.0) * (dX_46_u * t_3)) * dX_46_u)), (((t_1 * dY_46_v) * dY_46_v) - ((single(-1.0) * (dY_46_u * t_3)) * dY_46_u))) ^ single(-0.5))) * dX_46_v;
	else
		tmp = (single(1.0) / sqrt(max(t_5, t_7))) * t_6;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 75.3%

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

4. Applied rewrites 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\color{blue}{\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5}\right) \cdot dX.v}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

5. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dY.u\right)\right)}^{-0.5}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
6. Add Preprocessing

Alternative 3: 76.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \log \left(\left\lfloor h\right\rfloor \right) \cdot 2\\ t_2 := -1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ t_3 := -1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ t_4 := t\_3 \cdot t\_3 + t\_0 \cdot t\_0\\ t_5 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_6 := t\_2 \cdot t\_2 + t\_5 \cdot t\_5\\ t_7 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ \mathbf{if}\;t\_4 \geq t\_6:\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\left(\cosh t\_1 + \sinh t\_1\right) \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot t\_7\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot t\_7\right)\right) \cdot dY.u\right)\right)}^{-0.5}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(t\_4, t\_6\right)}} \cdot t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (log (floor h)) 2.0))
        (t_2 (* -1.0 (* (floor w) dY.u)))
        (t_3 (* -1.0 (* (floor w) dX.u)))
        (t_4 (+ (* t_3 t_3) (* t_0 t_0)))
        (t_5 (* (floor h) dY.v))
        (t_6 (+ (* t_2 t_2) (* t_5 t_5)))
        (t_7 (pow (floor w) 2.0)))
   (if (>= t_4 t_6)
     (*
      (*
       (floor h)
       (pow
        (fmax
         (-
          (* (* (+ (cosh t_1) (sinh t_1)) dX.v) dX.v)
          (* (* -1.0 (* dX.u t_7)) dX.u))
         (-
          (* (* (pow (floor h) 2.0) dY.v) dY.v)
          (* (* -1.0 (* dY.u t_7)) dY.u)))
        -0.5))
      dX.v)
     (* (/ 1.0 (sqrt (fmax t_4 t_6))) t_5))))

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 = logf(floorf(h)) * 2.0f;
	float t_2 = -1.0f * (floorf(w) * dY_46_u);
	float t_3 = -1.0f * (floorf(w) * dX_46_u);
	float t_4 = (t_3 * t_3) + (t_0 * t_0);
	float t_5 = floorf(h) * dY_46_v;
	float t_6 = (t_2 * t_2) + (t_5 * t_5);
	float t_7 = powf(floorf(w), 2.0f);
	float tmp;
	if (t_4 >= t_6) {
		tmp = (floorf(h) * powf(fmaxf(((((coshf(t_1) + sinhf(t_1)) * dX_46_v) * dX_46_v) - ((-1.0f * (dX_46_u * t_7)) * dX_46_u)), (((powf(floorf(h), 2.0f) * dY_46_v) * dY_46_v) - ((-1.0f * (dY_46_u * t_7)) * dY_46_u))), -0.5f)) * dX_46_v;
	} else {
		tmp = (1.0f / sqrtf(fmaxf(t_4, t_6))) * t_5;
	}
	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(log(floor(h)) * Float32(2.0))
	t_2 = Float32(Float32(-1.0) * Float32(floor(w) * dY_46_u))
	t_3 = Float32(Float32(-1.0) * Float32(floor(w) * dX_46_u))
	t_4 = Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0))
	t_5 = Float32(floor(h) * dY_46_v)
	t_6 = Float32(Float32(t_2 * t_2) + Float32(t_5 * t_5))
	t_7 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (t_4 >= t_6)
		tmp = Float32(Float32(floor(h) * (fmax(Float32(Float32(Float32(Float32(cosh(t_1) + sinh(t_1)) * dX_46_v) * dX_46_v) - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_7)) * dX_46_u)), Float32(Float32(Float32((floor(h) ^ Float32(2.0)) * dY_46_v) * dY_46_v) - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_7)) * dY_46_u))) ^ Float32(-0.5))) * dX_46_v);
	else
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(t_4, t_6))) * t_5);
	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 = log(floor(h)) * single(2.0);
	t_2 = single(-1.0) * (floor(w) * dY_46_u);
	t_3 = single(-1.0) * (floor(w) * dX_46_u);
	t_4 = (t_3 * t_3) + (t_0 * t_0);
	t_5 = floor(h) * dY_46_v;
	t_6 = (t_2 * t_2) + (t_5 * t_5);
	t_7 = floor(w) ^ single(2.0);
	tmp = single(0.0);
	if (t_4 >= t_6)
		tmp = (floor(h) * (max(((((cosh(t_1) + sinh(t_1)) * dX_46_v) * dX_46_v) - ((single(-1.0) * (dX_46_u * t_7)) * dX_46_u)), ((((floor(h) ^ single(2.0)) * dY_46_v) * dY_46_v) - ((single(-1.0) * (dY_46_u * t_7)) * dY_46_u))) ^ single(-0.5))) * dX_46_v;
	else
		tmp = (single(1.0) / sqrt(max(t_4, t_6))) * t_5;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 75.3%

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

4. Applied rewrites 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\color{blue}{\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5}\right) \cdot dX.v}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
5. Step-by-step derivation

   1. lift-floor.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left({\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{\frac{-1}{2}}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   2. lift-pow.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{\frac{-1}{2}}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   3. pow-to-exp N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2}} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{\frac{-1}{2}}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   4. sinh-+-cosh-rev N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\color{blue}{\left(\cosh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right) + \sinh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)\right)} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{\frac{-1}{2}}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   5. lower-+.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\color{blue}{\left(\cosh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right) + \sinh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)\right)} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{\frac{-1}{2}}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   6. lower-cosh.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\left(\color{blue}{\cosh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)} + \sinh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)\right) \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{\frac{-1}{2}}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   7. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\left(\cosh \color{blue}{\left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)} + \sinh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)\right) \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{\frac{-1}{2}}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   8. lower-log.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\left(\cosh \left(\color{blue}{\log \left(\left\lfloor h\right\rfloor \right)} \cdot 2\right) + \sinh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)\right) \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{\frac{-1}{2}}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   9. lift-floor.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\left(\cosh \left(\log \color{blue}{\left(\left\lfloor h\right\rfloor \right)} \cdot 2\right) + \sinh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)\right) \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{\frac{-1}{2}}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   10. lower-sinh.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\left(\cosh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right) + \color{blue}{\sinh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)}\right) \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{\frac{-1}{2}}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   11. lower-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\left(\cosh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right) + \sinh \color{blue}{\left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)}\right) \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{\frac{-1}{2}}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   12. lower-log.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\left(\cosh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right) + \sinh \left(\color{blue}{\log \left(\left\lfloor h\right\rfloor \right)} \cdot 2\right)\right) \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{\frac{-1}{2}}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   13. lift-floor.f32 74.9

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\left(\cosh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right) + \sinh \left(\log \color{blue}{\left(\left\lfloor h\right\rfloor \right)} \cdot 2\right)\right) \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

6. Applied rewrites 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\color{blue}{\left(\cosh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right) + \sinh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)\right)} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

7. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor h\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\left(\cosh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right) + \sinh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)\right) \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dY.u\right)\right)}^{-0.5}\right) \cdot dX.v\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
8. Add Preprocessing

Alternative 4: 66.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ t_1 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_2 := \left(t\_1 \cdot dY.v\right) \cdot dY.v\\ t_3 := \left(t\_1 \cdot dX.v\right) \cdot dX.v\\ t_4 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_5 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_6 := -1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ t_7 := t\_6 \cdot t\_6 + t\_4 \cdot t\_4\\ t_8 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_9 := t\_5 \cdot dY.u\\ t_10 := t\_9 \cdot dY.u\\ t_11 := t\_0 \cdot t\_0 + t\_8 \cdot t\_8\\ \mathbf{if}\;dX.v \leq -0.5 \lor \neg \left(dX.v \leq 40000000000\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_7 \geq t\_10:\\ \;\;\;\;\frac{dX.v \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left(t\_3, t\_10\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(t\_7, t\_11\right)}} \cdot t\_8\\ \end{array}\\ \mathbf{elif}\;t\_7 \geq t\_11:\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{max}\left(\left(\left(t\_5 \cdot dX.u\right) \cdot -1\right) \cdot \left(-1 \cdot dX.u\right), t\_2 - \left(-1 \cdot t\_9\right) \cdot dY.u\right)}\right) \cdot -1} \cdot t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.v \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left(t\_3 - \left(-1 \cdot \left(dX.u \cdot t\_5\right)\right) \cdot dX.u, t\_2 - \left(-1 \cdot \left(dY.u \cdot t\_5\right)\right) \cdot dY.u\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* -1.0 (* (floor w) dY.u)))
        (t_1 (pow (floor h) 2.0))
        (t_2 (* (* t_1 dY.v) dY.v))
        (t_3 (* (* t_1 dX.v) dX.v))
        (t_4 (* (floor h) dX.v))
        (t_5 (pow (floor w) 2.0))
        (t_6 (* -1.0 (* (floor w) dX.u)))
        (t_7 (+ (* t_6 t_6) (* t_4 t_4)))
        (t_8 (* (floor h) dY.v))
        (t_9 (* t_5 dY.u))
        (t_10 (* t_9 dY.u))
        (t_11 (+ (* t_0 t_0) (* t_8 t_8))))
   (if (or (<= dX.v -0.5) (not (<= dX.v 40000000000.0)))
     (if (>= t_7 t_10)
       (/ (* dX.v (floor h)) (sqrt (fmax t_3 t_10)))
       (* (/ 1.0 (sqrt (fmax t_7 t_11))) t_8))
     (if (>= t_7 t_11)
       (*
        (exp
         (*
          (log
           (sqrt
            (fmax
             (* (* (* t_5 dX.u) -1.0) (* -1.0 dX.u))
             (- t_2 (* (* -1.0 t_9) dY.u)))))
          -1.0))
        t_4)
       (/
        (* dY.v (floor h))
        (sqrt
         (fmax
          (- t_3 (* (* -1.0 (* dX.u t_5)) dX.u))
          (- t_2 (* (* -1.0 (* dY.u t_5)) 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 = -1.0f * (floorf(w) * dY_46_u);
	float t_1 = powf(floorf(h), 2.0f);
	float t_2 = (t_1 * dY_46_v) * dY_46_v;
	float t_3 = (t_1 * dX_46_v) * dX_46_v;
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = powf(floorf(w), 2.0f);
	float t_6 = -1.0f * (floorf(w) * dX_46_u);
	float t_7 = (t_6 * t_6) + (t_4 * t_4);
	float t_8 = floorf(h) * dY_46_v;
	float t_9 = t_5 * dY_46_u;
	float t_10 = t_9 * dY_46_u;
	float t_11 = (t_0 * t_0) + (t_8 * t_8);
	float tmp_1;
	if ((dX_46_v <= -0.5f) || !(dX_46_v <= 40000000000.0f)) {
		float tmp_2;
		if (t_7 >= t_10) {
			tmp_2 = (dX_46_v * floorf(h)) / sqrtf(fmaxf(t_3, t_10));
		} else {
			tmp_2 = (1.0f / sqrtf(fmaxf(t_7, t_11))) * t_8;
		}
		tmp_1 = tmp_2;
	} else if (t_7 >= t_11) {
		tmp_1 = expf((logf(sqrtf(fmaxf((((t_5 * dX_46_u) * -1.0f) * (-1.0f * dX_46_u)), (t_2 - ((-1.0f * t_9) * dY_46_u))))) * -1.0f)) * t_4;
	} else {
		tmp_1 = (dY_46_v * floorf(h)) / sqrtf(fmaxf((t_3 - ((-1.0f * (dX_46_u * t_5)) * dX_46_u)), (t_2 - ((-1.0f * (dY_46_u * t_5)) * dY_46_u))));
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(Float32(-1.0) * Float32(floor(w) * dY_46_u))
	t_1 = floor(h) ^ Float32(2.0)
	t_2 = Float32(Float32(t_1 * dY_46_v) * dY_46_v)
	t_3 = Float32(Float32(t_1 * dX_46_v) * dX_46_v)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = floor(w) ^ Float32(2.0)
	t_6 = Float32(Float32(-1.0) * Float32(floor(w) * dX_46_u))
	t_7 = Float32(Float32(t_6 * t_6) + Float32(t_4 * t_4))
	t_8 = Float32(floor(h) * dY_46_v)
	t_9 = Float32(t_5 * dY_46_u)
	t_10 = Float32(t_9 * dY_46_u)
	t_11 = Float32(Float32(t_0 * t_0) + Float32(t_8 * t_8))
	tmp_1 = Float32(0.0)
	if ((dX_46_v <= Float32(-0.5)) || !(dX_46_v <= Float32(40000000000.0)))
		tmp_2 = Float32(0.0)
		if (t_7 >= t_10)
			tmp_2 = Float32(Float32(dX_46_v * floor(h)) / sqrt(fmax(t_3, t_10)));
		else
			tmp_2 = Float32(Float32(Float32(1.0) / sqrt(fmax(t_7, t_11))) * t_8);
		end
		tmp_1 = tmp_2;
	elseif (t_7 >= t_11)
		tmp_1 = Float32(exp(Float32(log(sqrt(fmax(Float32(Float32(Float32(t_5 * dX_46_u) * Float32(-1.0)) * Float32(Float32(-1.0) * dX_46_u)), Float32(t_2 - Float32(Float32(Float32(-1.0) * t_9) * dY_46_u))))) * Float32(-1.0))) * t_4);
	else
		tmp_1 = Float32(Float32(dY_46_v * floor(h)) / sqrt(fmax(Float32(t_3 - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_5)) * dX_46_u)), Float32(t_2 - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_5)) * dY_46_u)))));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = single(-1.0) * (floor(w) * dY_46_u);
	t_1 = floor(h) ^ single(2.0);
	t_2 = (t_1 * dY_46_v) * dY_46_v;
	t_3 = (t_1 * dX_46_v) * dX_46_v;
	t_4 = floor(h) * dX_46_v;
	t_5 = floor(w) ^ single(2.0);
	t_6 = single(-1.0) * (floor(w) * dX_46_u);
	t_7 = (t_6 * t_6) + (t_4 * t_4);
	t_8 = floor(h) * dY_46_v;
	t_9 = t_5 * dY_46_u;
	t_10 = t_9 * dY_46_u;
	t_11 = (t_0 * t_0) + (t_8 * t_8);
	tmp_2 = single(0.0);
	if ((dX_46_v <= single(-0.5)) || ~((dX_46_v <= single(40000000000.0))))
		tmp_3 = single(0.0);
		if (t_7 >= t_10)
			tmp_3 = (dX_46_v * floor(h)) / sqrt(max(t_3, t_10));
		else
			tmp_3 = (single(1.0) / sqrt(max(t_7, t_11))) * t_8;
		end
		tmp_2 = tmp_3;
	elseif (t_7 >= t_11)
		tmp_2 = exp((log(sqrt(max((((t_5 * dX_46_u) * single(-1.0)) * (single(-1.0) * dX_46_u)), (t_2 - ((single(-1.0) * t_9) * dY_46_u))))) * single(-1.0))) * t_4;
	else
		tmp_2 = (dY_46_v * floor(h)) / sqrt(max((t_3 - ((single(-1.0) * (dX_46_u * t_5)) * dX_46_u)), (t_2 - ((single(-1.0) * (dY_46_u * t_5)) * dY_46_u))));
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < -0.5 or 4e10 < dX.v

   1. Initial program 63.8%

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

   3. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      2. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      3. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot \color{blue}{dX.v}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot \color{blue}{dX.v}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      5. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      6. lower-pow.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      7. lift-floor.f32 58.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   5. Applied rewrites 58.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   6. Taylor expanded in dY.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   8. Applied rewrites 58.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   9. Taylor expanded in dY.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{{dY.u}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       2. unpow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot \color{blue}{dY.u}\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       3. associate-*l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot \color{blue}{dY.u}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       4. lift-pow.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       5. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       6. lift-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       7. lift-*.f32 58.3

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot \color{blue}{dY.u}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   11. Applied rewrites 58.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   13. Applied rewrites 58.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\color{blue}{\frac{\left(\left(1 \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(1 \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot dY.u\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   if -0.5 < dX.v < 4e10

   1. Initial program 81.5%

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

   4. Applied rewrites 81.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)}}\\ \end{array} \]

   5. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)}}\\ \end{array} \]

   7. Applied rewrites 77.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)}}\\ \end{array} \]
   8. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)}}\\ \end{array} \]

      2. lift-pow.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)}}\\ \end{array} \]

      3. pow-to-exp N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(e^{\log \left(\left\lfloor w\right\rfloor \right) \cdot 2} \cdot dX.u\right) \cdot dX.u, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)}}\\ \end{array} \]

      4. lower-exp.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(e^{\log \left(\left\lfloor w\right\rfloor \right) \cdot 2} \cdot dX.u\right) \cdot dX.u, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)}}\\ \end{array} \]

      5. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(e^{\log \left(\left\lfloor w\right\rfloor \right) \cdot 2} \cdot dX.u\right) \cdot dX.u, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)}}\\ \end{array} \]

      6. lower-log.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(e^{\log \left(\left\lfloor w\right\rfloor \right) \cdot 2} \cdot dX.u\right) \cdot dX.u, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)}}\\ \end{array} \]

      7. lift-floor.f32 76.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(e^{\log \left(\left\lfloor w\right\rfloor \right) \cdot 2} \cdot dX.u\right) \cdot dX.u, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)}}\\ \end{array} \]

   9. Applied rewrites 76.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(e^{\log \left(\left\lfloor w\right\rfloor \right) \cdot 2} \cdot dX.u\right) \cdot dX.u, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)}}\\ \end{array} \]

   11. Applied rewrites 75.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\color{blue}{e^{\log \left(\sqrt{\mathsf{max}\left(\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot -1\right) \cdot dX.u\right) \cdot -1\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot dY.u\right)}\right) \cdot -1}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot -1}{-\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)}}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq -0.5 \lor \neg \left(dX.v \leq 40000000000\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\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(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{dX.v \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\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 h\right\rfloor \cdot dY.v\right)\\ \end{array}\\ \mathbf{elif}\;\left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{max}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot -1\right) \cdot \left(-1 \cdot dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot dY.u\right)}\right) \cdot -1} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.v \cdot \left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dY.u\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := -1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_4 := \log t\_3\\ t_5 := \cosh t\_4\\ t_6 := -1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ t_7 := t\_6 \cdot t\_6 + t\_0 \cdot t\_0\\ t_8 := \sinh t\_4\\ \mathbf{if}\;t\_7 \geq \left(t\_3 \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\frac{t\_5 \cdot t\_5 - t\_8 \cdot t\_8}{t\_5 - t\_8} \cdot \left(-1 \cdot dY.u\right)\right) \cdot \left(-1 \cdot dY.u\right)\right)}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(t\_7, t\_1 \cdot t\_1 + t\_2 \cdot t\_2\right)}} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* -1.0 (* (floor w) dY.u)))
        (t_2 (* (floor h) dY.v))
        (t_3 (pow (floor w) 2.0))
        (t_4 (log t_3))
        (t_5 (cosh t_4))
        (t_6 (* -1.0 (* (floor w) dX.u)))
        (t_7 (+ (* t_6 t_6) (* t_0 t_0)))
        (t_8 (sinh t_4)))
   (if (>= t_7 (* (* t_3 dY.u) dY.u))
     (*
      (/
       1.0
       (sqrt
        (fmax
         (* (* (pow (floor h) 2.0) dX.v) dX.v)
         (*
          (* (/ (- (* t_5 t_5) (* t_8 t_8)) (- t_5 t_8)) (* -1.0 dY.u))
          (* -1.0 dY.u)))))
      t_0)
     (* (/ 1.0 (sqrt (fmax t_7 (+ (* t_1 t_1) (* t_2 t_2))))) t_2))))

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 = -1.0f * (floorf(w) * dY_46_u);
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = powf(floorf(w), 2.0f);
	float t_4 = logf(t_3);
	float t_5 = coshf(t_4);
	float t_6 = -1.0f * (floorf(w) * dX_46_u);
	float t_7 = (t_6 * t_6) + (t_0 * t_0);
	float t_8 = sinhf(t_4);
	float tmp;
	if (t_7 >= ((t_3 * dY_46_u) * dY_46_u)) {
		tmp = (1.0f / sqrtf(fmaxf(((powf(floorf(h), 2.0f) * dX_46_v) * dX_46_v), (((((t_5 * t_5) - (t_8 * t_8)) / (t_5 - t_8)) * (-1.0f * dY_46_u)) * (-1.0f * dY_46_u))))) * t_0;
	} else {
		tmp = (1.0f / sqrtf(fmaxf(t_7, ((t_1 * t_1) + (t_2 * t_2))))) * t_2;
	}
	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(Float32(-1.0) * Float32(floor(w) * dY_46_u))
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = floor(w) ^ Float32(2.0)
	t_4 = log(t_3)
	t_5 = cosh(t_4)
	t_6 = Float32(Float32(-1.0) * Float32(floor(w) * dX_46_u))
	t_7 = Float32(Float32(t_6 * t_6) + Float32(t_0 * t_0))
	t_8 = sinh(t_4)
	tmp = Float32(0.0)
	if (t_7 >= Float32(Float32(t_3 * dY_46_u) * dY_46_u))
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v), Float32(Float32(Float32(Float32(Float32(t_5 * t_5) - Float32(t_8 * t_8)) / Float32(t_5 - t_8)) * Float32(Float32(-1.0) * dY_46_u)) * Float32(Float32(-1.0) * dY_46_u))))) * t_0);
	else
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(t_7, Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2))))) * t_2);
	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 = single(-1.0) * (floor(w) * dY_46_u);
	t_2 = floor(h) * dY_46_v;
	t_3 = floor(w) ^ single(2.0);
	t_4 = log(t_3);
	t_5 = cosh(t_4);
	t_6 = single(-1.0) * (floor(w) * dX_46_u);
	t_7 = (t_6 * t_6) + (t_0 * t_0);
	t_8 = sinh(t_4);
	tmp = single(0.0);
	if (t_7 >= ((t_3 * dY_46_u) * dY_46_u))
		tmp = (single(1.0) / sqrt(max((((floor(h) ^ single(2.0)) * dX_46_v) * dX_46_v), (((((t_5 * t_5) - (t_8 * t_8)) / (t_5 - t_8)) * (single(-1.0) * dY_46_u)) * (single(-1.0) * dY_46_u))))) * t_0;
	else
		tmp = (single(1.0) / sqrt(max(t_7, ((t_1 * t_1) + (t_2 * t_2))))) * t_2;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 75.3%

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

3. Taylor expanded in dX.u around 0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   2. unpow2 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   3. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot \color{blue}{dX.v}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   4. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot \color{blue}{dX.v}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   5. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   6. lower-pow.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   7. lift-floor.f32 57.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

5. Applied rewrites 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

6. Taylor expanded in dY.u around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

8. Applied rewrites 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

9. Taylor expanded in dY.u around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{{dY.u}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    2. unpow2 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot \color{blue}{dY.u}\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    3. associate-*l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot \color{blue}{dY.u}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    4. lift-pow.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    5. lift-floor.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    6. lift-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    7. lift-*.f32 50.9

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot \color{blue}{dY.u}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

11. Applied rewrites 50.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
12. Step-by-step derivation

    1. lift-floor.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    2. lift-pow.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    3. pow-to-exp N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(e^{\log \left(\left\lfloor w\right\rfloor \right) \cdot 2} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    4. sinh-+-cosh-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left(\cosh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right) + \sinh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right)\right) \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    5. flip-+ N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\frac{\cosh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right) \cdot \cosh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right) - \sinh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right) \cdot \sinh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right)}{\cosh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right) - \sinh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right)} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    6. lower-/.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\frac{\cosh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right) \cdot \cosh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right) - \sinh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right) \cdot \sinh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right)}{\cosh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right) - \sinh \left(\log \left(\left\lfloor w\right\rfloor \right) \cdot 2\right)} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

13. Applied rewrites 53.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\frac{\cosh \log \left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \cosh \log \left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right) - \sinh \log \left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \sinh \log \left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}{\cosh \log \left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right) - \sinh \log \left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)} \cdot dY.u\right) \cdot dY.u\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

14. Final simplification 53.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\frac{\cosh \log \left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \cosh \log \left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right) - \sinh \log \left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \sinh \log \left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}{\cosh \log \left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right) - \sinh \log \left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)} \cdot \left(-1 \cdot dY.u\right)\right) \cdot \left(-1 \cdot dY.u\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right) \cdot \left(-1 \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (line direction, v)"
  :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 h) dX.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 h) dY.v))))