Anisotropic x16 LOD (line direction, u)

Percentage Accurate: 76.8% → 99.2%

Time: 34.6s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_3 := t_2 \cdot t_2 + t_0 \cdot t_0\\ t_4 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_5 := t_1 \cdot t_1 + t_4 \cdot t_4\\ t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t_3, t_5\right)}}\\ \mathbf{if}\;t_3 \geq t_5:\\ \;\;\;\;t_6 \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot t_1\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_3 := t_2 \cdot t_2 + t_0 \cdot t_0\\ t_4 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_5 := t_1 \cdot t_1 + t_4 \cdot t_4\\ t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t_3, t_5\right)}}\\ \mathbf{if}\;t_3 \geq t_5:\\ \;\;\;\;t_6 \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot t_1\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \langle \left( \begin{array}{l} t_0 := \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)\\ t_1 := \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right)\\ t_2 := \sqrt{\mathsf{max}\left(t_1, t_0\right)}\\ \mathbf{if}\;t_1 \geq t_0:\\ \;\;\;\;\frac{dX.u}{\frac{t_2}{\left\lfloorw\right\rfloor}}\\ \mathbf{else}:\\ \;\;\;\;dY.u \cdot \frac{\left\lfloorw\right\rfloor}{t_2}\\ \end{array} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (let* ((t_0
           (fma
            (floor w)
            (* dY.u (* (floor w) dY.u))
            (* (floor h) (* (floor h) (* dY.v dY.v)))))
          (t_1
           (fma
            (floor w)
            (* (floor w) (* dX.u dX.u))
            (* (floor h) (* (floor h) (* dX.v dX.v)))))
          (t_2 (sqrt (fmax t_1 t_0))))
     (if (>= t_1 t_0)
       (/ dX.u (/ t_2 (floor w)))
       (* dY.u (/ (floor w) 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) {
	double t_0_2 = fma(floor(w), (((double) dY_46_u) * (floor(w) * ((double) dY_46_u))), (floor(h) * (floor(h) * (((double) dY_46_v) * ((double) dY_46_v)))));
	double t_1_3 = fma(floor(w), (floor(w) * (((double) dX_46_u) * ((double) dX_46_u))), (floor(h) * (floor(h) * (((double) dX_46_v) * ((double) dX_46_v)))));
	double t_2_4 = sqrt(fmax(t_1_3, t_0_2));
	double tmp_5;
	if (t_1_3 >= t_0_2) {
		tmp_5 = ((double) dX_46_u) / (t_2_4 / floor(w));
	} else {
		tmp_5 = ((double) dY_46_u) * (floor(w) / t_2_4);
	}
	double tmp_1 = tmp_5;
	return (float) tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0_2 = fma(floor(w), Float64(Float64(dY_46_u) * Float64(floor(w) * Float64(dY_46_u))), Float64(floor(h) * Float64(floor(h) * Float64(Float64(dY_46_v) * Float64(dY_46_v)))))
	t_1_3 = fma(floor(w), Float64(floor(w) * Float64(Float64(dX_46_u) * Float64(dX_46_u))), Float64(floor(h) * Float64(floor(h) * Float64(Float64(dX_46_v) * Float64(dX_46_v)))))
	t_2_4 = sqrt(((t_1_3 != t_1_3) ? t_0_2 : ((t_0_2 != t_0_2) ? t_1_3 : max(t_1_3, t_0_2))))
	tmp_5 = 0.0
	if (t_1_3 >= t_0_2)
		tmp_5 = Float64(Float64(dX_46_u) / Float64(t_2_4 / floor(w)));
	else
		tmp_5 = Float64(Float64(dY_46_u) * Float64(floor(w) / t_2_4));
	end
	tmp_1 = tmp_5
	return Float32(tmp_1)
end

Derivation

1. Initial program 99.3%

   \[\langle \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;\frac{dX.u}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \mathbf{else}:\\ \;\;\;\;dY.u \cdot \frac{\left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)}}\\ \end{array} \rangle_{\text{binary64}} \]

2. Final simplification 99.3%

   \[\leadsto \langle \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right):\\ \;\;\;\;\frac{dX.u}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \mathbf{else}:\\ \;\;\;\;dY.u \cdot \frac{\left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)\right)}}\\ \end{array} \rangle_{\text{binary64}} \]

Alternative 2: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \langle \left( \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot t_0, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)\\ t_2 := \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right)\\ \mathbf{if}\;t_2 \geq t_1:\\ \;\;\;\;\frac{dX.u}{\frac{\langle \left( \langle \left( \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {t_0}^{2}\right)} \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}}{\left\lfloorw\right\rfloor}}\\ \mathbf{else}:\\ \;\;\;\;dY.u \cdot \frac{\left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left(t_2, t_1\right)}}\\ \end{array} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (let* ((t_0 (* (floor w) dY.u))
          (t_1
           (fma
            (floor w)
            (* dY.u t_0)
            (* (floor h) (* (floor h) (* dY.v dY.v)))))
          (t_2
           (fma
            (floor w)
            (* (floor w) (* dX.u dX.u))
            (* (floor h) (* (floor h) (* dX.v dX.v))))))
     (if (>= t_2 t_1)
       (/
        dX.u
        (/
         (cast
          (!
           :precision
           binary32
           (cast
            (!
             :precision
             binary64
             (sqrt
              (fmax
               (+ (pow (* dX.u (floor w)) 2.0) (pow (* (floor h) dX.v) 2.0))
               (+ (pow (* (floor h) dY.v) 2.0) (pow t_0 2.0))))))))
         (floor w)))
       (* dY.u (/ (floor w) (sqrt (fmax t_2 t_1)))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	double t_0_8 = floor(w) * ((double) dY_46_u);
	double t_1_9 = fma(floor(w), (((double) dY_46_u) * t_0_8), (floor(h) * (floor(h) * (((double) dY_46_v) * ((double) dY_46_v)))));
	double t_2_10 = fma(floor(w), (floor(w) * (((double) dX_46_u) * ((double) dX_46_u))), (floor(h) * (floor(h) * (((double) dX_46_v) * ((double) dX_46_v)))));
	double tmp_14;
	if (t_2_10 >= t_1_9) {
		double tmp_17 = sqrt(fmax((pow((((double) dX_46_u) * floor(w)), 2.0) + pow((floor(h) * ((double) dX_46_v)), 2.0)), (pow((floor(h) * ((double) dY_46_v)), 2.0) + pow(t_0_8, 2.0))));
		double tmp_16 = (float) tmp_17;
		tmp_14 = ((double) dX_46_u) / (((double) tmp_16) / floor(w));
	} else {
		tmp_14 = ((double) dY_46_u) * (floor(w) / sqrt(fmax(t_2_10, t_1_9)));
	}
	double tmp_7 = tmp_14;
	return (float) tmp_7;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0_8 = Float64(floor(w) * Float64(dY_46_u))
	t_1_9 = fma(floor(w), Float64(Float64(dY_46_u) * t_0_8), Float64(floor(h) * Float64(floor(h) * Float64(Float64(dY_46_v) * Float64(dY_46_v)))))
	t_2_10 = fma(floor(w), Float64(floor(w) * Float64(Float64(dX_46_u) * Float64(dX_46_u))), Float64(floor(h) * Float64(floor(h) * Float64(Float64(dX_46_v) * Float64(dX_46_v)))))
	tmp_14 = 0.0
	if (t_2_10 >= t_1_9)
		tmp_17 = sqrt(((Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (Float64(floor(h) * Float64(dX_46_v)) ^ 2.0)) != Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (Float64(floor(h) * Float64(dX_46_v)) ^ 2.0))) ? Float64((Float64(floor(h) * Float64(dY_46_v)) ^ 2.0) + (t_0_8 ^ 2.0)) : ((Float64((Float64(floor(h) * Float64(dY_46_v)) ^ 2.0) + (t_0_8 ^ 2.0)) != Float64((Float64(floor(h) * Float64(dY_46_v)) ^ 2.0) + (t_0_8 ^ 2.0))) ? Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (Float64(floor(h) * Float64(dX_46_v)) ^ 2.0)) : max(Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (Float64(floor(h) * Float64(dX_46_v)) ^ 2.0)), Float64((Float64(floor(h) * Float64(dY_46_v)) ^ 2.0) + (t_0_8 ^ 2.0))))))
		tmp_16 = Float32(tmp_17)
		tmp_14 = Float64(Float64(dX_46_u) / Float64(Float64(tmp_16) / floor(w)));
	else
		tmp_14 = Float64(Float64(dY_46_u) * Float64(floor(w) / sqrt(((t_2_10 != t_2_10) ? t_1_9 : ((t_1_9 != t_1_9) ? t_2_10 : max(t_2_10, t_1_9))))));
	end
	tmp_7 = tmp_14
	return Float32(tmp_7)
end

Derivation

1. Initial program 99.3%

   \[\langle \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;\frac{dX.u}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \mathbf{else}:\\ \;\;\;\;dY.u \cdot \frac{\left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)}}\\ \end{array} \rangle_{\text{binary64}} \]
2. Step-by-step derivation

   1. rewrite-binary64/binary32-simplify 99.0%

      \[\leadsto \color{blue}{\langle \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;\frac{dX.u}{\frac{\langle \left( \langle \left( \sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)} \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}}{\left\lfloorw\right\rfloor}}\\ \mathbf{else}:\\ \;\;\;\;dY.u \cdot \frac{\left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)}}\\ } \end{array}} \rangle_{\text{binary64}}} \]

3. Applied rewrite-once 99.0%

   \[\leadsto \langle \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, \color{blue}{dY.u} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;\frac{dX.u}{\frac{\langle \left( \langle \left( \sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)} \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}}{\left\lfloorw\right\rfloor}}\\ \mathbf{else}:\\ \;\;\;\;dY.u \cdot \frac{\left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)}}\\ \end{array} \rangle_{\text{binary64}} \]

5. Simplified 99.0%

   \[\leadsto \langle \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, \color{blue}{dY.u} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;\frac{dX.u}{\frac{\langle \left( \langle \left( \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)} \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}}{\left\lfloorw\right\rfloor}}\\ \mathbf{else}:\\ \;\;\;\;dY.u \cdot \frac{\left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)}}\\ \end{array} \rangle_{\text{binary64}} \]

6. Final simplification 99.0%

   \[\leadsto \langle \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right):\\ \;\;\;\;\frac{dX.u}{\frac{\langle \left( \langle \left( \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)} \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}}{\left\lfloorw\right\rfloor}}\\ \mathbf{else}:\\ \;\;\;\;dY.u \cdot \frac{\left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)\right)}}\\ \end{array} \rangle_{\text{binary64}} \]

Alternative 3: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.u \cdot \left\lfloorw\right\rfloor\\ t_1 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_2 := {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\\ t_3 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_4 := {t_3}^{2}\\ t_5 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_6 := {t_5}^{2}\\ t_7 := \sqrt{\mathsf{max}\left({t_0}^{2} + t_6, t_2 + t_4\right)}\\ t_8 := \mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot t_1, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_5\right)\right)\\ t_9 := \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right), \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot t_3\right)\right)\\ t_10 := \sqrt{\mathsf{max}\left(t_8, t_9\right)}\\ \mathbf{if}\;dY.v \leq 10000000272564224:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t_6 + {t_1}^{2} \geq t_4 + t_2:\\ \;\;\;\;\langle \left( \frac{t_0}{t_7} \right)_{\text{binary64}} \rangle_{\text{binary32}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{t_10}\\ \end{array}\\ \mathbf{elif}\;t_8 \geq t_9:\\ \;\;\;\;\frac{t_1}{t_10}\\ \mathbf{else}:\\ \;\;\;\;\langle \left( \frac{t_3}{t_7} \right)_{\text{binary64}} \rangle_{\text{binary32}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.u (floor w)))
        (t_1 (* (floor w) dX.u))
        (t_2 (pow (* (floor h) dY.v) 2.0))
        (t_3 (* (floor w) dY.u))
        (t_4 (pow t_3 2.0))
        (t_5 (* (floor h) dX.v))
        (t_6 (pow t_5 2.0))
        (t_7 (sqrt (fmax (+ (pow t_0 2.0) t_6) (+ t_2 t_4))))
        (t_8 (fma (floor w) (* dX.u t_1) (* (floor h) (* dX.v t_5))))
        (t_9
         (fma
          (floor h)
          (* (floor h) (* dY.v dY.v))
          (* (floor w) (* dY.u t_3))))
        (t_10 (sqrt (fmax t_8 t_9))))
   (if (<= dY.v 10000000272564224.0)
     (if (>= (+ t_6 (pow t_1 2.0)) (+ t_4 t_2))
       (cast (! :precision binary64 (/ t_0 t_7)))
       (/ t_3 t_10))
     (if (>= t_8 t_9)
       (/ t_1 t_10)
       (cast (! :precision binary64 (/ t_3 t_7)))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dX_46_u * floorf(w);
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = powf((floorf(h) * dY_46_v), 2.0f);
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = powf(t_3, 2.0f);
	float t_5 = floorf(h) * dX_46_v;
	float t_6 = powf(t_5, 2.0f);
	float t_7 = sqrtf(fmaxf((powf(t_0, 2.0f) + t_6), (t_2 + t_4)));
	float t_8 = fmaf(floorf(w), (dX_46_u * t_1), (floorf(h) * (dX_46_v * t_5)));
	float t_9 = fmaf(floorf(h), (floorf(h) * (dY_46_v * dY_46_v)), (floorf(w) * (dY_46_u * t_3)));
	float t_10 = sqrtf(fmaxf(t_8, t_9));
	double tmp_3;
	if (dY_46_v <= 10000000272564224.0f) {
		double tmp_5;
		if ((t_6 + powf(t_1, 2.0f)) >= (t_4 + t_2)) {
			double tmp_6 = ((double) t_0) / ((double) t_7);
			tmp_5 = (float) tmp_6;
		} else {
			tmp_5 = ((double) t_3) / ((double) t_10);
		}
		tmp_3 = tmp_5;
	} else if (t_8 >= t_9) {
		tmp_3 = ((double) t_1) / ((double) t_10);
	} else {
		double tmp_7 = ((double) t_3) / ((double) t_7);
		tmp_3 = (double) tmp_7;
	}
	return tmp_3;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dX_46_u * floor(w))
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = Float32(floor(h) * dY_46_v) ^ Float32(2.0)
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = t_3 ^ Float32(2.0)
	t_5 = Float32(floor(h) * dX_46_v)
	t_6 = t_5 ^ Float32(2.0)
	t_7 = sqrt(((Float32((t_0 ^ Float32(2.0)) + t_6) != Float32((t_0 ^ Float32(2.0)) + t_6)) ? Float32(t_2 + t_4) : ((Float32(t_2 + t_4) != Float32(t_2 + t_4)) ? Float32((t_0 ^ Float32(2.0)) + t_6) : max(Float32((t_0 ^ Float32(2.0)) + t_6), Float32(t_2 + t_4)))))
	t_8 = fma(floor(w), Float32(dX_46_u * t_1), Float32(floor(h) * Float32(dX_46_v * t_5)))
	t_9 = fma(floor(h), Float32(floor(h) * Float32(dY_46_v * dY_46_v)), Float32(floor(w) * Float32(dY_46_u * t_3)))
	t_10 = sqrt(((t_8 != t_8) ? t_9 : ((t_9 != t_9) ? t_8 : max(t_8, t_9))))
	tmp_3 = 0.0
	if (dY_46_v <= Float32(10000000272564224.0))
		tmp_5 = 0.0
		if (Float32(t_6 + (t_1 ^ Float32(2.0))) >= Float32(t_4 + t_2))
			tmp_6 = Float64(Float64(t_0) / Float64(t_7))
			tmp_5 = Float32(tmp_6);
		else
			tmp_5 = Float64(Float64(t_3) / Float64(t_10));
		end
		tmp_3 = tmp_5;
	elseif (t_8 >= t_9)
		tmp_3 = Float64(Float64(t_1) / Float64(t_10));
	else
		tmp_7 = Float64(Float64(t_3) / Float64(t_7))
		tmp_3 = Float64(tmp_7);
	end
	return tmp_3
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 1.00000003e16

   1. Initial program 76.7%

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

   3. Simplified 76.8%

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

      1. rewrite-binary32/binary64 91.4%

         \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right):\\ \;\;\;\;\langle \frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ } \end{array}} \]

   5. Applied rewrite-once 91.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right):\\ \;\;\;\;\color{blue}{\langle \color{blue}{\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}} \rangle_{\text{binary64}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

   7. Simplified 91.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right):\\ \;\;\;\;\color{blue}{\langle \color{blue}{\frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}}} \rangle_{\text{binary64}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

   8. Taylor expanded in w around 0 91.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\mathsf{fma}\left(\left\lfloorw\right\rfloor, {dX.u}^{2} \cdot \left\lfloorw\right\rfloor, {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}:\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]
   9. Step-by-step derivation

      1. unpow2 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \color{blue}{\left(dX.u \cdot dX.u\right)} \cdot \left\lfloorw\right\rfloor, {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      2. associate-*r* 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \color{blue}{dX.u \cdot \left(dX.u \cdot \left\lfloorw\right\rfloor\right)}, {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      3. *-commutative 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \color{blue}{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}, {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      4. *-commutative 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      5. unpow2 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dX.v}^{2}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      6. unpow2 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      7. swap-sqr 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      8. fma-def 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)} \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      9. associate-*l* 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      10. unpow2 91.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      11. unpow2 91.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}} \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      12. *-commutative 91.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + {\color{blue}{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}}^{2} \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      13. fma-udef 91.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} \geq \color{blue}{\left\lfloorh\right\rfloor \cdot \left({dY.v}^{2} \cdot \left\lfloorh\right\rfloor\right) + {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}}:\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

   10. Simplified 91.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}}:\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

   if 1.00000003e16 < dY.v

   1. Initial program 39.8%

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

   3. Simplified 39.8%

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

      1. rewrite-binary32/binary64 84.9%

         \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\langle \frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}} \rangle_{\text{binary64}}\\ } \end{array}} \]

   5. Applied rewrite-once 84.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\langle \frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}} \rangle_{\text{binary64}}\\ \end{array} \]

   7. Simplified 84.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\langle \frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 10000000272564224:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}:\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right), \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array}\\ \mathbf{elif}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right), \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right), \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\langle \frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \end{array} \]

Alternative 4: 91.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.u \cdot \left\lfloorw\right\rfloor\\ t_1 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_2 := {t_1}^{2}\\ t_3 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_4 := {t_3}^{2}\\ t_5 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_6 := t_5 \cdot t_5\\ t_7 := {t_5}^{2}\\ t_8 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_9 := {t_8}^{2}\\ t_10 := \sqrt{\mathsf{max}\left({t_0}^{2} + t_7, t_9 + t_4\right)}\\ t_11 := t_4 + t_9\\ \mathbf{if}\;dY.v \leq 10000000272564224:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t_7 + t_2 \geq t_11:\\ \;\;\;\;\langle \left( \frac{t_0}{t_10} \right)_{\text{binary64}} \rangle_{\text{binary32}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot t_1, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_5\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right), \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot t_3\right)\right)\right)}}\\ \end{array}\\ \mathbf{elif}\;t_2 + t_6 \geq t_11:\\ \;\;\;\;t_1 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_6 + t_1 \cdot t_1, t_3 \cdot t_3 + t_8 \cdot t_8\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{1}{\langle \left( t_10 \right)_{\text{binary64}} \rangle_{\text{binary32}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.u (floor w)))
        (t_1 (* (floor w) dX.u))
        (t_2 (pow t_1 2.0))
        (t_3 (* (floor w) dY.u))
        (t_4 (pow t_3 2.0))
        (t_5 (* (floor h) dX.v))
        (t_6 (* t_5 t_5))
        (t_7 (pow t_5 2.0))
        (t_8 (* (floor h) dY.v))
        (t_9 (pow t_8 2.0))
        (t_10 (sqrt (fmax (+ (pow t_0 2.0) t_7) (+ t_9 t_4))))
        (t_11 (+ t_4 t_9)))
   (if (<= dY.v 10000000272564224.0)
     (if (>= (+ t_7 t_2) t_11)
       (cast (! :precision binary64 (/ t_0 t_10)))
       (/
        t_3
        (sqrt
         (fmax
          (fma (floor w) (* dX.u t_1) (* (floor h) (* dX.v t_5)))
          (fma
           (floor h)
           (* (floor h) (* dY.v dY.v))
           (* (floor w) (* dY.u t_3)))))))
     (if (>= (+ t_2 t_6) t_11)
       (*
        t_1
        (/ 1.0 (sqrt (fmax (+ t_6 (* t_1 t_1)) (+ (* t_3 t_3) (* t_8 t_8))))))
       (* t_3 (/ 1.0 (cast (! :precision binary64 t_10))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dX_46_u * floorf(w);
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = powf(t_1, 2.0f);
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = powf(t_3, 2.0f);
	float t_5 = floorf(h) * dX_46_v;
	float t_6 = t_5 * t_5;
	float t_7 = powf(t_5, 2.0f);
	float t_8 = floorf(h) * dY_46_v;
	float t_9 = powf(t_8, 2.0f);
	float t_10 = sqrtf(fmaxf((powf(t_0, 2.0f) + t_7), (t_9 + t_4)));
	float t_11 = t_4 + t_9;
	double tmp_3;
	if (dY_46_v <= 10000000272564224.0f) {
		double tmp_5;
		if ((t_7 + t_2) >= t_11) {
			double tmp_6 = ((double) t_0) / ((double) t_10);
			tmp_5 = (float) tmp_6;
		} else {
			tmp_5 = ((double) t_3) / sqrt(fmax(fma(floor(w), (((double) dX_46_u) * ((double) t_1)), (floor(h) * (((double) dX_46_v) * ((double) t_5)))), fma(floor(h), (floor(h) * (((double) dY_46_v) * ((double) dY_46_v))), (floor(w) * (((double) dY_46_u) * ((double) t_3))))));
		}
		tmp_3 = tmp_5;
	} else if ((t_2 + t_6) >= t_11) {
		tmp_3 = ((double) t_1) * (1.0 / sqrt(fmax((((double) t_6) + (((double) t_1) * ((double) t_1))), ((((double) t_3) * ((double) t_3)) + (((double) t_8) * ((double) t_8))))));
	} else {
		float tmp_7 = t_10;
		tmp_3 = ((double) t_3) * (1.0 / ((double) ((double) tmp_7)));
	}
	return tmp_3;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dX_46_u * floor(w))
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = t_1 ^ Float32(2.0)
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = t_3 ^ Float32(2.0)
	t_5 = Float32(floor(h) * dX_46_v)
	t_6 = Float32(t_5 * t_5)
	t_7 = t_5 ^ Float32(2.0)
	t_8 = Float32(floor(h) * dY_46_v)
	t_9 = t_8 ^ Float32(2.0)
	t_10 = sqrt(((Float32((t_0 ^ Float32(2.0)) + t_7) != Float32((t_0 ^ Float32(2.0)) + t_7)) ? Float32(t_9 + t_4) : ((Float32(t_9 + t_4) != Float32(t_9 + t_4)) ? Float32((t_0 ^ Float32(2.0)) + t_7) : max(Float32((t_0 ^ Float32(2.0)) + t_7), Float32(t_9 + t_4)))))
	t_11 = Float32(t_4 + t_9)
	tmp_3 = 0.0
	if (dY_46_v <= Float32(10000000272564224.0))
		tmp_5 = 0.0
		if (Float32(t_7 + t_2) >= t_11)
			tmp_6 = Float64(Float64(t_0) / Float64(t_10))
			tmp_5 = Float32(tmp_6);
		else
			tmp_5 = Float64(Float64(t_3) / sqrt(((fma(floor(w), Float64(Float64(dX_46_u) * Float64(t_1)), Float64(floor(h) * Float64(Float64(dX_46_v) * Float64(t_5)))) != fma(floor(w), Float64(Float64(dX_46_u) * Float64(t_1)), Float64(floor(h) * Float64(Float64(dX_46_v) * Float64(t_5))))) ? fma(floor(h), Float64(floor(h) * Float64(Float64(dY_46_v) * Float64(dY_46_v))), Float64(floor(w) * Float64(Float64(dY_46_u) * Float64(t_3)))) : ((fma(floor(h), Float64(floor(h) * Float64(Float64(dY_46_v) * Float64(dY_46_v))), Float64(floor(w) * Float64(Float64(dY_46_u) * Float64(t_3)))) != fma(floor(h), Float64(floor(h) * Float64(Float64(dY_46_v) * Float64(dY_46_v))), Float64(floor(w) * Float64(Float64(dY_46_u) * Float64(t_3))))) ? fma(floor(w), Float64(Float64(dX_46_u) * Float64(t_1)), Float64(floor(h) * Float64(Float64(dX_46_v) * Float64(t_5)))) : max(fma(floor(w), Float64(Float64(dX_46_u) * Float64(t_1)), Float64(floor(h) * Float64(Float64(dX_46_v) * Float64(t_5)))), fma(floor(h), Float64(floor(h) * Float64(Float64(dY_46_v) * Float64(dY_46_v))), Float64(floor(w) * Float64(Float64(dY_46_u) * Float64(t_3)))))))));
		end
		tmp_3 = tmp_5;
	elseif (Float32(t_2 + t_6) >= t_11)
		tmp_3 = Float64(Float64(t_1) * Float64(1.0 / sqrt(((Float64(Float64(t_6) + Float64(Float64(t_1) * Float64(t_1))) != Float64(Float64(t_6) + Float64(Float64(t_1) * Float64(t_1)))) ? Float64(Float64(Float64(t_3) * Float64(t_3)) + Float64(Float64(t_8) * Float64(t_8))) : ((Float64(Float64(Float64(t_3) * Float64(t_3)) + Float64(Float64(t_8) * Float64(t_8))) != Float64(Float64(Float64(t_3) * Float64(t_3)) + Float64(Float64(t_8) * Float64(t_8)))) ? Float64(Float64(t_6) + Float64(Float64(t_1) * Float64(t_1))) : max(Float64(Float64(t_6) + Float64(Float64(t_1) * Float64(t_1))), Float64(Float64(Float64(t_3) * Float64(t_3)) + Float64(Float64(t_8) * Float64(t_8)))))))));
	else
		tmp_7 = t_10
		tmp_3 = Float64(Float64(t_3) * Float64(1.0 / Float64(Float64(tmp_7))));
	end
	return tmp_3
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 1.00000003e16

   1. Initial program 76.7%

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

   3. Simplified 76.8%

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

      1. rewrite-binary32/binary64 91.4%

         \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right):\\ \;\;\;\;\langle \frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ } \end{array}} \]

   5. Applied rewrite-once 91.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right):\\ \;\;\;\;\color{blue}{\langle \color{blue}{\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}} \rangle_{\text{binary64}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

   7. Simplified 91.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right):\\ \;\;\;\;\color{blue}{\langle \color{blue}{\frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}}} \rangle_{\text{binary64}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

   8. Taylor expanded in w around 0 91.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\mathsf{fma}\left(\left\lfloorw\right\rfloor, {dX.u}^{2} \cdot \left\lfloorw\right\rfloor, {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}:\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]
   9. Step-by-step derivation

      1. unpow2 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \color{blue}{\left(dX.u \cdot dX.u\right)} \cdot \left\lfloorw\right\rfloor, {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      2. associate-*r* 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \color{blue}{dX.u \cdot \left(dX.u \cdot \left\lfloorw\right\rfloor\right)}, {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      3. *-commutative 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \color{blue}{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}, {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      4. *-commutative 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      5. unpow2 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dX.v}^{2}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      6. unpow2 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      7. swap-sqr 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)}\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      8. fma-def 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)} \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      9. associate-*l* 91.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      10. unpow2 91.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      11. unpow2 91.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}} \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      12. *-commutative 91.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + {\color{blue}{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}}^{2} \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, {dY.v}^{2} \cdot \left\lfloorh\right\rfloor, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right):\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

      13. fma-udef 91.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} \geq \color{blue}{\left\lfloorh\right\rfloor \cdot \left({dY.v}^{2} \cdot \left\lfloorh\right\rfloor\right) + {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}}:\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

   10. Simplified 91.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}}:\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

   if 1.00000003e16 < dY.v

   1. Initial program 39.8%

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

   2. Taylor expanded in w around 0 39.8%

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

      1. unpow2 39.8%

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

      2. unpow2 39.8%

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

      3. swap-sqr 39.8%

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

      4. unpow2 39.8%

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

   4. Simplified 39.8%

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

   5. Taylor expanded in w around 0 39.8%

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

      1. *-commutative 39.8%

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

      2. unpow2 39.8%

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

      3. unpow2 39.8%

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

      4. swap-sqr 39.8%

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

      5. unpow2 39.8%

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

   7. Simplified 39.8%

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

   8. Taylor expanded in h around 0 39.8%

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

      1. *-commutative 39.8%

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

      2. unpow2 39.8%

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

      3. unpow2 39.8%

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

      4. swap-sqr 39.8%

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

      5. unpow2 39.8%

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

      6. *-commutative 39.8%

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

   10. Simplified 39.8%

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

       1. rewrite-binary32/binary64 84.1%

          \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\langle \sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)} \rangle_{\text{binary64}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ } \end{array}} \]

   12. Applied rewrite-once 84.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\langle \sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)} \rangle_{\text{binary64}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   13. Step-by-step derivation

   14. Simplified 84.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\langle \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)} \rangle_{\text{binary64}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 10000000272564224:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}:\\ \;\;\;\;\langle \frac{dX.u \cdot \left\lfloorw\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right), \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array}\\ \mathbf{elif}\;{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) + \left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \frac{1}{\langle \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)} \rangle_{\text{binary64}}}\\ \end{array} \]

Alternative 5: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := t_2 \cdot t_2\\ t_4 := \mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot t_0, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_2\right)\right)\\ t_5 := {t_1}^{2}\\ t_6 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_7 := \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right), \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot t_6\right)\right)\\ t_8 := {t_6}^{2}\\ t_9 := \langle \left( \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {t_2}^{2}, t_5 + t_8\right)} \right)_{\text{binary64}} \rangle_{\text{binary32}}\\ \mathbf{if}\;dY.v \leq 10000000272564224:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t_4 \geq t_7:\\ \;\;\;\;\frac{-\left\lfloorw\right\rfloor}{\frac{t_9}{-dX.u}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_6}{\sqrt{\mathsf{max}\left(t_4, t_7\right)}}\\ \end{array}\\ \mathbf{elif}\;{t_0}^{2} + t_3 \geq t_8 + t_5:\\ \;\;\;\;t_0 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_3 + t_0 \cdot t_0, t_6 \cdot t_6 + t_1 \cdot t_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot \frac{1}{t_9}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor h) dX.v))
        (t_3 (* t_2 t_2))
        (t_4 (fma (floor w) (* dX.u t_0) (* (floor h) (* dX.v t_2))))
        (t_5 (pow t_1 2.0))
        (t_6 (* (floor w) dY.u))
        (t_7
         (fma
          (floor h)
          (* (floor h) (* dY.v dY.v))
          (* (floor w) (* dY.u t_6))))
        (t_8 (pow t_6 2.0))
        (t_9
         (cast
          (!
           :precision
           binary64
           (sqrt
            (fmax
             (+ (pow (* dX.u (floor w)) 2.0) (pow t_2 2.0))
             (+ t_5 t_8)))))))
   (if (<= dY.v 10000000272564224.0)
     (if (>= t_4 t_7)
       (/ (- (floor w)) (/ t_9 (- dX.u)))
       (/ t_6 (sqrt (fmax t_4 t_7))))
     (if (>= (+ (pow t_0 2.0) t_3) (+ t_8 t_5))
       (*
        t_0
        (/ 1.0 (sqrt (fmax (+ t_3 (* t_0 t_0)) (+ (* t_6 t_6) (* t_1 t_1))))))
       (* t_6 (/ 1.0 t_9))))))

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(w) * dX_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = t_2 * t_2;
	float t_4 = fmaf(floorf(w), (dX_46_u * t_0), (floorf(h) * (dX_46_v * t_2)));
	float t_5 = powf(t_1, 2.0f);
	float t_6 = floorf(w) * dY_46_u;
	float t_7 = fmaf(floorf(h), (floorf(h) * (dY_46_v * dY_46_v)), (floorf(w) * (dY_46_u * t_6)));
	float t_8 = powf(t_6, 2.0f);
	double tmp = sqrt(fmax((pow((((double) dX_46_u) * floor(w)), 2.0) + pow(t_2, 2.0)), (((double) t_5) + ((double) t_8))));
	double t_9 = (float) tmp;
	float tmp_2;
	if (dY_46_v <= 10000000272564224.0f) {
		float tmp_3;
		if (t_4 >= t_7) {
			tmp_3 = -floorf(w) / (t_9 / -dX_46_u);
		} else {
			tmp_3 = t_6 / sqrtf(fmaxf(t_4, t_7));
		}
		tmp_2 = tmp_3;
	} else if ((powf(t_0, 2.0f) + t_3) >= (t_8 + t_5)) {
		tmp_2 = t_0 * (1.0f / sqrtf(fmaxf((t_3 + (t_0 * t_0)), ((t_6 * t_6) + (t_1 * t_1)))));
	} else {
		tmp_2 = t_6 * (1.0f / t_9);
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(t_2 * t_2)
	t_4 = fma(floor(w), Float32(dX_46_u * t_0), Float32(floor(h) * Float32(dX_46_v * t_2)))
	t_5 = t_1 ^ Float32(2.0)
	t_6 = Float32(floor(w) * dY_46_u)
	t_7 = fma(floor(h), Float32(floor(h) * Float32(dY_46_v * dY_46_v)), Float32(floor(w) * Float32(dY_46_u * t_6)))
	t_8 = t_6 ^ Float32(2.0)
	tmp = sqrt(((Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (t_2 ^ 2.0)) != Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (t_2 ^ 2.0))) ? Float64(Float64(t_5) + Float64(t_8)) : ((Float64(Float64(t_5) + Float64(t_8)) != Float64(Float64(t_5) + Float64(t_8))) ? Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (t_2 ^ 2.0)) : max(Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (t_2 ^ 2.0)), Float64(Float64(t_5) + Float64(t_8))))))
	t_9 = Float32(tmp)
	tmp_2 = Float32(0.0)
	if (dY_46_v <= Float32(10000000272564224.0))
		tmp_3 = Float32(0.0)
		if (t_4 >= t_7)
			tmp_3 = Float32(Float32(-floor(w)) / Float32(t_9 / Float32(-dX_46_u)));
		else
			tmp_3 = Float32(t_6 / sqrt(((t_4 != t_4) ? t_7 : ((t_7 != t_7) ? t_4 : max(t_4, t_7)))));
		end
		tmp_2 = tmp_3;
	elseif (Float32((t_0 ^ Float32(2.0)) + t_3) >= Float32(t_8 + t_5))
		tmp_2 = Float32(t_0 * Float32(Float32(1.0) / sqrt(((Float32(t_3 + Float32(t_0 * t_0)) != Float32(t_3 + Float32(t_0 * t_0))) ? Float32(Float32(t_6 * t_6) + Float32(t_1 * t_1)) : ((Float32(Float32(t_6 * t_6) + Float32(t_1 * t_1)) != Float32(Float32(t_6 * t_6) + Float32(t_1 * t_1))) ? Float32(t_3 + Float32(t_0 * t_0)) : max(Float32(t_3 + Float32(t_0 * t_0)), Float32(Float32(t_6 * t_6) + Float32(t_1 * t_1))))))));
	else
		tmp_2 = Float32(t_6 * Float32(Float32(1.0) / t_9));
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 1.00000003e16

   1. Initial program 76.7%

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

   3. Simplified 76.8%

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

   5. Applied egg-rr 76.8%

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

      1. rewrite-binary32/binary64 91.0%

         \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right):\\ \;\;\;\;-\frac{\left\lfloorw\right\rfloor}{\frac{\langle \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}\right)} \rangle_{\text{binary64}}}{-dX.u}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ } \end{array}} \]

   7. Applied rewrite-once 91.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right):\\ \;\;\;\;-\frac{\left\lfloorw\right\rfloor}{\frac{\color{blue}{\langle \color{blue}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}\right)}} \rangle_{\text{binary64}}}}{-dX.u}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array} \]

   if 1.00000003e16 < dY.v

   1. Initial program 39.8%

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

   2. Taylor expanded in w around 0 39.8%

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

      1. unpow2 39.8%

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

      2. unpow2 39.8%

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

      3. swap-sqr 39.8%

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

      4. unpow2 39.8%

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

   4. Simplified 39.8%

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

   5. Taylor expanded in w around 0 39.8%

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

      1. *-commutative 39.8%

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

      2. unpow2 39.8%

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

      3. unpow2 39.8%

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

      4. swap-sqr 39.8%

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

      5. unpow2 39.8%

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

   7. Simplified 39.8%

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

   8. Taylor expanded in h around 0 39.8%

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

      1. *-commutative 39.8%

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

      2. unpow2 39.8%

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

      3. unpow2 39.8%

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

      4. swap-sqr 39.8%

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

      5. unpow2 39.8%

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

      6. *-commutative 39.8%

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

   10. Simplified 39.8%

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

       1. rewrite-binary32/binary64 84.1%

          \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\langle \sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)} \rangle_{\text{binary64}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ } \end{array}} \]

   12. Applied rewrite-once 84.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\langle \sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)} \rangle_{\text{binary64}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   13. Step-by-step derivation

   14. Simplified 84.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\langle \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)} \rangle_{\text{binary64}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 10000000272564224:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right), \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right):\\ \;\;\;\;\frac{-\left\lfloorw\right\rfloor}{\frac{\langle \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)} \rangle_{\text{binary64}}}{-dX.u}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right), \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)}}\\ \end{array}\\ \mathbf{elif}\;{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) + \left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \frac{1}{\langle \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)} \rangle_{\text{binary64}}}\\ \end{array} \]

Alternative 6: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := t_0 \cdot t_0\\ t_2 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_3 := {t_2}^{2}\\ t_4 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_5 := {t_4}^{2}\\ t_6 := \frac{1}{\langle \left( \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {t_0}^{2}, t_5 + t_3\right)} \right)_{\text{binary64}} \rangle_{\text{binary32}}}\\ t_7 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_8 := {t_7}^{2} + t_1 \geq t_3 + t_5\\ t_9 := \frac{1}{\sqrt{\mathsf{max}\left(t_1 + t_7 \cdot t_7, t_2 \cdot t_2 + t_4 \cdot t_4\right)}}\\ \mathbf{if}\;dY.v \leq 10000000272564224:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t_8:\\ \;\;\;\;t_7 \cdot t_6\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_9\\ \end{array}\\ \mathbf{elif}\;t_8:\\ \;\;\;\;t_7 \cdot t_9\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_6\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* t_0 t_0))
        (t_2 (* (floor w) dY.u))
        (t_3 (pow t_2 2.0))
        (t_4 (* (floor h) dY.v))
        (t_5 (pow t_4 2.0))
        (t_6
         (/
          1.0
          (cast
           (!
            :precision
            binary64
            (sqrt
             (fmax
              (+ (pow (* dX.u (floor w)) 2.0) (pow t_0 2.0))
              (+ t_5 t_3)))))))
        (t_7 (* (floor w) dX.u))
        (t_8 (>= (+ (pow t_7 2.0) t_1) (+ t_3 t_5)))
        (t_9
         (/
          1.0
          (sqrt (fmax (+ t_1 (* t_7 t_7)) (+ (* t_2 t_2) (* t_4 t_4)))))))
   (if (<= dY.v 10000000272564224.0)
     (if t_8 (* t_7 t_6) (* t_2 t_9))
     (if t_8 (* t_7 t_9) (* t_2 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 = t_0 * t_0;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = powf(t_2, 2.0f);
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = powf(t_4, 2.0f);
	double tmp = sqrt(fmax((pow((((double) dX_46_u) * floor(w)), 2.0) + pow(t_0, 2.0)), (((double) t_5) + ((double) t_3))));
	float t_6 = 1.0f / ((float) tmp);
	float t_7 = floorf(w) * dX_46_u;
	int t_8 = (powf(t_7, 2.0f) + t_1) >= (t_3 + t_5);
	float t_9 = 1.0f / sqrtf(fmaxf((t_1 + (t_7 * t_7)), ((t_2 * t_2) + (t_4 * t_4))));
	float tmp_2;
	if (dY_46_v <= 10000000272564224.0f) {
		float tmp_3;
		if (t_8) {
			tmp_3 = t_7 * t_6;
		} else {
			tmp_3 = t_2 * t_9;
		}
		tmp_2 = tmp_3;
	} else if (t_8) {
		tmp_2 = t_7 * t_9;
	} else {
		tmp_2 = t_2 * t_6;
	}
	return tmp_2;
}

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(t_0 * t_0)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = t_2 ^ Float32(2.0)
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = t_4 ^ Float32(2.0)
	tmp = sqrt(((Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (t_0 ^ 2.0)) != Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (t_0 ^ 2.0))) ? Float64(Float64(t_5) + Float64(t_3)) : ((Float64(Float64(t_5) + Float64(t_3)) != Float64(Float64(t_5) + Float64(t_3))) ? Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (t_0 ^ 2.0)) : max(Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (t_0 ^ 2.0)), Float64(Float64(t_5) + Float64(t_3))))))
	t_6 = Float32(Float32(1.0) / Float32(tmp))
	t_7 = Float32(floor(w) * dX_46_u)
	t_8 = Float32((t_7 ^ Float32(2.0)) + t_1) >= Float32(t_3 + t_5)
	t_9 = Float32(Float32(1.0) / sqrt(((Float32(t_1 + Float32(t_7 * t_7)) != Float32(t_1 + Float32(t_7 * t_7))) ? Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4)) : ((Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4)) != Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4))) ? Float32(t_1 + Float32(t_7 * t_7)) : max(Float32(t_1 + Float32(t_7 * t_7)), Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4)))))))
	tmp_2 = Float32(0.0)
	if (dY_46_v <= Float32(10000000272564224.0))
		tmp_3 = Float32(0.0)
		if (t_8)
			tmp_3 = Float32(t_7 * t_6);
		else
			tmp_3 = Float32(t_2 * t_9);
		end
		tmp_2 = tmp_3;
	elseif (t_8)
		tmp_2 = Float32(t_7 * t_9);
	else
		tmp_2 = Float32(t_2 * t_6);
	end
	return tmp_2
end

MATLAB

function tmp_5 = 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 = t_0 * t_0;
	t_2 = floor(w) * dY_46_u;
	t_3 = t_2 ^ single(2.0);
	t_4 = floor(h) * dY_46_v;
	t_5 = t_4 ^ single(2.0);
	tmp = sqrt(max((((double(dX_46_u) * floor(w)) ^ 2.0) + (double(t_0) ^ 2.0)), (double(t_5) + double(t_3))));
	t_6 = single((double(single(1.0)) / single(tmp)));
	t_7 = floor(w) * dX_46_u;
	t_8 = ((t_7 ^ single(2.0)) + t_1) >= (t_3 + t_5);
	t_9 = single(1.0) / sqrt(max((t_1 + (t_7 * t_7)), ((t_2 * t_2) + (t_4 * t_4))));
	tmp_3 = single(0.0);
	if (dY_46_v <= single(10000000272564224.0))
		tmp_4 = single(0.0);
		if (t_8)
			tmp_4 = t_7 * t_6;
		else
			tmp_4 = t_2 * t_9;
		end
		tmp_3 = tmp_4;
	elseif (t_8)
		tmp_3 = t_7 * t_9;
	else
		tmp_3 = t_2 * t_6;
	end
	tmp_5 = tmp_3;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 1.00000003e16

   1. Initial program 76.7%

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

   2. Taylor expanded in w around 0 76.7%

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

      1. unpow2 76.7%

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

      2. unpow2 76.7%

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

      3. swap-sqr 76.7%

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

      4. unpow2 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in w around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. unpow2 76.7%

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

      3. unpow2 76.7%

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

      4. swap-sqr 76.7%

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

      5. unpow2 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in h around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. unpow2 76.7%

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

      3. unpow2 76.7%

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

      4. swap-sqr 76.7%

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

      5. unpow2 76.7%

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

      6. *-commutative 76.7%

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

   10. Simplified 76.7%

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

       1. rewrite-binary32/binary64 90.9%

          \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\langle \sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)} \rangle_{\text{binary64}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ } \end{array}} \]

   12. Applied rewrite-once 90.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\color{blue}{\langle \color{blue}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \rangle_{\text{binary64}}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   13. Step-by-step derivation

   14. Simplified 90.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\color{blue}{\langle \color{blue}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   if 1.00000003e16 < dY.v

   1. Initial program 39.8%

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

   2. Taylor expanded in w around 0 39.8%

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

      1. unpow2 39.8%

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

      2. unpow2 39.8%

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

      3. swap-sqr 39.8%

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

      4. unpow2 39.8%

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

   4. Simplified 39.8%

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

   5. Taylor expanded in w around 0 39.8%

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

      1. *-commutative 39.8%

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

      2. unpow2 39.8%

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

      3. unpow2 39.8%

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

      4. swap-sqr 39.8%

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

      5. unpow2 39.8%

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

   7. Simplified 39.8%

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

   8. Taylor expanded in h around 0 39.8%

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

      1. *-commutative 39.8%

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

      2. unpow2 39.8%

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

      3. unpow2 39.8%

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

      4. swap-sqr 39.8%

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

      5. unpow2 39.8%

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

      6. *-commutative 39.8%

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

   10. Simplified 39.8%

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

       1. rewrite-binary32/binary64 84.1%

          \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\langle \sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)} \rangle_{\text{binary64}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ } \end{array}} \]

   12. Applied rewrite-once 84.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\langle \sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)} \rangle_{\text{binary64}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   13. Step-by-step derivation

   14. Simplified 84.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\langle \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)} \rangle_{\text{binary64}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 10000000272564224:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \frac{1}{\langle \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)} \rangle_{\text{binary64}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) + \left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array}\\ \mathbf{elif}\;{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) + \left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \frac{1}{\langle \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)} \rangle_{\text{binary64}}}\\ \end{array} \]

Alternative 7: 88.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := t_0 \cdot t_0\\ t_2 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_3 := {t_2}^{2}\\ t_4 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_5 := {t_4}^{2}\\ t_6 := \left\lfloorw\right\rfloor \cdot dX.u\\ \mathbf{if}\;{t_6}^{2} + t_1 \geq t_3 + t_5:\\ \;\;\;\;t_6 \cdot \frac{1}{\langle \left( \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {t_0}^{2}, t_5 + t_3\right)} \right)_{\text{binary64}} \rangle_{\text{binary32}}}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_1 + t_6 \cdot t_6, t_2 \cdot t_2 + t_4 \cdot t_4\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* t_0 t_0))
        (t_2 (* (floor w) dY.u))
        (t_3 (pow t_2 2.0))
        (t_4 (* (floor h) dY.v))
        (t_5 (pow t_4 2.0))
        (t_6 (* (floor w) dX.u)))
   (if (>= (+ (pow t_6 2.0) t_1) (+ t_3 t_5))
     (*
      t_6
      (/
       1.0
       (cast
        (!
         :precision
         binary64
         (sqrt
          (fmax
           (+ (pow (* dX.u (floor w)) 2.0) (pow t_0 2.0))
           (+ t_5 t_3)))))))
     (*
      t_2
      (/ 1.0 (sqrt (fmax (+ t_1 (* t_6 t_6)) (+ (* t_2 t_2) (* t_4 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 = t_0 * t_0;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = powf(t_2, 2.0f);
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = powf(t_4, 2.0f);
	float t_6 = floorf(w) * dX_46_u;
	float tmp_1;
	if ((powf(t_6, 2.0f) + t_1) >= (t_3 + t_5)) {
		double tmp_2 = sqrt(fmax((pow((((double) dX_46_u) * floor(w)), 2.0) + pow(t_0, 2.0)), (((double) t_5) + ((double) t_3))));
		tmp_1 = t_6 * (1.0f / ((float) tmp_2));
	} else {
		tmp_1 = t_2 * (1.0f / sqrtf(fmaxf((t_1 + (t_6 * t_6)), ((t_2 * t_2) + (t_4 * t_4)))));
	}
	return tmp_1;
}

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(t_0 * t_0)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = t_2 ^ Float32(2.0)
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = t_4 ^ Float32(2.0)
	t_6 = Float32(floor(w) * dX_46_u)
	tmp_1 = Float32(0.0)
	if (Float32((t_6 ^ Float32(2.0)) + t_1) >= Float32(t_3 + t_5))
		tmp_2 = sqrt(((Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (t_0 ^ 2.0)) != Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (t_0 ^ 2.0))) ? Float64(Float64(t_5) + Float64(t_3)) : ((Float64(Float64(t_5) + Float64(t_3)) != Float64(Float64(t_5) + Float64(t_3))) ? Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (t_0 ^ 2.0)) : max(Float64((Float64(Float64(dX_46_u) * floor(w)) ^ 2.0) + (t_0 ^ 2.0)), Float64(Float64(t_5) + Float64(t_3))))))
		tmp_1 = Float32(t_6 * Float32(Float32(1.0) / Float32(tmp_2)));
	else
		tmp_1 = Float32(t_2 * Float32(Float32(1.0) / sqrt(((Float32(t_1 + Float32(t_6 * t_6)) != Float32(t_1 + Float32(t_6 * t_6))) ? Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4)) : ((Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4)) != Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4))) ? Float32(t_1 + Float32(t_6 * t_6)) : max(Float32(t_1 + Float32(t_6 * t_6)), Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4))))))));
	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 = floor(h) * dX_46_v;
	t_1 = t_0 * t_0;
	t_2 = floor(w) * dY_46_u;
	t_3 = t_2 ^ single(2.0);
	t_4 = floor(h) * dY_46_v;
	t_5 = t_4 ^ single(2.0);
	t_6 = floor(w) * dX_46_u;
	tmp_2 = single(0.0);
	if (((t_6 ^ single(2.0)) + t_1) >= (t_3 + t_5))
		tmp_3 = sqrt(max((((double(dX_46_u) * floor(w)) ^ 2.0) + (double(t_0) ^ 2.0)), (double(t_5) + double(t_3))));
		tmp_2 = t_6 * single((double(single(1.0)) / single(tmp_3)));
	else
		tmp_2 = t_2 * (single(1.0) / sqrt(max((t_1 + (t_6 * t_6)), ((t_2 * t_2) + (t_4 * t_4)))));
	end
	tmp_4 = tmp_2;
end

Derivation

1. Initial program 75.0%

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

2. Taylor expanded in w around 0 75.0%

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

   1. unpow2 75.0%

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

   2. unpow2 75.0%

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

   3. swap-sqr 75.0%

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

   4. unpow2 75.0%

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

4. Simplified 75.0%

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

5. Taylor expanded in w around 0 75.0%

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

   1. *-commutative 75.0%

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

   2. unpow2 75.0%

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

   3. unpow2 75.0%

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

   4. swap-sqr 75.0%

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

   5. unpow2 75.0%

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

7. Simplified 75.0%

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

8. Taylor expanded in h around 0 75.0%

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

   1. *-commutative 75.0%

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

   2. unpow2 75.0%

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

   3. unpow2 75.0%

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

   4. swap-sqr 75.0%

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

   5. unpow2 75.0%

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

   6. *-commutative 75.0%

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

10. Simplified 75.0%

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

    1. rewrite-binary32/binary64 89.2%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\langle \sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)} \rangle_{\text{binary64}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ } \end{array}} \]

12. Applied rewrite-once 89.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\color{blue}{\langle \color{blue}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \rangle_{\text{binary64}}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
13. Step-by-step derivation

14. Simplified 89.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}:\\ \;\;\;\;\frac{1}{\color{blue}{\langle \color{blue}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}} \rangle_{\text{binary64}}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

15. Final simplification 89.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \frac{1}{\langle \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)} \rangle_{\text{binary64}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) + \left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]

Alternative 8: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right), \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot t_1\right)\right)\\ t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_4 := \mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot t_3, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_0\right)\right)\\ \mathbf{if}\;t_4 \geq t_2:\\ \;\;\;\;\frac{-\left\lfloorw\right\rfloor}{\frac{\sqrt{\mathsf{max}\left({t_0}^{2} + {t_3}^{2}, {t_1}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)}}{-dX.u}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\sqrt{\mathsf{max}\left(t_4, t_2\right)}}\\ \end{array} \end{array} \]

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
         (fma
          (floor h)
          (* (floor h) (* dY.v dY.v))
          (* (floor w) (* dY.u t_1))))
        (t_3 (* (floor w) dX.u))
        (t_4 (fma (floor w) (* dX.u t_3) (* (floor h) (* dX.v t_0)))))
   (if (>= t_4 t_2)
     (/
      (- (floor w))
      (/
       (sqrt
        (fmax
         (+ (pow t_0 2.0) (pow t_3 2.0))
         (+ (pow t_1 2.0) (pow (* (floor h) dY.v) 2.0))))
       (- dX.u)))
     (/ t_1 (sqrt (fmax t_4 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 = floorf(w) * dY_46_u;
	float t_2 = fmaf(floorf(h), (floorf(h) * (dY_46_v * dY_46_v)), (floorf(w) * (dY_46_u * t_1)));
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = fmaf(floorf(w), (dX_46_u * t_3), (floorf(h) * (dX_46_v * t_0)));
	float tmp;
	if (t_4 >= t_2) {
		tmp = -floorf(w) / (sqrtf(fmaxf((powf(t_0, 2.0f) + powf(t_3, 2.0f)), (powf(t_1, 2.0f) + powf((floorf(h) * dY_46_v), 2.0f)))) / -dX_46_u);
	} else {
		tmp = t_1 / sqrtf(fmaxf(t_4, 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(floor(w) * dY_46_u)
	t_2 = fma(floor(h), Float32(floor(h) * Float32(dY_46_v * dY_46_v)), Float32(floor(w) * Float32(dY_46_u * t_1)))
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = fma(floor(w), Float32(dX_46_u * t_3), Float32(floor(h) * Float32(dX_46_v * t_0)))
	tmp = Float32(0.0)
	if (t_4 >= t_2)
		tmp = Float32(Float32(-floor(w)) / Float32(sqrt(((Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))) != Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))) ? Float32((t_1 ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) : ((Float32((t_1 ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) != Float32((t_1 ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))) ? Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))) : max(Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))), Float32((t_1 ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))))))) / Float32(-dX_46_u)));
	else
		tmp = Float32(t_1 / sqrt(((t_4 != t_4) ? t_2 : ((t_2 != t_2) ? t_4 : max(t_4, t_2)))));
	end
	return tmp
end

Derivation

1. Initial program 75.0%

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

3. Simplified 75.1%

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

5. Applied egg-rr 75.1%

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

6. Final simplification 75.1%

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

Alternative 9: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dY.v\right), \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot t_1\right)\right)\\ t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_4 := \mathsf{fma}\left(\left\lfloorw\right\rfloor, dX.u \cdot t_3, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_0\right)\right)\\ \mathbf{if}\;t_4 \geq t_2:\\ \;\;\;\;\frac{t_3}{\sqrt{\mathsf{max}\left(t_4, t_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left({t_0}^{2} + {t_3}^{2}, {t_1}^{2} + {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{-0.5} \cdot {\left(\frac{1}{t_1}\right)}^{-1}\\ \end{array} \end{array} \]

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
         (fma
          (floor h)
          (* (floor h) (* dY.v dY.v))
          (* (floor w) (* dY.u t_1))))
        (t_3 (* (floor w) dX.u))
        (t_4 (fma (floor w) (* dX.u t_3) (* (floor h) (* dX.v t_0)))))
   (if (>= t_4 t_2)
     (/ t_3 (sqrt (fmax t_4 t_2)))
     (*
      (pow
       (fmax
        (+ (pow t_0 2.0) (pow t_3 2.0))
        (+ (pow t_1 2.0) (pow (* (floor h) dY.v) 2.0)))
       -0.5)
      (pow (/ 1.0 t_1) -1.0)))))

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 = fmaf(floorf(h), (floorf(h) * (dY_46_v * dY_46_v)), (floorf(w) * (dY_46_u * t_1)));
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = fmaf(floorf(w), (dX_46_u * t_3), (floorf(h) * (dX_46_v * t_0)));
	float tmp;
	if (t_4 >= t_2) {
		tmp = t_3 / sqrtf(fmaxf(t_4, t_2));
	} else {
		tmp = powf(fmaxf((powf(t_0, 2.0f) + powf(t_3, 2.0f)), (powf(t_1, 2.0f) + powf((floorf(h) * dY_46_v), 2.0f))), -0.5f) * powf((1.0f / t_1), -1.0f);
	}
	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 = fma(floor(h), Float32(floor(h) * Float32(dY_46_v * dY_46_v)), Float32(floor(w) * Float32(dY_46_u * t_1)))
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = fma(floor(w), Float32(dX_46_u * t_3), Float32(floor(h) * Float32(dX_46_v * t_0)))
	tmp = Float32(0.0)
	if (t_4 >= t_2)
		tmp = Float32(t_3 / sqrt(((t_4 != t_4) ? t_2 : ((t_2 != t_2) ? t_4 : max(t_4, t_2)))));
	else
		tmp = Float32((((Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))) != Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))) ? Float32((t_1 ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) : ((Float32((t_1 ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) != Float32((t_1 ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))) ? Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))) : max(Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))), Float32((t_1 ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))) ^ Float32(-0.5)) * (Float32(Float32(1.0) / t_1) ^ Float32(-1.0)));
	end
	return tmp
end

Derivation

1. Initial program 75.0%

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

3. Simplified 75.1%

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

5. Applied egg-rr 75.1%

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

6. Final simplification 75.1%

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

Alternative 10: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := t_0 \cdot t_0\\ t_2 := {\left(\left\lfloorh\right\rfloor\right)}^{2}\\ t_3 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_4 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_5 := {t_4}^{2}\\ t_6 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_7 := t_3 \cdot t_3 + t_6 \cdot t_6\\ \mathbf{if}\;t_5 + t_1 \geq t_7:\\ \;\;\;\;t_4 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_5 + {dX.v}^{2} \cdot t_2, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + t_2 \cdot {dY.v}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t_1 + t_4 \cdot t_4, t_7\right)}}\\ \end{array} \end{array} \]

FPCore

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

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

Derivation

1. Initial program 75.0%

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

2. Taylor expanded in w around 0 75.0%

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

   1. unpow2 75.0%

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

   2. unpow2 75.0%

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

   3. swap-sqr 75.0%

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

   4. unpow2 75.0%

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

4. Simplified 75.0%

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

   1. pow2 75.0%

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

   2. pow-to-exp 57.0%

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

   3. *-commutative 57.0%

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

   4. *-commutative 57.0%

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

6. Applied egg-rr 57.0%

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

7. Taylor expanded in dX.u around 0 75.0%

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

8. Final simplification 75.0%

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

Alternative 11: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := t_0 \cdot t_0\\ t_2 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_3 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_4 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_5 := \frac{1}{\sqrt{\mathsf{max}\left(t_1 + t_4 \cdot t_4, t_2 \cdot t_2 + t_3 \cdot t_3\right)}}\\ \mathbf{if}\;{t_4}^{2} + t_1 \geq {t_2}^{2} + {t_3}^{2}:\\ \;\;\;\;t_4 \cdot t_5\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_5\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* t_0 t_0))
        (t_2 (* (floor w) dY.u))
        (t_3 (* (floor h) dY.v))
        (t_4 (* (floor w) dX.u))
        (t_5
         (/
          1.0
          (sqrt (fmax (+ t_1 (* t_4 t_4)) (+ (* t_2 t_2) (* t_3 t_3)))))))
   (if (>= (+ (pow t_4 2.0) t_1) (+ (pow t_2 2.0) (pow t_3 2.0)))
     (* t_4 t_5)
     (* t_2 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 = t_0 * t_0;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = floorf(h) * dY_46_v;
	float t_4 = floorf(w) * dX_46_u;
	float t_5 = 1.0f / sqrtf(fmaxf((t_1 + (t_4 * t_4)), ((t_2 * t_2) + (t_3 * t_3))));
	float tmp;
	if ((powf(t_4, 2.0f) + t_1) >= (powf(t_2, 2.0f) + powf(t_3, 2.0f))) {
		tmp = t_4 * t_5;
	} else {
		tmp = t_2 * 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(t_0 * t_0)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(floor(h) * dY_46_v)
	t_4 = Float32(floor(w) * dX_46_u)
	t_5 = Float32(Float32(1.0) / sqrt(((Float32(t_1 + Float32(t_4 * t_4)) != Float32(t_1 + Float32(t_4 * t_4))) ? Float32(Float32(t_2 * t_2) + Float32(t_3 * t_3)) : ((Float32(Float32(t_2 * t_2) + Float32(t_3 * t_3)) != Float32(Float32(t_2 * t_2) + Float32(t_3 * t_3))) ? Float32(t_1 + Float32(t_4 * t_4)) : max(Float32(t_1 + Float32(t_4 * t_4)), Float32(Float32(t_2 * t_2) + Float32(t_3 * t_3)))))))
	tmp = Float32(0.0)
	if (Float32((t_4 ^ Float32(2.0)) + t_1) >= Float32((t_2 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))))
		tmp = Float32(t_4 * t_5);
	else
		tmp = Float32(t_2 * 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 = t_0 * t_0;
	t_2 = floor(w) * dY_46_u;
	t_3 = floor(h) * dY_46_v;
	t_4 = floor(w) * dX_46_u;
	t_5 = single(1.0) / sqrt(max((t_1 + (t_4 * t_4)), ((t_2 * t_2) + (t_3 * t_3))));
	tmp = single(0.0);
	if (((t_4 ^ single(2.0)) + t_1) >= ((t_2 ^ single(2.0)) + (t_3 ^ single(2.0))))
		tmp = t_4 * t_5;
	else
		tmp = t_2 * t_5;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 75.0%

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

2. Taylor expanded in w around 0 75.0%

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

   1. unpow2 75.0%

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

   2. unpow2 75.0%

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

   3. swap-sqr 75.0%

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

   4. unpow2 75.0%

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

4. Simplified 75.0%

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

5. Taylor expanded in w around 0 75.0%

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

   1. *-commutative 75.0%

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

   2. unpow2 75.0%

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

   3. unpow2 75.0%

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

   4. swap-sqr 75.0%

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

   5. unpow2 75.0%

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

7. Simplified 75.0%

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

8. Taylor expanded in h around 0 75.0%

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

   1. *-commutative 75.0%

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

   2. unpow2 75.0%

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

   3. unpow2 75.0%

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

   4. swap-sqr 75.0%

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

   5. unpow2 75.0%

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

   6. *-commutative 75.0%

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

10. Simplified 75.0%

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

11. Final simplification 75.0%

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

Reproduce

herbie shell --seed 2023297 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (line direction, u)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (== maxAniso 16.0))
  (if (>= (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor w) dX.u)) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor w) dY.u))))