UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 10.5s

Alternatives: 16

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
       (t_1 (sqrt (- 1.0 (* t_0 t_0))))
       (t_2 (* (* uy 2.0) PI)))
  (+
   (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi))
   (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Initial Program: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
       (t_1 (sqrt (- 1.0 (* t_0 t_0))))
       (t_2 (* (* uy 2.0) PI)))
  (+
   (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi))
   (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := maxCos \cdot \left(1 - ux\right)\\ t_1 := \pi \cdot \left(uy + uy\right)\\ t_2 := \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot t\_0, ux, 1\right)}\\ \mathsf{fma}\left(t\_2 \cdot \cos t\_1, xi, \mathsf{fma}\left(zi \cdot t\_0, ux, yi \cdot \left(\sin t\_1 \cdot t\_2\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* maxCos (- 1.0 ux)))
       (t_1 (* PI (+ uy uy)))
       (t_2 (sqrt (fma (* (* (- ux 1.0) (* maxCos ux)) t_0) ux 1.0))))
  (fma
   (* t_2 (cos t_1))
   xi
   (fma (* zi t_0) ux (* yi (* (sin t_1) t_2))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = maxCos * (1.0f - ux);
	float t_1 = ((float) M_PI) * (uy + uy);
	float t_2 = sqrtf(fmaf((((ux - 1.0f) * (maxCos * ux)) * t_0), ux, 1.0f));
	return fmaf((t_2 * cosf(t_1)), xi, fmaf((zi * t_0), ux, (yi * (sinf(t_1) * t_2))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(maxCos * Float32(Float32(1.0) - ux))
	t_1 = Float32(Float32(pi) * Float32(uy + uy))
	t_2 = sqrt(fma(Float32(Float32(Float32(ux - Float32(1.0)) * Float32(maxCos * ux)) * t_0), ux, Float32(1.0)))
	return fma(Float32(t_2 * cos(t_1)), xi, fma(Float32(zi * t_0), ux, Float32(yi * Float32(sin(t_1) * t_2))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(maxCos \cdot \left(1 - ux\right)\right), ux, 1\right)} \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right), xi, \mathsf{fma}\left(zi \cdot \left(maxCos \cdot \left(1 - ux\right)\right), ux, yi \cdot \left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(maxCos \cdot \left(1 - ux\right)\right), ux, 1\right)}\right)\right)\right)} \]
4. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := maxCos \cdot \left(1 - ux\right)\\ t_1 := \pi \cdot \left(uy + uy\right)\\ t_2 := \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot t\_0, ux, 1\right)}\\ \mathsf{fma}\left(\sin t\_1 \cdot t\_2, yi, \mathsf{fma}\left(xi \cdot \cos t\_1, t\_2, zi \cdot \left(t\_0 \cdot ux\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* maxCos (- 1.0 ux)))
       (t_1 (* PI (+ uy uy)))
       (t_2 (sqrt (fma (* (* (- ux 1.0) (* maxCos ux)) t_0) ux 1.0))))
  (fma
   (* (sin t_1) t_2)
   yi
   (fma (* xi (cos t_1)) t_2 (* zi (* t_0 ux))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = maxCos * (1.0f - ux);
	float t_1 = ((float) M_PI) * (uy + uy);
	float t_2 = sqrtf(fmaf((((ux - 1.0f) * (maxCos * ux)) * t_0), ux, 1.0f));
	return fmaf((sinf(t_1) * t_2), yi, fmaf((xi * cosf(t_1)), t_2, (zi * (t_0 * ux))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(maxCos * Float32(Float32(1.0) - ux))
	t_1 = Float32(Float32(pi) * Float32(uy + uy))
	t_2 = sqrt(fma(Float32(Float32(Float32(ux - Float32(1.0)) * Float32(maxCos * ux)) * t_0), ux, Float32(1.0)))
	return fma(Float32(sin(t_1) * t_2), yi, fma(Float32(xi * cos(t_1)), t_2, Float32(zi * Float32(t_0 * ux))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(maxCos \cdot \left(1 - ux\right)\right), ux, 1\right)}, yi, \mathsf{fma}\left(xi \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(maxCos \cdot \left(1 - ux\right)\right), ux, 1\right)}, zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right)} \]
4. Add Preprocessing

Alternative 3: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0}\right) \cdot xi + \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux)))
  (+
   (+
    (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0)))) xi)
    (* (sin (* 2.0 (* uy PI))) yi))
   (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	return (((cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)))) * xi) + (sinf((2.0f * (uy * ((float) M_PI)))) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	return Float32(Float32(Float32(Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))) * xi) + Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	tmp = (((cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)))) * xi) + (sin((single(2.0) * (uy * single(pi)))) * yi)) + (t_0 * zi);
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in ux around 0

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \color{blue}{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
3. Step-by-step derivation

   1. lower-sin.f32 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. lower-*.f32 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. lower-*.f32 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. lower-PI.f32 98.8%

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

4. Applied rewrites 98.8%

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \color{blue}{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* 2.0 (* uy PI))))
  (fma
   maxCos
   (* ux (* zi (- 1.0 ux)))
   (fma xi (cos t_0) (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return fmaf(maxCos, (ux * (zi * (1.0f - ux))), fmaf(xi, cosf(t_0), (yi * sinf(t_0))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), fma(xi, cos(t_0), Float32(yi * sin(t_0))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.7%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.7%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

8. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
9. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

10. Applied rewrites 98.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]
11. Add Preprocessing

Alternative 5: 95.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* 2.0 (* uy PI))))
  (fma maxCos (* ux zi) (fma xi (cos t_0) (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return fmaf(maxCos, (ux * zi), fmaf(xi, cosf(t_0), (yi * sinf(t_0))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return fma(maxCos, Float32(ux * zi), fma(xi, cos(t_0), Float32(yi * sin(t_0))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. lower-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

4. Applied rewrites 95.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]
5. Add Preprocessing

Alternative 6: 93.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ t_1 := \cos t\_0\\ \mathbf{if}\;uy \leq 0.00039999998989515007:\\ \;\;\;\;zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot t\_1}{zi} + \frac{2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)}{zi}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, t\_1, yi \cdot \sin t\_0\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* 2.0 (* uy PI))) (t_1 (cos t_0)))
  (if (<= uy 0.00039999998989515007)
    (*
     zi
     (fma
      maxCos
      ux
      (+ (/ (* xi t_1) zi) (/ (* 2.0 (* uy (* yi PI))) zi))))
    (fma xi t_1 (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float t_1 = cosf(t_0);
	float tmp;
	if (uy <= 0.00039999998989515007f) {
		tmp = zi * fmaf(maxCos, ux, (((xi * t_1) / zi) + ((2.0f * (uy * (yi * ((float) M_PI)))) / zi)));
	} else {
		tmp = fmaf(xi, t_1, (yi * sinf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	t_1 = cos(t_0)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00039999998989515007))
		tmp = Float32(zi * fma(maxCos, ux, Float32(Float32(Float32(xi * t_1) / zi) + Float32(Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi)))) / zi))));
	else
		tmp = fma(xi, t_1, Float32(yi * sin(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.9999999e-4

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in zi around inf

      \[\leadsto \color{blue}{zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(\frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)\right)} \]

   4. Applied rewrites 98.1%

      \[\leadsto \color{blue}{zi \cdot \mathsf{fma}\left(maxCos, ux \cdot \left(1 - ux\right), \frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)} \]

   5. Taylor expanded in ux around 0

      \[\leadsto zi \cdot \left(maxCos \cdot ux + \color{blue}{\left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

      2. lower-+.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

   7. Applied rewrites 95.1%

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, \color{blue}{ux}, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]

   8. Taylor expanded in uy around 0

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)}{zi}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

      4. lower-PI.f32 86.4%

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)}{zi}\right) \]

   10. Applied rewrites 86.4%

       \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)}{zi}\right) \]

   if 3.9999999e-4 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(maxCos \cdot \left(1 - ux\right)\right), ux, 1\right)} \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right), xi, \mathsf{fma}\left(zi \cdot \left(maxCos \cdot \left(1 - ux\right)\right), ux, yi \cdot \left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(ux - 1\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \left(maxCos \cdot \left(1 - ux\right)\right), ux, 1\right)}\right)\right)\right)} \]

   4. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
   5. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lower-PI.f32 90.4%

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   6. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 85.0% accurate, 2.6× speedup

Math

\[zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (*
 zi
 (fma maxCos ux (+ (/ xi zi) (/ (* yi (sin (* 2.0 (* uy PI)))) zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return zi * fmaf(maxCos, ux, ((xi / zi) + ((yi * sinf((2.0f * (uy * ((float) M_PI))))) / zi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(zi * fma(maxCos, ux, Float32(Float32(xi / zi) + Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) / zi))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(\frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)\right)} \]

4. Applied rewrites 98.1%

   \[\leadsto \color{blue}{zi \cdot \mathsf{fma}\left(maxCos, ux \cdot \left(1 - ux\right), \frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto zi \cdot \left(maxCos \cdot ux + \color{blue}{\left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)}\right) \]
6. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

   2. lower-+.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

7. Applied rewrites 95.1%

   \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, \color{blue}{ux}, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]

8. Taylor expanded in uy around 0

   \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]
9. Step-by-step derivation

   1. lower-/.f32 85.0%

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]

10. Applied rewrites 85.0%

    \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]
11. Add Preprocessing

Alternative 8: 81.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.009999999776482582:\\ \;\;\;\;zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \pi\right)}{zi}, \frac{xi}{zi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (if (<= uy 0.009999999776482582)
  (* zi (fma maxCos ux (fma 2.0 (/ (* uy (* yi PI)) zi) (/ xi zi))))
  (* zi (fma maxCos ux (/ (* yi (sin (* 2.0 (* uy PI)))) zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.009999999776482582f) {
		tmp = zi * fmaf(maxCos, ux, fmaf(2.0f, ((uy * (yi * ((float) M_PI))) / zi), (xi / zi)));
	} else {
		tmp = zi * fmaf(maxCos, ux, ((yi * sinf((2.0f * (uy * ((float) M_PI))))) / zi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.009999999776482582))
		tmp = Float32(zi * fma(maxCos, ux, fma(Float32(2.0), Float32(Float32(uy * Float32(yi * Float32(pi))) / zi), Float32(xi / zi))));
	else
		tmp = Float32(zi * fma(maxCos, ux, Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) / zi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00999999978

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in zi around inf

      \[\leadsto \color{blue}{zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(\frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)\right)} \]

   4. Applied rewrites 98.1%

      \[\leadsto \color{blue}{zi \cdot \mathsf{fma}\left(maxCos, ux \cdot \left(1 - ux\right), \frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)} \]

   5. Taylor expanded in ux around 0

      \[\leadsto zi \cdot \left(maxCos \cdot ux + \color{blue}{\left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

      2. lower-+.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

   7. Applied rewrites 95.1%

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, \color{blue}{ux}, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]

   8. Taylor expanded in uy around 0

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, 2 \cdot \frac{uy \cdot \left(yi \cdot \pi\right)}{zi} + \frac{xi}{zi}\right) \]
   9. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{zi}, \frac{xi}{zi}\right)\right) \]

      2. lower-/.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{zi}, \frac{xi}{zi}\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{zi}, \frac{xi}{zi}\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{zi}, \frac{xi}{zi}\right)\right) \]

      5. lower-PI.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \pi\right)}{zi}, \frac{xi}{zi}\right)\right) \]

      6. lower-/.f32 78.2%

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \pi\right)}{zi}, \frac{xi}{zi}\right)\right) \]

   10. Applied rewrites 78.2%

       \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \pi\right)}{zi}, \frac{xi}{zi}\right)\right) \]

   if 0.00999999978 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in zi around inf

      \[\leadsto \color{blue}{zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(\frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)\right)} \]

   4. Applied rewrites 98.1%

      \[\leadsto \color{blue}{zi \cdot \mathsf{fma}\left(maxCos, ux \cdot \left(1 - ux\right), \frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)} \]

   5. Taylor expanded in ux around 0

      \[\leadsto zi \cdot \left(maxCos \cdot ux + \color{blue}{\left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

      2. lower-+.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

   7. Applied rewrites 95.1%

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, \color{blue}{ux}, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]

   8. Taylor expanded in xi around 0

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]
   9. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

      3. lower-sin.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

      6. lower-PI.f32 42.2%

         \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]

   10. Applied rewrites 42.2%

       \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 81.0% accurate, 2.7× speedup

Math

\[zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(uy, \mathsf{fma}\left(-2, \frac{uy \cdot \left(xi \cdot {\pi}^{2}\right)}{zi}, 2 \cdot \frac{yi \cdot \pi}{zi}\right), \frac{xi}{zi}\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (*
 zi
 (fma
  maxCos
  ux
  (fma
   uy
   (fma
    -2.0
    (/ (* uy (* xi (pow PI 2.0))) zi)
    (* 2.0 (/ (* yi PI) zi)))
   (/ xi zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return zi * fmaf(maxCos, ux, fmaf(uy, fmaf(-2.0f, ((uy * (xi * powf(((float) M_PI), 2.0f))) / zi), (2.0f * ((yi * ((float) M_PI)) / zi))), (xi / zi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(zi * fma(maxCos, ux, fma(uy, fma(Float32(-2.0), Float32(Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0)))) / zi), Float32(Float32(2.0) * Float32(Float32(yi * Float32(pi)) / zi))), Float32(xi / zi))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(\frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)\right)} \]

4. Applied rewrites 98.1%

   \[\leadsto \color{blue}{zi \cdot \mathsf{fma}\left(maxCos, ux \cdot \left(1 - ux\right), \frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto zi \cdot \left(maxCos \cdot ux + \color{blue}{\left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)}\right) \]
6. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

   2. lower-+.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

7. Applied rewrites 95.1%

   \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, \color{blue}{ux}, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]

8. Taylor expanded in uy around 0

   \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, uy \cdot \left(-2 \cdot \frac{uy \cdot \left(xi \cdot {\pi}^{2}\right)}{zi} + 2 \cdot \frac{yi \cdot \pi}{zi}\right) + \frac{xi}{zi}\right) \]
9. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(uy, -2 \cdot \frac{uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{zi} + 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{zi}, \frac{xi}{zi}\right)\right) \]

10. Applied rewrites 81.8%

    \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(uy, \mathsf{fma}\left(-2, \frac{uy \cdot \left(xi \cdot {\pi}^{2}\right)}{zi}, 2 \cdot \frac{yi \cdot \pi}{zi}\right), \frac{xi}{zi}\right)\right) \]
11. Add Preprocessing

Alternative 10: 78.2% accurate, 5.8× speedup

Math

\[zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \pi\right)}{zi}, \frac{xi}{zi}\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (* zi (fma maxCos ux (fma 2.0 (/ (* uy (* yi PI)) zi) (/ xi zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return zi * fmaf(maxCos, ux, fmaf(2.0f, ((uy * (yi * ((float) M_PI))) / zi), (xi / zi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(zi * fma(maxCos, ux, fma(Float32(2.0), Float32(Float32(uy * Float32(yi * Float32(pi))) / zi), Float32(xi / zi))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(\frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)\right)} \]

4. Applied rewrites 98.1%

   \[\leadsto \color{blue}{zi \cdot \mathsf{fma}\left(maxCos, ux \cdot \left(1 - ux\right), \frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto zi \cdot \left(maxCos \cdot ux + \color{blue}{\left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)}\right) \]
6. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

   2. lower-+.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

7. Applied rewrites 95.1%

   \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, \color{blue}{ux}, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]

8. Taylor expanded in uy around 0

   \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, 2 \cdot \frac{uy \cdot \left(yi \cdot \pi\right)}{zi} + \frac{xi}{zi}\right) \]
9. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{zi}, \frac{xi}{zi}\right)\right) \]

   2. lower-/.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{zi}, \frac{xi}{zi}\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{zi}, \frac{xi}{zi}\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{zi}, \frac{xi}{zi}\right)\right) \]

   5. lower-PI.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \pi\right)}{zi}, \frac{xi}{zi}\right)\right) \]

   6. lower-/.f32 78.2%

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \pi\right)}{zi}, \frac{xi}{zi}\right)\right) \]

10. Applied rewrites 78.2%

    \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \mathsf{fma}\left(2, \frac{uy \cdot \left(yi \cdot \pi\right)}{zi}, \frac{xi}{zi}\right)\right) \]
11. Add Preprocessing

Alternative 11: 51.8% accurate, 10.5× speedup

Math

\[xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (+ xi (* maxCos (* ux (* zi (- 1.0 ux))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (maxCos * (ux * (zi * (1.0f - ux))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = xi + (maxcos * (ux * (zi * (1.0e0 - ux))))
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (maxCos * (ux * (zi * (single(1.0) - ux))));
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.7%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.7%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

8. Taylor expanded in xi around 0

   \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

   2. lower-*.f32 12.0%

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

10. Applied rewrites 12.0%

    \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

11. Taylor expanded in maxCos around 0

    \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
12. Step-by-step derivation

    1. lower-+.f32 N/A

       \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

    2. lower-*.f32 N/A

       \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}\right) \]

    3. lower-*.f32 N/A

       \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right)\right) \]

    4. lower-*.f32 N/A

       \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right)\right) \]

    5. lower--.f32 51.8%

       \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]

13. Applied rewrites 51.8%

    \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
14. Add Preprocessing

Alternative 12: 49.8% accurate, 16.6× speedup

Math

\[xi + maxCos \cdot \left(ux \cdot zi\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (+ xi (* maxCos (* ux zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (maxCos * (ux * zi));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = xi + (maxcos * (ux * zi))
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + Float32(maxCos * Float32(ux * zi)))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (maxCos * (ux * zi));
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.7%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.7%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
8. Add Preprocessing

Alternative 13: 49.7% accurate, 17.7× speedup

Math

\[\mathsf{fma}\left(zi \cdot maxCos, ux, xi\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (fma (* zi maxCos) ux xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((zi * maxCos), ux, xi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(zi * maxCos), ux, xi)
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.7%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.7%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. +-commutative N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   3. lift-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   4. lift-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   5. *-commutative N/A

      \[\leadsto maxCos \cdot \left(zi \cdot ux\right) + xi \]

   6. associate-*r* N/A

      \[\leadsto \left(maxCos \cdot zi\right) \cdot ux + xi \]

   7. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot zi, ux, xi\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(zi \cdot maxCos, ux, xi\right) \]

   9. lower-*.f32 49.8%

      \[\leadsto \mathsf{fma}\left(zi \cdot maxCos, ux, xi\right) \]

9. Applied rewrites 49.8%

   \[\leadsto \mathsf{fma}\left(zi \cdot maxCos, ux, xi\right) \]
10. Add Preprocessing

Alternative 14: 49.7% accurate, 17.7× speedup

Math

\[\mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (fma (* maxCos ux) zi xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((maxCos * ux), zi, xi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos * ux), zi, xi)
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.7%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.7%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. +-commutative N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   3. lift-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   4. lift-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + xi \]

   5. associate-*r* N/A

      \[\leadsto \left(maxCos \cdot ux\right) \cdot zi + xi \]

   6. lift-*.f32 N/A

      \[\leadsto \left(maxCos \cdot ux\right) \cdot zi + xi \]

   7. lower-fma.f32 49.7%

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \]

9. Applied rewrites 49.7%

   \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \]
10. Add Preprocessing

Alternative 15: 12.0% accurate, 22.8× speedup

Math

\[zi \cdot \left(maxCos \cdot ux\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (* zi (* maxCos ux)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return zi * (maxCos * ux);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = zi * (maxcos * ux)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(zi * Float32(maxCos * ux))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = zi * (maxCos * ux);
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(\frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)\right)} \]

4. Applied rewrites 98.1%

   \[\leadsto \color{blue}{zi \cdot \mathsf{fma}\left(maxCos, ux \cdot \left(1 - ux\right), \frac{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{zi}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto zi \cdot \left(maxCos \cdot ux + \color{blue}{\left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)}\right) \]
6. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

   2. lower-+.f32 N/A

      \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{zi}\right) \]

7. Applied rewrites 95.1%

   \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, \color{blue}{ux}, \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right) \]

8. Taylor expanded in zi around inf

   \[\leadsto zi \cdot \left(maxCos \cdot ux\right) \]
9. Step-by-step derivation

   1. lower-*.f32 12.0%

      \[\leadsto zi \cdot \left(maxCos \cdot ux\right) \]

10. Applied rewrites 12.0%

    \[\leadsto zi \cdot \left(maxCos \cdot ux\right) \]
11. Add Preprocessing

Alternative 16: 12.0% accurate, 22.8× speedup

Math

\[maxCos \cdot \left(ux \cdot zi\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (* maxCos (* ux zi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (ux * zi);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (ux * zi)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(ux * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * (ux * zi);
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.7%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.7%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

8. Taylor expanded in xi around 0

   \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

   2. lower-*.f32 12.0%

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

10. Applied rewrites 12.0%

    \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (and (<= 2.328306437e-10 ux) (<= ux 1.0))) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (+ (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))