UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 8.4s

Alternatives: 15

Speedup: 1.2×

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
  :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)))
  (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

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

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
  :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)))
  (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

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 1.2× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 99.0%

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

Alternative 3: 98.9% accurate, 1.3× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 99.0%

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

4. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(zi \cdot \left(1 - ux\right), maxCos \cdot ux, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(maxCos \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 98.9%

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

Alternative 4: 98.8% accurate, 1.6× speedup

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

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(pi) + Float32(pi)) * uy)
	return fma(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux), zi, fma(sin(t_0), yi, Float32(cos(t_0) * 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 xi around 0

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

4. Applied rewrites 44.0%

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

5. Taylor expanded in maxCos around 0

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

7. Applied rewrites 98.8%

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

9. Applied rewrites 98.8%

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

Alternative 5: 97.3% accurate, 1.7× speedup

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

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;uy \leq 0.007000000216066837:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :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)))
  (let* ((t_0 (* 2.0 (* uy PI))))
  (if (<= uy 0.007000000216066837)
    (fma
     maxCos
     (* ux (* zi (- 1.0 ux)))
     (+
      xi
      (*
       uy
       (fma
        2.0
        (* yi PI)
        (*
         uy
         (fma
          -2.0
          (* xi (pow PI 2.0))
          (* -1.3333333333333333 (* uy (* yi (pow PI 3.0))))))))))
    (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));
	float tmp;
	if (uy <= 0.007000000216066837f) {
		tmp = fmaf(maxCos, (ux * (zi * (1.0f - ux))), (xi + (uy * fmaf(2.0f, (yi * ((float) M_PI)), (uy * fmaf(-2.0f, (xi * powf(((float) M_PI), 2.0f)), (-1.3333333333333333f * (uy * (yi * powf(((float) M_PI), 3.0f))))))))));
	} else {
		tmp = fmaf(xi, cosf(t_0), (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)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.007000000216066837))
		tmp = fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(xi + Float32(uy * fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * fma(Float32(-2.0), Float32(xi * (Float32(pi) ^ Float32(2.0))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * (Float32(pi) ^ Float32(3.0)))))))))));
	else
		tmp = fma(xi, cos(t_0), Float32(yi * sin(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00700000022

   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 xi around 0

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

   4. Applied rewrites 44.0%

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

   5. Taylor expanded in maxCos around 0

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(2 \cdot \left(yi \cdot \pi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\pi}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 89.6%

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right) \]

   if 0.00700000022 < 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 xi around 0

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

   4. Applied rewrites 44.0%

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

   5. Taylor expanded in ux around 0

      \[\leadsto xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 90.8%

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

Alternative 6: 95.8% accurate, 1.6× speedup

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

Math

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

FPCore

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

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

4. Applied rewrites 95.8%

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

Alternative 7: 90.2% accurate, 1.9× speedup

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

Math

\[\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :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)))
  (fma
 maxCos
 (* ux (* zi (- 1.0 ux)))
 (+
  xi
  (*
   uy
   (fma
    2.0
    (* yi PI)
    (*
     uy
     (fma
      -2.0
      (* xi (pow PI 2.0))
      (* -1.3333333333333333 (* uy (* yi (pow PI 3.0)))))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(maxCos, (ux * (zi * (1.0f - ux))), (xi + (uy * fmaf(2.0f, (yi * ((float) M_PI)), (uy * fmaf(-2.0f, (xi * powf(((float) M_PI), 2.0f)), (-1.3333333333333333f * (uy * (yi * powf(((float) M_PI), 3.0f))))))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(xi + Float32(uy * fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * fma(Float32(-2.0), Float32(xi * (Float32(pi) ^ Float32(2.0))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * (Float32(pi) ^ Float32(3.0)))))))))))
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 xi around 0

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

4. Applied rewrites 44.0%

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

5. Taylor expanded in maxCos around 0

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(2 \cdot \left(yi \cdot \pi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\pi}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 89.6%

    \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 8: 89.6% accurate, 2.3× speedup

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

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :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)))
  (fma
 maxCos
 (* ux (* zi (- 1.0 ux)))
 (fma xi (cos (* 2.0 (* uy PI))) (* 2.0 (* uy (* yi PI))))))

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), fma(xi, cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))), Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi))))))
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 xi around 0

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

4. Applied rewrites 44.0%

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

5. Taylor expanded in maxCos around 0

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in uy around 0

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

10. Applied rewrites 90.2%

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

Alternative 9: 86.1% accurate, 2.9× speedup

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

Math

\[xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :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)))
  (+
 xi
 (fma
  maxCos
  (* ux (* zi (- 1.0 ux)))
  (* uy (fma -2.0 (* uy (* xi (pow PI 2.0))) (* 2.0 (* yi PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + fmaf(maxCos, (ux * (zi * (1.0f - ux))), (uy * fmaf(-2.0f, (uy * (xi * powf(((float) M_PI), 2.0f))), (2.0f * (yi * ((float) M_PI))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))))))
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 xi around 0

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

4. Applied rewrites 44.0%

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

5. Taylor expanded in maxCos around 0

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in uy around 0

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

10. Applied rewrites 86.1%

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

Alternative 10: 81.8% accurate, 6.1× speedup

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

Math

\[xi + \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :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)))
  (+ xi (fma 2.0 (* uy (* yi PI)) (* maxCos (* ux (* zi (- 1.0 ux)))))))

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + fma(Float32(2.0), Float32(uy * Float32(yi * Float32(pi))), Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(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 xi around 0

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

4. Applied rewrites 44.0%

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

5. Taylor expanded in maxCos around 0

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in uy around 0

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

10. Applied rewrites 81.8%

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

Alternative 11: 81.8% accurate, 6.1× speedup

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

Math

\[\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :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)))
  (fma maxCos (* ux (* zi (- 1.0 ux))) (+ xi (* 2.0 (* uy (* yi PI))))))

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi))))))
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 xi around 0

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

4. Applied rewrites 44.0%

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

5. Taylor expanded in maxCos around 0

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in uy around 0

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

10. Applied rewrites 81.8%

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

Alternative 12: 51.9% accurate, 10.5× speedup

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

Math

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

FPCore

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

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

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

4. Applied rewrites 52.0%

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

5. Taylor expanded in maxCos around 0

   \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 51.9%

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

Alternative 13: 49.8% accurate, 17.7× speedup

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

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :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)))
  (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

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

4. Applied rewrites 52.0%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 49.8%

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

9. Applied rewrites 49.8%

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

Alternative 14: 11.8% accurate, 22.8× speedup

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

Math

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

FPCore

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

C

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

Fortran

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(Float32(maxCos * 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

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

4. Applied rewrites 52.0%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 49.8%

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

8. Taylor expanded in xi around 0

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

10. Applied rewrites 11.8%

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

12. Applied rewrites 11.8%

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

Alternative 15: 11.8% accurate, 22.8× speedup

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

Math

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

FPCore

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

C

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

Fortran

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

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

4. Applied rewrites 52.0%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 49.8%

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

8. Taylor expanded in xi around 0

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

10. Applied rewrites 11.8%

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

Reproduce

herbie shell --seed 2025359 
(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)))