UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 8.5s

Alternatives: 19

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, 1.2× 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

\[\mathsf{fma}\left(ux \cdot \left(1 - ux\right), maxCos \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \sin \left(\mathsf{fma}\left(-2 \cdot uy, \pi, \pi \cdot 0.5\right)\right) \cdot xi\right)\right) \]

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
 (* ux (- 1.0 ux))
 (* maxCos zi)
 (*
  (sqrt
   (fma
    (* (- ux 1.0) maxCos)
    (* (* (* maxCos (- 1.0 ux)) ux) ux)
    1.0))
  (fma
   (sin (* PI (+ uy uy)))
   yi
   (* (sin (fma (* -2.0 uy) PI (* PI 0.5))) xi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((ux * (1.0f - ux)), (maxCos * zi), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * fmaf(sinf((((float) M_PI) * (uy + uy))), yi, (sinf(fmaf((-2.0f * uy), ((float) M_PI), (((float) M_PI) * 0.5f))) * xi))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(ux * Float32(Float32(1.0) - ux)), Float32(maxCos * zi), 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(Float32(Float32(pi) * Float32(uy + uy))), yi, Float32(sin(fma(Float32(Float32(-2.0) * uy), Float32(pi), Float32(Float32(pi) * Float32(0.5)))) * 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%

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

5. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(ux \cdot \left(1 - ux\right), maxCos \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \sin \left(\mathsf{fma}\left(-2 \cdot uy, \pi, \pi \cdot 0.5\right)\right) \cdot xi\right)\right) \]
6. 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

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

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
   (* ux (- 1.0 ux))
   (* maxCos zi)
   (*
    (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((ux * (1.0f - ux)), (maxCos * zi), (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(ux * Float32(Float32(1.0) - ux)), Float32(maxCos * zi), 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%

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

Alternative 3: 98.8% accurate, 1.5× 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

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

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

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(xi, cosf(t_0), (yi * sinf(t_0))) + ((((1.0f - ux) * maxCos) * ux) * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return Float32(fma(xi, cos(t_0), Float32(yi * sin(t_0))) + Float32(Float32(Float32(Float32(Float32(1.0) - ux) * 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 ux around 0

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

4. Applied rewrites 98.8%

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

Alternative 4: 98.8% accurate, 1.6× speedup

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

Math

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

FPCore

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

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

4. Applied rewrites 52.2%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 50.1%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 98.8%

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

Alternative 5: 97.4% 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

\[\begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;uy \leq 0.018200000748038292:\\ \;\;\;\;\mathsf{fma}\left(ux \cdot \left(1 - ux\right), maxCos \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\left(yi + yi\right) \cdot uy, \pi, \mathsf{fma}\left(uy \cdot uy, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot xi, -2, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy\right) \cdot -1.3333333333333333\right), xi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\\ \end{array} \]

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 0.018200000748038292)
    (fma
     (* ux (- 1.0 ux))
     (* maxCos zi)
     (*
      (sqrt
       (fma
        (* (- ux 1.0) maxCos)
        (* (* (* maxCos (- 1.0 ux)) ux) ux)
        1.0))
      (fma
       (* (+ yi yi) uy)
       PI
       (fma
        (* uy uy)
        (fma
         (* (* PI PI) xi)
         -2.0
         (* (* (* (* (* PI PI) PI) yi) uy) -1.3333333333333333))
        xi))))
    (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 <= 0.018200000748038292f) {
		tmp = fmaf((ux * (1.0f - ux)), (maxCos * zi), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * fmaf(((yi + yi) * uy), ((float) M_PI), fmaf((uy * uy), fmaf(((((float) M_PI) * ((float) M_PI)) * xi), -2.0f, (((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * yi) * uy) * -1.3333333333333333f)), xi))));
	} 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(0.018200000748038292))
		tmp = fma(Float32(ux * Float32(Float32(1.0) - ux)), Float32(maxCos * zi), 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(Float32(Float32(yi + yi) * uy), Float32(pi), fma(Float32(uy * uy), fma(Float32(Float32(Float32(pi) * Float32(pi)) * xi), Float32(-2.0), Float32(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * yi) * uy) * Float32(-1.3333333333333333))), xi))));
	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 < 0.0182000007

   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%

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

   4. Taylor expanded in uy around 0

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

   6. Applied rewrites 89.4%

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

   8. Applied rewrites 89.4%

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

   if 0.0182000007 < 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 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 52.2%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 50.1%

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

   8. Taylor expanded in ux around 0

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

   10. Applied rewrites 90.5%

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

Alternative 6: 95.7% 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

\[\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(-2, uy \cdot \pi, 0.5 \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]

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)
 (fma
  xi
  (sin (fma -2.0 (* uy PI) (* 0.5 PI)))
  (* yi (sin (* 2.0 (* uy PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(maxCos, (ux * zi), fmaf(xi, sinf(fmaf(-2.0f, (uy * ((float) M_PI)), (0.5f * ((float) M_PI)))), (yi * sinf((2.0f * (uy * ((float) M_PI)))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * zi), fma(xi, sin(fma(Float32(-2.0), Float32(uy * Float32(pi)), Float32(Float32(0.5) * Float32(pi)))), Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))))
end

Derivation

1. Initial program 98.9%

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

3. Applied rewrites 99.0%

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

5. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(ux \cdot \left(1 - ux\right), maxCos \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \sin \left(\mathsf{fma}\left(-2 \cdot uy, \pi, \pi \cdot 0.5\right)\right) \cdot xi\right)\right) \]

6. Taylor expanded in ux around 0

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

8. Applied rewrites 95.7%

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(-2, uy \cdot \pi, 0.5 \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 7: 95.7% 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

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

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 (* 2.0 (* uy PI))))
  (+ (+ (* (cos t_0) xi) (* (sin t_0) yi)) (* (* maxCos ux) zi))))

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 ((cosf(t_0) * xi) + (sinf(t_0) * yi)) + ((maxCos * ux) * zi);
}

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (uy * single(pi));
	tmp = ((cos(t_0) * xi) + (sin(t_0) * yi)) + ((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 ux around 0

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 95.7%

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

8. Taylor expanded in ux around 0

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

10. Applied rewrites 95.7%

    \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi + \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) + \left(maxCos \cdot ux\right) \cdot zi \]
11. Add Preprocessing

Alternative 8: 95.7% 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 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 52.2%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 95.7%

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

Alternative 9: 89.5% accurate, 1.8× 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

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

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
 (* ux (- 1.0 ux))
 (* maxCos zi)
 (*
  (sqrt
   (fma
    (* (- ux 1.0) maxCos)
    (* (* (* maxCos (- 1.0 ux)) ux) ux)
    1.0))
  (fma
   (* (+ yi yi) uy)
   PI
   (fma
    (* uy uy)
    (fma
     (* (* PI PI) xi)
     -2.0
     (* (* (* (* (* PI PI) PI) yi) uy) -1.3333333333333333))
    xi)))))

C

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

Julia

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

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

4. Taylor expanded in uy around 0

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

6. Applied rewrites 89.4%

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

8. Applied rewrites 89.4%

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

Alternative 10: 89.4% accurate, 1.8× 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

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

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
 (* ux (- 1.0 ux))
 (* maxCos zi)
 (*
  (sqrt
   (fma
    (* (- ux 1.0) maxCos)
    (* (* (* maxCos (- 1.0 ux)) ux) ux)
    1.0))
  (+
   xi
   (*
    uy
    (fma
     PI
     yi
     (fma
      yi
      PI
      (*
       (fma
        (* (* PI PI) xi)
        -2.0
        (* (* (* (* (* PI PI) PI) yi) uy) -1.3333333333333333))
       uy))))))))

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(ux * Float32(Float32(1.0) - ux)), Float32(maxCos * zi), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * Float32(xi + Float32(uy * fma(Float32(pi), yi, fma(yi, Float32(pi), Float32(fma(Float32(Float32(Float32(pi) * Float32(pi)) * xi), Float32(-2.0), Float32(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * yi) * uy) * Float32(-1.3333333333333333))) * uy)))))))
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%

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

4. Taylor expanded in uy around 0

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

6. Applied rewrites 89.4%

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

8. Applied rewrites 89.4%

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

Alternative 11: 89.4% 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

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

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
 (* ux (- 1.0 ux))
 (* maxCos zi)
 (*
  (sqrt
   (fma
    (* (- ux 1.0) maxCos)
    (* (* (* maxCos (- 1.0 ux)) ux) ux)
    1.0))
  (+
   xi
   (*
    uy
    (fma
     (fma
      (* (* PI PI) xi)
      -2.0
      (* (* (* (* (* PI PI) PI) yi) uy) -1.3333333333333333))
     uy
     (* (+ PI PI) yi)))))))

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(ux * Float32(Float32(1.0) - ux)), Float32(maxCos * zi), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * Float32(xi + Float32(uy * fma(fma(Float32(Float32(Float32(pi) * Float32(pi)) * xi), Float32(-2.0), Float32(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * yi) * uy) * Float32(-1.3333333333333333))), uy, Float32(Float32(Float32(pi) + Float32(pi)) * yi))))))
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%

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

4. Taylor expanded in uy around 0

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

6. Applied rewrites 89.4%

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

8. Applied rewrites 89.4%

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

Alternative 12: 89.4% 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

\[\begin{array}{l} t_0 := \left(1 - ux\right) \cdot ux\\ \mathsf{fma}\left(t\_0, zi \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, t\_0, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\pi + \pi, yi, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot xi, -2, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy\right) \cdot -1.3333333333333333\right) \cdot uy\right), uy, xi\right)\right) \end{array} \]

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 (* (- 1.0 ux) ux)))
  (fma
   t_0
   (* zi maxCos)
   (*
    (sqrt (fma (* (* (* (- ux 1.0) maxCos) ux) maxCos) t_0 1.0))
    (fma
     (fma
      (+ PI PI)
      yi
      (*
       (fma
        (* (* PI PI) xi)
        -2.0
        (* (* (* (* (* PI PI) PI) yi) uy) -1.3333333333333333))
       uy))
     uy
     xi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) * ux;
	return fmaf(t_0, (zi * maxCos), (sqrtf(fmaf(((((ux - 1.0f) * maxCos) * ux) * maxCos), t_0, 1.0f)) * fmaf(fmaf((((float) M_PI) + ((float) M_PI)), yi, (fmaf(((((float) M_PI) * ((float) M_PI)) * xi), -2.0f, (((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * yi) * uy) * -1.3333333333333333f)) * uy)), uy, xi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) * ux)
	return fma(t_0, Float32(zi * maxCos), Float32(sqrt(fma(Float32(Float32(Float32(Float32(ux - Float32(1.0)) * maxCos) * ux) * maxCos), t_0, Float32(1.0))) * fma(fma(Float32(Float32(pi) + Float32(pi)), yi, Float32(fma(Float32(Float32(Float32(pi) * Float32(pi)) * xi), Float32(-2.0), Float32(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * yi) * uy) * Float32(-1.3333333333333333))) * uy)), uy, 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%

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

4. Taylor expanded in uy around 0

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

6. Applied rewrites 89.4%

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

8. Applied rewrites 89.5%

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

Alternative 13: 83.0% accurate, 5.0× 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

\[xi + \mathsf{fma}\left(maxCos, ux \cdot zi, uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot 9.869604110717773\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]

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)
  (*
   uy
   (fma -2.0 (* uy (* xi 9.869604110717773)) (* 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), (uy * fmaf(-2.0f, (uy * (xi * 9.869604110717773f)), (2.0f * (yi * ((float) M_PI))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + fma(maxCos, Float32(ux * zi), Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(9.869604110717773))), 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 uy around 0

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

4. Applied rewrites 85.9%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 83.0%

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

8. Evaluated real constant 83.0%

   \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot zi, uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot 9.869604110717773\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]
9. Add Preprocessing

Alternative 14: 78.8% accurate, 7.7× 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

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

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)))))

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)));
}

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 * 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

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

4. Applied rewrites 85.9%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 83.0%

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

8. Taylor expanded in xi around 0

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

10. Applied rewrites 78.8%

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

Alternative 15: 61.0% accurate, 6.2× 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

\[\begin{array}{l} t_0 := \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), maxCos \cdot \left(ux \cdot zi\right)\right)\\ \mathbf{if}\;yi \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

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 (fma 2.0 (* uy (* yi PI)) (* maxCos (* ux zi)))))
  (if (<= yi -5.0000000843119176e-17)
    t_0
    (if (<= yi 4.999999980020986e-13)
      (+ xi (* maxCos (* ux (* zi (- 1.0 ux)))))
      t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = fmaf(2.0f, (uy * (yi * ((float) M_PI))), (maxCos * (ux * zi)));
	float tmp;
	if (yi <= -5.0000000843119176e-17f) {
		tmp = t_0;
	} else if (yi <= 4.999999980020986e-13f) {
		tmp = xi + (maxCos * (ux * (zi * (1.0f - ux))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = fma(Float32(2.0), Float32(uy * Float32(yi * Float32(pi))), Float32(maxCos * Float32(ux * zi)))
	tmp = Float32(0.0)
	if (yi <= Float32(-5.0000000843119176e-17))
		tmp = t_0;
	elseif (yi <= Float32(4.999999980020986e-13))
		tmp = Float32(xi + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if yi < -5.00000008e-17 or 4.99999998e-13 < yi

   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

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

   4. Applied rewrites 85.9%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 83.0%

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

   8. Taylor expanded in zi around inf

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

   10. Applied rewrites 82.4%

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

   11. Taylor expanded in xi around 0

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

   13. Applied rewrites 35.0%

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

   if -5.00000008e-17 < yi < 4.99999998e-13

   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 52.2%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 50.1%

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

   8. Taylor expanded in maxCos around 0

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

   10. Applied rewrites 52.2%

       \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 52.2% 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 52.2%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 50.1%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 52.2%

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

Alternative 17: 50.1% 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 52.2%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 50.1%

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

9. Applied rewrites 50.1%

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

Alternative 18: 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

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

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)))
  (* zi (* maxCos ux)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return zi * (maxCos * 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 = 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 uy around 0

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

4. Applied rewrites 85.9%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 83.0%

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

8. Taylor expanded in zi around inf

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

10. Applied rewrites 82.4%

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

11. Taylor expanded in zi around inf

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

13. Applied rewrites 11.8%

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

Alternative 19: 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 52.2%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 50.1%

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

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 2026035 +o sampling:rival3
(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)))