UniformSampleCone 2

Percentage Accurate: 99.0% → 99.0%

Time: 10.8s

Alternatives: 24

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
  (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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (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 99.0%

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

3. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (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 (* yi (sin t_1)) t_2 (fma (* xi t_2) (cos 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((yi * sinf(t_1)), t_2, fmaf((xi * t_2), cosf(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(yi * sin(t_1)), t_2, fma(Float32(xi * t_2), cos(t_1), Float32(zi * t_0)))
end

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 99.0%

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

Alternative 3: 99.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* PI (+ uy uy))))
  (fma
   (- 1.0 ux)
   (* (* maxCos ux) 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((1.0f - ux), ((maxCos * ux) * 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(Float32(1.0) - ux), Float32(Float32(maxCos * ux) * 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 99.0%

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

3. Applied rewrites 99.0%

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

Alternative 4: 98.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

4. Applied rewrites 82.3%

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

6. Applied rewrites 82.4%

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

7. Taylor expanded in maxCos around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-fma.f32 N/A

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

9. Applied rewrites 98.7%

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

Alternative 5: 95.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\pi + \pi\right) \cdot uy\\ \mathbf{if}\;uy \leq 0.0001500000071246177:\\ \;\;\;\;\mathsf{fma}\left(\left(uy + uy\right) \cdot \pi, yi, \mathsf{fma}\left(maxCos \cdot \left(zi \cdot \left(1 - ux\right)\right), ux, \sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)} \cdot xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_0, xi, \sin t\_0 \cdot yi\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* (+ PI PI) uy)))
  (if (<= uy 0.0001500000071246177)
    (fma
     (* (+ uy uy) PI)
     yi
     (fma
      (* maxCos (* zi (- 1.0 ux)))
      ux
      (*
       (sqrt
        (fma
         (* (* (* (- ux 1.0) maxCos) ux) maxCos)
         (* (- 1.0 ux) ux)
         1.0))
       xi)))
    (fma (cos t_0) xi (* (sin t_0) yi)))))

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;
	float tmp;
	if (uy <= 0.0001500000071246177f) {
		tmp = fmaf(((uy + uy) * ((float) M_PI)), yi, fmaf((maxCos * (zi * (1.0f - ux))), ux, (sqrtf(fmaf(((((ux - 1.0f) * maxCos) * ux) * maxCos), ((1.0f - ux) * ux), 1.0f)) * xi)));
	} else {
		tmp = fmaf(cosf(t_0), xi, (sinf(t_0) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(pi) + Float32(pi)) * uy)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0001500000071246177))
		tmp = fma(Float32(Float32(uy + uy) * Float32(pi)), yi, fma(Float32(maxCos * Float32(zi * Float32(Float32(1.0) - ux))), ux, Float32(sqrt(fma(Float32(Float32(Float32(Float32(ux - Float32(1.0)) * maxCos) * ux) * maxCos), Float32(Float32(Float32(1.0) - ux) * ux), Float32(1.0))) * xi)));
	else
		tmp = fma(cos(t_0), xi, Float32(sin(t_0) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 1.50000007e-4

   1. Initial program 99.0%

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

   2. Taylor expanded in uy around 0

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

      1. lower-fma.f32 N/A

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

   4. Applied rewrites 82.3%

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

   6. Applied rewrites 82.4%

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

   7. Taylor expanded in ux around 0

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

      1. lower-PI.f32 82.3%

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

   9. Applied rewrites 82.3%

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

   if 1.50000007e-4 < uy

   1. Initial program 99.0%

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

   2. Taylor expanded in zi around inf

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sin.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 90.0%

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

   7. Applied rewrites 90.0%

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

      1. lift-fma.f32 N/A

         \[\leadsto xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi + \color{blue}{yi} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \cos \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot xi + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \cos \left(\left(2 \cdot \pi\right) \cdot uy\right) \cdot xi + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

      7. count-2 N/A

         \[\leadsto \cos \left(\left(\pi + \pi\right) \cdot uy\right) \cdot xi + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

      8. lift-+.f32 N/A

         \[\leadsto \cos \left(\left(\pi + \pi\right) \cdot uy\right) \cdot xi + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

      9. lift-*.f32 N/A

         \[\leadsto \cos \left(\left(\pi + \pi\right) \cdot uy\right) \cdot xi + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

      10. lower-fma.f32 90.0%

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

   9. Applied rewrites 90.0%

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

Alternative 6: 95.7% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (fma
 maxCos
 (* ux zi)
 (fma
  xi
  (sin (* PI (+ 0.5 (* -2.0 uy))))
  (* 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((((float) M_PI) * (0.5f + (-2.0f * uy)))), (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(Float32(Float32(pi) * Float32(Float32(0.5) + Float32(Float32(-2.0) * uy)))), Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))))
end

Derivation

1. Initial program 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. lift-PI.f32 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. distribute-lft-out N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. distribute-lft-neg-in N/A

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

   16. lower-fma.f32 N/A

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

   17. metadata-eval 99.0%

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

3. Applied rewrites 99.0%

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

4. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fma.f32 N/A

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

6. Applied rewrites 95.8%

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

Alternative 7: 95.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-cos.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 95.7%

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

Alternative 8: 87.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in zi around inf

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-+.f32 N/A

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

7. Applied rewrites 95.1%

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

8. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 87.2%

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

10. Applied rewrites 87.2%

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

Alternative 9: 85.1% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (*
 zi
 (fma
  (/ 1.0 zi)
  (* yi (sin (* (+ PI PI) uy)))
  (fma maxCos ux (/ xi zi)))))

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(zi * fma(Float32(Float32(1.0) / zi), Float32(yi * sin(Float32(Float32(Float32(pi) + Float32(pi)) * uy))), fma(maxCos, ux, Float32(xi / zi))))
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in zi around inf

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-+.f32 N/A

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

7. Applied rewrites 95.1%

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

   1. lift-fma.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. mult-flip N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

9. Applied rewrites 94.7%

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

10. Taylor expanded in uy around 0

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

    1. lower-fma.f32 N/A

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

    2. lower-/.f32 85.0%

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

12. Applied rewrites 85.0%

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

Alternative 10: 85.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in zi around inf

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-+.f32 N/A

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

7. Applied rewrites 95.1%

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

8. Taylor expanded in uy around 0

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

   1. lower-/.f32 85.1%

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

10. Applied rewrites 85.1%

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

Alternative 11: 82.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in zi around inf

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-+.f32 N/A

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

7. Applied rewrites 95.1%

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

8. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

10. Applied rewrites 82.5%

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

Alternative 12: 82.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

4. Applied rewrites 82.3%

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

6. Applied rewrites 82.4%

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

7. Taylor expanded in ux around 0

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

   1. lower-PI.f32 82.3%

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

9. Applied rewrites 82.3%

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

Alternative 13: 82.2% accurate, 6.1× speedup

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
  (+ 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 99.0%

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

2. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

4. Applied rewrites 82.3%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 82.2%

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

7. Applied rewrites 82.2%

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

Alternative 14: 79.5% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

4. Applied rewrites 82.3%

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

6. Applied rewrites 82.4%

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

7. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 79.5%

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

9. Applied rewrites 79.5%

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

    1. lift-fma.f32 N/A

       \[\leadsto xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)}\right) \]

    2. lift-*.f32 N/A

       \[\leadsto xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

    3. associate-*r* N/A

       \[\leadsto xi + \left(\left(2 \cdot uy\right) \cdot \left(yi \cdot \pi\right) + maxCos \cdot \left(\color{blue}{ux} \cdot zi\right)\right) \]

    4. count-2 N/A

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

    5. lift-+.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. lift-+.f32 N/A

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

    10. count-2 N/A

        \[\leadsto xi + \left(\left(\left(2 \cdot uy\right) \cdot \pi\right) \cdot yi + maxCos \cdot \left(ux \cdot zi\right)\right) \]

    11. associate-*r* N/A

        \[\leadsto xi + \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + maxCos \cdot \left(ux \cdot zi\right)\right) \]

    12. lift-*.f32 N/A

        \[\leadsto xi + \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + maxCos \cdot \left(ux \cdot zi\right)\right) \]

    13. lift-*.f32 N/A

        \[\leadsto xi + \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + maxCos \cdot \left(ux \cdot zi\right)\right) \]

    14. lower-fma.f32 79.5%

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

    15. lift-*.f32 N/A

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

    16. lift-*.f32 N/A

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

    17. *-commutative N/A

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

    18. associate-*l* N/A

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

    19. count-2 N/A

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

    20. lift-+.f32 N/A

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

    21. lift-*.f32 79.5%

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

    22. lift-*.f32 N/A

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

    23. *-commutative N/A

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

    24. lower-*.f32 79.5%

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

11. Applied rewrites 79.5%

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

Alternative 15: 79.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

4. Applied rewrites 82.3%

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

6. Applied rewrites 82.4%

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

7. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 79.5%

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

9. Applied rewrites 79.5%

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

    1. lift-+.f32 N/A

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

    2. +-commutative N/A

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

    3. lift-fma.f32 N/A

       \[\leadsto \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) + xi \]

    4. associate-+l+ N/A

       \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + \left(maxCos \cdot \left(ux \cdot zi\right) + \color{blue}{xi}\right) \]

    5. lift-*.f32 N/A

       \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + \left(maxCos \cdot \left(ux \cdot zi\right) + xi\right) \]

    6. associate-*r* N/A

       \[\leadsto \left(2 \cdot uy\right) \cdot \left(yi \cdot \pi\right) + \left(maxCos \cdot \left(ux \cdot zi\right) + xi\right) \]

    7. count-2 N/A

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

    8. lift-+.f32 N/A

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

    9. lift-*.f32 N/A

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

    10. *-commutative N/A

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

    11. associate-*r* N/A

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

    12. lift-+.f32 N/A

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

    13. count-2 N/A

        \[\leadsto \left(\left(2 \cdot uy\right) \cdot \pi\right) \cdot yi + \left(maxCos \cdot \left(ux \cdot zi\right) + xi\right) \]

    14. associate-*r* N/A

        \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + \left(maxCos \cdot \left(ux \cdot zi\right) + xi\right) \]

    15. lift-*.f32 N/A

        \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + \left(maxCos \cdot \left(ux \cdot zi\right) + xi\right) \]

    16. lift-*.f32 N/A

        \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + \left(maxCos \cdot \left(ux \cdot zi\right) + xi\right) \]

    17. +-commutative N/A

        \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + \left(xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)}\right) \]

    18. lift-*.f32 N/A

        \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + \left(xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right)\right) \]

    19. lift-*.f32 N/A

        \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + \left(xi + maxCos \cdot \left(ux \cdot zi\right)\right) \]

    20. lower-fma.f32 N/A

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

11. Applied rewrites 79.5%

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

Alternative 16: 79.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

4. Applied rewrites 82.3%

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

6. Applied rewrites 82.4%

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

7. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 79.5%

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

9. Applied rewrites 79.5%

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

    1. lift-+.f32 N/A

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

    2. +-commutative N/A

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

    3. lift-fma.f32 N/A

       \[\leadsto \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) + xi \]

    4. associate-+l+ N/A

       \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + \left(maxCos \cdot \left(ux \cdot zi\right) + \color{blue}{xi}\right) \]

    5. lift-*.f32 N/A

       \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + \left(maxCos \cdot \left(ux \cdot zi\right) + xi\right) \]

    6. associate-*r* N/A

       \[\leadsto \left(2 \cdot uy\right) \cdot \left(yi \cdot \pi\right) + \left(maxCos \cdot \left(ux \cdot zi\right) + xi\right) \]

    7. count-2 N/A

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

    8. lift-+.f32 N/A

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

    9. lift-*.f32 N/A

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

    10. associate-*r* N/A

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

    11. +-commutative N/A

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

    12. lift-*.f32 N/A

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

    13. lift-*.f32 N/A

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

    14. lower-fma.f32 N/A

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

    15. lower-*.f32 N/A

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

    16. lift-*.f32 N/A

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

    17. lift-*.f32 N/A

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

    18. +-commutative N/A

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

    19. lift-*.f32 N/A

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

    20. *-commutative N/A

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

    21. lower-fma.f32 79.5%

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

11. Applied rewrites 79.5%

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

Alternative 17: 74.3% accurate, 8.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;zi \leq -0.00019999999494757503:\\ \;\;\;\;xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  (if (<= zi -0.00019999999494757503)
  (+ xi (* 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) {
	float tmp;
	if (zi <= -0.00019999999494757503f) {
		tmp = xi + (maxCos * (ux * (zi * (1.0f - ux))));
	} else {
		tmp = xi + (2.0f * (uy * (yi * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (zi <= Float32(-0.00019999999494757503))
		tmp = Float32(xi + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))));
	else
		tmp = Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (zi <= single(-0.00019999999494757503))
		tmp = xi + (maxCos * (ux * (zi * (single(1.0) - ux))));
	else
		tmp = xi + (single(2.0) * (uy * (yi * single(pi))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if zi < -1.99999995e-4

   1. Initial program 99.0%

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

   2. Taylor expanded in uy around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. lower--.f32 N/A

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 52.2%

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

   7. Applied rewrites 52.2%

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

   if -1.99999995e-4 < zi

   1. Initial program 99.0%

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

   2. Taylor expanded in uy around 0

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

      1. lower-fma.f32 N/A

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in ux around 0

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

      1. lower-+.f32 N/A

         \[\leadsto xi + 2 \cdot \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto xi + 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto xi + 2 \cdot \left(uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. lower-PI.f32 74.3%

         \[\leadsto xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) \]

   7. Applied rewrites 74.3%

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

Alternative 18: 74.1% accurate, 12.7× speedup

Math

\[xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (single(2.0) * (uy * (yi * single(pi))));
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

4. Applied rewrites 82.3%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

      \[\leadsto xi + 2 \cdot \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto xi + 2 \cdot \left(uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. lower-PI.f32 74.3%

      \[\leadsto xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) \]

7. Applied rewrites 74.3%

   \[\leadsto xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)} \]
8. Add Preprocessing

Alternative 19: 50.1% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower--.f32 N/A

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

4. Applied rewrites 52.3%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 50.1%

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

7. Applied rewrites 50.1%

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

Alternative 20: 50.1% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower--.f32 N/A

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

4. Applied rewrites 52.3%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 50.1%

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

7. Applied rewrites 50.1%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 50.1%

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 50.1%

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

9. Applied rewrites 50.1%

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

Alternative 21: 50.1% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower--.f32 N/A

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

4. Applied rewrites 52.3%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 50.1%

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

7. Applied rewrites 50.1%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 50.1%

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

9. Applied rewrites 50.1%

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

Alternative 22: 50.1% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower--.f32 N/A

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

4. Applied rewrites 52.3%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 50.1%

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

7. Applied rewrites 50.1%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-*.f32 50.1%

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

9. Applied rewrites 50.1%

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

Alternative 23: 12.3% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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(Float32(zi * maxCos) * ux)
end

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower--.f32 N/A

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

4. Applied rewrites 52.3%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 50.1%

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

7. Applied rewrites 50.1%

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

8. Taylor expanded in xi around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 12.3%

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

10. Applied rewrites 12.3%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. lower-*.f32 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 12.3%

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

12. Applied rewrites 12.3%

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

Alternative 24: 12.3% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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 99.0%

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

2. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower--.f32 N/A

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

4. Applied rewrites 52.3%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 50.1%

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

7. Applied rewrites 50.1%

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

8. Taylor expanded in xi around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 12.3%

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

10. Applied rewrites 12.3%

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

Reproduce

herbie shell --seed 2025323 
(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)))