UniformSampleCone 2

Percentage Accurate: 98.9% → 98.8%

Time: 28.4s

Alternatives: 10

Speedup: N/A×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

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: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\\ \left(\left(\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{1 + t\_0 \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}\right) \cdot xi + yi \cdot \sin \left(\sqrt[3]{{\left(uy \cdot 2\right)}^{3} \cdot {\pi}^{3}}\right)\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (* (- 1.0 ux) maxCos))))
   (+
    (+
     (*
      (*
       (cos (* PI (* uy 2.0)))
       (sqrt (+ 1.0 (* t_0 (* ux (* maxCos (+ ux -1.0)))))))
      xi)
     (* yi (sin (cbrt (* (pow (* uy 2.0) 3.0) (pow PI 3.0))))))
    (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ux * ((1.0f - ux) * maxCos);
	return (((cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf((1.0f + (t_0 * (ux * (maxCos * (ux + -1.0f))))))) * xi) + (yi * sinf(cbrtf((powf((uy * 2.0f), 3.0f) * powf(((float) M_PI), 3.0f)))))) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos))
	return Float32(Float32(Float32(Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(Float32(1.0) + Float32(t_0 * Float32(ux * Float32(maxCos * Float32(ux + Float32(-1.0)))))))) * xi) + Float32(yi * sin(cbrt(Float32((Float32(uy * Float32(2.0)) ^ Float32(3.0)) * (Float32(pi) ^ Float32(3.0))))))) + Float32(t_0 * zi))
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

   1. add-cbrt-cube 98.7%

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

   2. add-cbrt-cube 98.7%

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

   3. cbrt-unprod 98.8%

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

   4. pow3 98.8%

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

   5. pow3 98.8%

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

7. Applied egg-rr 98.8%

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

8. Final simplification 98.8%

   \[\leadsto \left(\left(\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{1 + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}\right) \cdot xi + yi \cdot \sin \left(\sqrt[3]{{\left(uy \cdot 2\right)}^{3} \cdot {\pi}^{3}}\right)\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi \]
9. Add Preprocessing

Alternative 2: 98.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in maxCos around 0 98.7%

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

4. Taylor expanded in ux around 0 98.7%

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

   1. associate-*r* 98.7%

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

6. Simplified 98.7%

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

7. Final simplification 98.7%

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

Alternative 3: 98.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 4: 90.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in uy around 0 86.6%

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

7. Final simplification 86.6%

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

Alternative 5: 90.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in uy around 0 86.6%

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

   1. *-commutative 86.6%

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

   2. associate-*l* 86.5%

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

   3. *-commutative 86.5%

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

   4. associate-*r* 86.7%

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

8. Simplified 86.7%

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

9. Final simplification 86.7%

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

Alternative 6: 52.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, xi \cdot \sqrt{1 + ux \cdot \left(ux \cdot \left(\left(maxCos \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right)\right)\right)}, maxCos \cdot \left(ux \cdot zi - ux \cdot \left(ux \cdot zi\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in uy around 0 52.5%

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

   1. associate-*r* 52.5%

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

   2. sub-neg 52.5%

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

   3. distribute-rgt-in 52.6%

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

   4. *-un-lft-identity 52.6%

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

7. Applied egg-rr 52.6%

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

8. Taylor expanded in uy around 0 45.8%

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

9. Final simplification 45.8%

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

Alternative 7: 52.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, xi \cdot \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot maxCos\right)\right)}, maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in uy around 0 52.5%

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

6. Taylor expanded in ux around 0 52.5%

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

7. Taylor expanded in uy around 0 45.8%

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

8. Final simplification 45.8%

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

Alternative 8: 52.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, xi \cdot \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot maxCos\right)\right)}, maxCos \cdot \left(ux \cdot \left(zi - ux \cdot zi\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in uy around 0 52.5%

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

6. Taylor expanded in ux around 0 52.5%

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

7. Taylor expanded in uy around 0 45.8%

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

8. Taylor expanded in ux around 0 45.8%

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

   1. mul-1-neg 45.8%

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

   2. *-commutative 45.8%

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

   3. unsub-neg 45.8%

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

10. Simplified 45.8%

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

11. Final simplification 45.8%

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

Alternative 9: 52.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, xi \cdot \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot maxCos\right)\right)}, maxCos \cdot \left(zi \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in uy around 0 52.5%

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

6. Taylor expanded in ux around 0 52.5%

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

7. Taylor expanded in uy around 0 45.8%

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

8. Taylor expanded in ux around 0 45.8%

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

   1. associate-*r* 45.8%

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

   2. fma-define 45.8%

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

   3. neg-mul-1 45.8%

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

   4. unpow2 45.8%

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

   5. distribute-lft-neg-in 45.8%

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

   6. fma-define 45.8%

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

   7. associate-*r* 45.8%

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

   8. distribute-lft1-in 45.8%

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

   9. associate-*r* 45.8%

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

   10. +-commutative 45.8%

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

   11. sub-neg 45.8%

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

   12. *-commutative 45.8%

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

   13. *-commutative 45.8%

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

10. Simplified 45.8%

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

11. Final simplification 45.8%

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

Alternative 10: 50.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, xi \cdot \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot maxCos\right)\right)}, maxCos \cdot \left(ux \cdot zi\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in uy around 0 52.5%

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

6. Taylor expanded in ux around 0 52.5%

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

7. Taylor expanded in uy around 0 45.8%

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

8. Taylor expanded in ux around 0 42.8%

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

9. Final simplification 42.8%

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

Reproduce

herbie shell --seed 2024046 
(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)))