UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 21.3s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos - Float32(maxCos * ux)), Float32(ux * zi), Float32(sqrt(Float32(Float32(1.0) - Float32(ux * Float32(ux * Float32(maxCos * Float32(Float32(Float32(maxCos * ux) - maxCos) * Float32(ux + Float32(-1.0)))))))) * Float32(Float32(xi * cos(Float32(Float32(pi) * Float32(uy * Float32(-2.0))))) + Float32(yi * sin(Float32(uy * Float32(Float32(pi) * Float32(2.0))))))))
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. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ux * (maxCos * (ux + single(-1.0)));
	t_1 = single(pi) * (uy * single(2.0));
	tmp = ((xi * (cos(t_1) * sqrt((single(1.0) - (t_0 * t_0))))) + (yi * sin(t_1))) + (zi * (ux * (maxCos * (single(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 ux around 0 98.9%

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

   1. associate-*r* 98.9%

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

4. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 3: 90.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos - Float32(maxCos * ux)), Float32(ux * zi), Float32(sqrt(Float32(Float32(1.0) - Float32(ux * Float32(ux * Float32(maxCos * Float32(Float32(Float32(maxCos * ux) - maxCos) * Float32(ux + Float32(-1.0)))))))) * Float32(Float32(xi * cos(Float32(Float32(pi) * Float32(uy * Float32(-2.0))))) + Float32(Float32(2.0) * Float32(Float32(pi) * Float32(uy * yi))))))
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. Simplified 99.0%

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

4. Taylor expanded in uy around 0 93.3%

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

   1. associate-*r* 93.3%

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

   2. *-commutative 93.3%

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

6. Simplified 93.3%

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

7. Final simplification 93.3%

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

Alternative 4: 90.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos - Float32(maxCos * ux)), Float32(ux * zi), Float32(sqrt(Float32(Float32(1.0) + Float32(ux * Float32(ux * Float32(maxCos * Float32(Float32(ux * Float32(maxCos * Float32(2.0))) - maxCos)))))) * Float32(Float32(xi * cos(Float32(Float32(pi) * Float32(uy * Float32(-2.0))))) + Float32(Float32(uy * Float32(2.0)) * Float32(Float32(pi) * yi)))))
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. Simplified 99.0%

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

4. Taylor expanded in uy around 0 93.3%

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

   1. associate-*r* 93.3%

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

   2. *-commutative 93.3%

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

6. Simplified 93.3%

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

7. Taylor expanded in ux around 0 93.2%

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

   1. neg-mul-1 93.2%

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

   2. +-commutative 93.2%

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

   3. unsub-neg 93.2%

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

   4. associate-*r* 93.2%

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

   5. *-commutative 93.2%

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

9. Simplified 93.2%

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

10. Final simplification 93.2%

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

Alternative 5: 90.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in uy around 0 93.3%

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

   1. associate-*r* 93.3%

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

   2. *-commutative 93.3%

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

6. Simplified 93.3%

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

7. Taylor expanded in ux around 0 93.2%

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

   1. neg-mul-1 93.2%

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

9. Simplified 93.2%

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

10. Final simplification 93.2%

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

Alternative 6: 59.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos \left(uy \cdot \left(\pi \cdot 2\right)\right), xi \cdot \sqrt{1 + \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot ux\right)\right)\right) \cdot \left(ux + -1\right)}, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Simplified 98.9%

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

4. Taylor expanded in uy around 0 60.6%

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

5. Taylor expanded in ux around 0 60.6%

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

   1. *-commutative 60.6%

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

   2. unpow2 60.6%

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

   3. associate-*l* 60.6%

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

   4. *-commutative 60.6%

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

7. Simplified 60.6%

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

8. Final simplification 60.6%

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

Alternative 7: 59.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos \left(uy \cdot \left(\pi \cdot 2\right)\right), xi \cdot \sqrt{1 + \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot ux\right)\right)\right) \cdot \left(ux + -1\right)}, maxCos \cdot \left(ux \cdot \left(zi - ux \cdot zi\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Simplified 98.9%

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

4. Taylor expanded in uy around 0 60.6%

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

5. Taylor expanded in ux around 0 60.6%

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

   1. *-commutative 60.6%

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

   2. unpow2 60.6%

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

   3. associate-*l* 60.6%

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

   4. *-commutative 60.6%

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

7. Simplified 60.6%

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

8. Taylor expanded in ux around 0 60.6%

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

   1. +-commutative 60.6%

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

   2. associate-*r* 60.6%

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

   3. distribute-rgt-out 60.6%

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

   4. *-lft-identity 60.6%

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

   5. unpow2 60.6%

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

   6. associate-*r* 60.6%

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

   7. neg-mul-1 60.6%

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

   8. distribute-rgt-in 60.6%

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

   9. sub-neg 60.6%

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

   10. associate-*l* 60.6%

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

   11. *-commutative 60.6%

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

   12. associate-*r* 60.6%

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

   13. *-commutative 60.6%

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

   14. sub-neg 60.6%

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

   15. +-commutative 60.6%

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

   16. distribute-rgt1-in 60.6%

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

   17. distribute-lft-neg-in 60.6%

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

   18. unsub-neg 60.6%

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

10. Simplified 60.6%

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

11. Final simplification 60.6%

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

Alternative 8: 57.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos \left(uy \cdot \left(\pi \cdot 2\right)\right), xi \cdot \sqrt{1 + \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot ux\right)\right)\right) \cdot \left(ux + -1\right)}, maxCos \cdot \left(ux \cdot zi\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Simplified 98.9%

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

4. Taylor expanded in uy around 0 60.6%

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

5. Taylor expanded in ux around 0 60.6%

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

   1. *-commutative 60.6%

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

   2. unpow2 60.6%

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

   3. associate-*l* 60.6%

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

   4. *-commutative 60.6%

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

7. Simplified 60.6%

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

8. Taylor expanded in ux around 0 58.5%

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

9. Final simplification 58.5%

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

Reproduce

herbie shell --seed 2023293 
(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)))