UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 21.7s

Alternatives: 12

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: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in maxCos around 0 99.2%

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

3. Final simplification 99.2%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 5: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in ux around 0 99.1%

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

3. Final simplification 99.1%

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

Alternative 6: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in maxCos around 0 99.2%

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

3. Taylor expanded in ux around 0 99.1%

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

4. Final simplification 99.1%

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

Alternative 7: 95.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in ux around 0 99.1%

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

3. Taylor expanded in ux around 0 96.8%

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

   1. associate-*r* 96.7%

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

   2. *-commutative 96.7%

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

   3. associate-*l* 96.8%

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

5. Simplified 96.8%

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

6. Final simplification 96.8%

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

Alternative 8: 90.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in ux around 0 99.1%

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

3. Taylor expanded in uy around 0 90.2%

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

   1. associate-*r* 90.3%

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

   2. *-commutative 90.3%

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

   3. associate-*r* 90.2%

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

5. Simplified 90.2%

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

6. Final simplification 90.2%

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

Alternative 9: 81.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in uy around 0 90.3%

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

   1. associate-*r* 90.3%

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

6. Simplified 90.3%

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

   1. add-cbrt-cube 79.0%

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

   2. pow3 79.0%

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

   3. associate-*l* 79.0%

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

8. Applied egg-rr 79.0%

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

9. Taylor expanded in uy around 0 71.9%

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

    1. rem-cbrt-cube 82.3%

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

    2. *-commutative 82.3%

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

    3. *-commutative 82.3%

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

11. Applied egg-rr 82.3%

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

12. Final simplification 82.3%

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

Alternative 10: 81.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in uy around 0 90.3%

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

   1. associate-*r* 90.3%

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

6. Simplified 90.3%

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

7. Taylor expanded in ux around 0 90.0%

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

8. Taylor expanded in uy around 0 82.0%

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

9. Taylor expanded in ux around 0 82.0%

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

    1. unpow2 82.0%

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

    2. unpow2 82.0%

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

11. Simplified 82.0%

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

12. Final simplification 82.0%

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

Alternative 11: 81.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in uy around 0 90.3%

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

   1. associate-*r* 90.3%

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

6. Simplified 90.3%

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

7. Taylor expanded in ux around 0 90.0%

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

8. Taylor expanded in uy around 0 82.0%

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

9. Taylor expanded in yi around 0 82.1%

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

10. Final simplification 82.1%

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

Alternative 12: 78.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in uy around 0 90.3%

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

   1. associate-*r* 90.3%

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

6. Simplified 90.3%

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

7. Taylor expanded in ux around 0 90.0%

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

8. Taylor expanded in uy around 0 82.0%

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

9. Taylor expanded in ux around 0 79.8%

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

10. Final simplification 79.8%

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

Reproduce

herbie shell --seed 2023182 
(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)))