UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 22.6s

Alternatives: 14

Speedup: 0.8×

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, 0.8× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI)))
        (t_1
         (sqrt
          (+
           1.0
           (* (- 1.0 ux) (* maxCos (* (* ux maxCos) (* ux (+ ux -1.0)))))))))
   (fma
    (cos t_0)
    (* t_1 xi)
    (fma (* t_1 (sin t_0)) yi (* ux (* (* (- 1.0 ux) maxCos) 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 = sqrtf((1.0f + ((1.0f - ux) * (maxCos * ((ux * maxCos) * (ux * (ux + -1.0f)))))));
	return fmaf(cosf(t_0), (t_1 * xi), fmaf((t_1 * sinf(t_0)), yi, (ux * (((1.0f - ux) * maxCos) * zi))));
}

Julia

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

Derivation

1. Initial program 98.9%

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

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. Final simplification 98.9%

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

Alternative 2: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

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 maxCos around 0 98.9%

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

   1. fma-def 98.9%

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

   2. associate-*r* 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 3: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in 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%

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

Alternative 4: 98.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in 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. Taylor expanded in ux around 0 98.7%

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

   1. *-commutative 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 5: 90.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

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. Taylor expanded in uy around 0 90.1%

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

   1. associate-*r* 90.1%

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

7. Simplified 90.1%

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

8. Final simplification 90.1%

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

Alternative 6: 90.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

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. Taylor expanded in uy around 0 90.1%

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

   1. associate-*r* 90.1%

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

   2. *-commutative 90.1%

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

   3. *-commutative 90.1%

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

   4. associate-*l* 90.1%

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

7. Simplified 90.1%

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

8. Final simplification 90.1%

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

Alternative 7: 58.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

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 62.8%

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

   1. fma-udef 62.8%

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

   2. associate-*r* 62.8%

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

   3. *-commutative 62.8%

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

   4. *-commutative 62.8%

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

   5. *-commutative 62.8%

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

   6. associate-*r* 62.8%

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

6. Applied egg-rr 62.8%

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

7. Final simplification 62.8%

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

Alternative 8: 58.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in 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. Step-by-step derivation

   1. *-commutative 98.9%

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

   2. *-commutative 98.9%

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

   3. add-cube-cbrt 98.3%

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

   4. pow3 98.4%

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

   5. *-commutative 98.4%

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

   6. *-commutative 98.4%

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

   7. associate-*l* 98.4%

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in uy around 0 62.8%

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \color{blue}{0}\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

8. Final simplification 62.8%

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

Alternative 9: 58.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

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 62.8%

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

5. Taylor expanded in maxCos around 0 62.6%

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

   1. fma-def 62.6%

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

   2. associate-*r* 62.6%

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

7. Simplified 62.6%

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

8. Final simplification 62.6%

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

Alternative 10: 58.8% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ (* (- 1.0 ux) (* maxCos (* ux zi))) (* xi (cos (* 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((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(xi * cos(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(2.0) * (uy * single(pi)))));
end

Derivation

1. Initial program 98.9%

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

3. 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 62.8%

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

5. Taylor expanded in maxCos around 0 62.6%

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

6. Final simplification 62.6%

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

Alternative 11: 56.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. 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 62.8%

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

5. Taylor expanded in ux around 0 61.8%

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

6. Final simplification 61.8%

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

Alternative 12: 52.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. 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 62.8%

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

5. Taylor expanded in ux around 0 58.7%

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

6. Final simplification 58.7%

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

Alternative 13: 49.5% accurate, 94.1× speedup

Math

\[\begin{array}{l} \\ xi + maxCos \cdot \left(ux \cdot zi\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. 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 62.8%

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

5. Taylor expanded in ux around 0 61.8%

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

6. Taylor expanded in uy around 0 51.9%

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

7. Final simplification 51.9%

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

Alternative 14: 11.8% accurate, 131.8× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(ux \cdot zi\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

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 62.8%

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

5. Taylor expanded in xi around 0 11.1%

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

   1. associate-*r* 11.1%

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

7. Simplified 11.1%

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

8. Taylor expanded in ux around 0 10.6%

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

9. Final simplification 10.6%

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

Reproduce

herbie shell --seed 2023268 
(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)))