UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 19.5s

Alternatives: 16

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, 0.7× speedup

Math

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

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) (* ux maxCos)) (* (* ux maxCos) (+ ux -1.0)))))))
   (fma
    (cos t_0)
    (* xi t_1)
    (fma (sin t_0) (* yi t_1) (* (- 1.0 ux) (* (* 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) * (ux * maxCos)) * ((ux * maxCos) * (ux + -1.0f)))));
	return fmaf(cosf(t_0), (xi * t_1), fmaf(sinf(t_0), (yi * t_1), ((1.0f - ux) * ((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(Float32(1.0) - ux) * Float32(ux * maxCos)) * Float32(Float32(ux * maxCos) * Float32(ux + Float32(-1.0))))))
	return fma(cos(t_0), Float32(xi * t_1), fma(sin(t_0), Float32(yi * t_1), Float32(Float32(Float32(1.0) - ux) * Float32(Float32(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. Step-by-step derivation

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.2%

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

3. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 98.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) * Float32(ux * maxCos))
	t_1 = Float32(uy * Float32(Float32(2.0) * Float32(pi)))
	t_2 = sqrt(Float32(Float32(1.0) + Float32(t_0 * Float32(Float32(ux * 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(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. Step-by-step derivation

3. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

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 ((xi * (cosf(t_0) * sqrtf((1.0f + (1.0f + (-1.0f - powf(((1.0f - ux) * (ux * maxCos)), 2.0f))))))) + (yi * sinf(t_0))) + (zi * (ux * ((1.0f - ux) * maxCos)));
}

Julia

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

MATLAB

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

3. Taylor expanded in ux around 0 99.2%

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

   1. associate-*r* 99.2%

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

5. Simplified 99.2%

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

   1. pow2 99.2%

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

   2. associate-*r* 99.2%

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

   3. *-commutative 99.2%

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

   4. pow2 99.2%

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

   5. expm1-log1p-u 99.2%

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\right)\right)}}\right) \cdot xi + yi \cdot \sin \left(\left(2 \cdot uy\right) \cdot \pi\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   6. expm1-undefine 99.2%

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\right)} - 1\right)}}\right) \cdot xi + yi \cdot \sin \left(\left(2 \cdot uy\right) \cdot \pi\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. log1p-undefine 99.2%

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

   8. add-exp-log 99.2%

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

   9. pow2 99.2%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 4: 98.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy \cdot 2\right)\\ zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) + \left(yi \cdot \sin t\_0 + xi \cdot \left(\cos t\_0 \cdot \left(1 + -0.5 \cdot {\left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)}^{2}\right)\right)\right) \end{array} \end{array} \]

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)
       (+ 1.0 (* -0.5 (pow (* (- 1.0 ux) (* ux maxCos)) 2.0)))))))))

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) * (1.0f + (-0.5f * powf(((1.0f - ux) * (ux * maxCos)), 2.0f))))));
}

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) * Float32(Float32(1.0) + Float32(Float32(-0.5) * (Float32(Float32(Float32(1.0) - ux) * Float32(ux * maxCos)) ^ Float32(2.0))))))))
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) * (single(1.0) + (single(-0.5) * (((single(1.0) - ux) * (ux * maxCos)) ^ single(2.0)))))));
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. Add Preprocessing

3. Taylor expanded in ux around 0 99.2%

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

   1. associate-*r* 99.2%

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

5. Simplified 99.2%

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

   1. pow2 99.2%

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

   2. associate-*r* 99.2%

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

   3. *-commutative 99.2%

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

   4. pow2 99.2%

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

   5. expm1-log1p-u 99.2%

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\right)\right)}}\right) \cdot xi + yi \cdot \sin \left(\left(2 \cdot uy\right) \cdot \pi\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   6. expm1-undefine 99.2%

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\right)} - 1\right)}}\right) \cdot xi + yi \cdot \sin \left(\left(2 \cdot uy\right) \cdot \pi\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. log1p-undefine 99.2%

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

   8. add-exp-log 99.2%

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

   9. pow2 99.2%

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

7. Applied egg-rr 99.2%

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

8. Taylor expanded in maxCos around 0 99.2%

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\left(1 + -0.5 \cdot \left({maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)\right)\right)}\right) \cdot xi + yi \cdot \sin \left(\left(2 \cdot uy\right) \cdot \pi\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
9. Step-by-step derivation

   1. associate-*r* 99.2%

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

   2. unpow2 99.2%

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

   3. unpow2 99.2%

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

   4. swap-sqr 99.2%

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

   5. unpow2 99.2%

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

   6. swap-sqr 99.2%

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

   7. *-commutative 99.2%

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

   8. *-commutative 99.2%

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

   9. *-commutative 99.2%

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

   10. *-commutative 99.2%

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

   11. unpow2 99.2%

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

   12. *-commutative 99.2%

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

10. Simplified 99.2%

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

11. Final simplification 99.2%

    \[\leadsto zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) + \left(yi \cdot \sin \left(\pi \cdot \left(uy \cdot 2\right)\right) + xi \cdot \left(\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \left(1 + -0.5 \cdot {\left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)}^{2}\right)\right)\right) \]
12. Add Preprocessing

Alternative 5: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy * Float32(2.0)))
	return Float32(Float32(zi * Float32(Float32(Float32(1.0) - ux) * Float32(ux * maxCos))) + fma(xi, cos(t_0), Float32(yi * sin(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. Step-by-step derivation

3. Simplified 99.2%

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

5. Taylor expanded in ux around 0 99.0%

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

   1. fma-define 99.0%

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

   2. associate-*r* 99.0%

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

   3. associate-*r* 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 6: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (uy * single(pi));
	tmp = (maxCos * (ux * ((single(1.0) - ux) * zi))) + ((xi * cos(t_0)) + (yi * sin(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. Step-by-step derivation

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in maxCos around 0 99.0%

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

6. Final simplification 99.0%

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

Alternative 7: 95.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (uy * single(pi));
	tmp = ((xi * cos(t_0)) + (yi * sin(t_0))) + (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. Step-by-step derivation

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in ux around 0 96.1%

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

6. Final simplification 96.1%

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

Alternative 8: 85.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (maxCos * (ux * zi)) + (xi + (yi * (single(2.0) * (sin((uy * single(pi))) * cos((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. Step-by-step derivation

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in ux around 0 96.1%

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

6. Taylor expanded in uy around 0 85.6%

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

   1. sin-2 85.6%

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

8. Applied egg-rr 85.6%

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

Alternative 9: 82.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot \left(ux \cdot zi\right)\\ \mathbf{if}\;uy \leq 0.006099999882280827:\\ \;\;\;\;xi + \left(t\_0 + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* maxCos (* ux zi))))
   (if (<= uy 0.006099999882280827)
     (+ xi (+ t_0 (* 2.0 (* uy (* PI yi)))))
     (+ (* xi (cos (* 2.0 (* uy PI)))) t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = maxCos * (ux * zi);
	float tmp;
	if (uy <= 0.006099999882280827f) {
		tmp = xi + (t_0 + (2.0f * (uy * (((float) M_PI) * yi))));
	} else {
		tmp = (xi * cosf((2.0f * (uy * ((float) M_PI))))) + t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(maxCos * Float32(ux * zi))
	tmp = Float32(0.0)
	if (uy <= Float32(0.006099999882280827))
		tmp = Float32(xi + Float32(t_0 + Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi)))));
	else
		tmp = Float32(Float32(xi * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = maxCos * (ux * zi);
	tmp = single(0.0);
	if (uy <= single(0.006099999882280827))
		tmp = xi + (t_0 + (single(2.0) * (uy * (single(pi) * yi))));
	else
		tmp = (xi * cos((single(2.0) * (uy * single(pi))))) + t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00609999988

   1. Initial program 99.4%

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

      1. associate-+l+ 99.4%

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

      2. associate-*l* 99.4%

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

      3. fma-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ux around 0 96.1%

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

      1. add-cube-cbrt 96.0%

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

      2. pow3 96.0%

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

   7. Applied egg-rr 96.0%

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

   8. Taylor expanded in uy around 0 92.8%

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

   if 0.00609999988 < uy

   1. Initial program 98.2%

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

      1. associate-+l+ 98.2%

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

      2. associate-*l* 98.2%

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

      3. fma-define 98.3%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in ux around 0 95.9%

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

      1. add-cube-cbrt 95.1%

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

      2. pow3 95.1%

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

   7. Applied egg-rr 95.1%

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

   8. Taylor expanded in yi around 0 60.4%

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.006099999882280827:\\ \;\;\;\;xi + \left(maxCos \cdot \left(ux \cdot zi\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = ((ux * maxCos) * zi) + (xi + (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. Step-by-step derivation

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in ux around 0 96.1%

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

6. Taylor expanded in uy around 0 85.6%

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

   1. pow1 85.6%

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

   2. associate-*r* 85.6%

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

8. Applied egg-rr 85.6%

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

   1. unpow1 85.6%

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

   2. *-commutative 85.6%

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

   3. *-commutative 85.6%

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

10. Simplified 85.6%

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

11. Final simplification 85.6%

    \[\leadsto \left(ux \cdot maxCos\right) \cdot zi + \left(xi + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
12. Add Preprocessing

Alternative 11: 85.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (maxCos * (ux * zi)) + (xi + (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. Step-by-step derivation

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in ux around 0 96.1%

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

6. Taylor expanded in uy around 0 85.6%

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

Alternative 12: 79.1% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + ((maxCos * (ux * zi)) + (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. Step-by-step derivation

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in ux around 0 96.1%

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

   1. add-cube-cbrt 95.8%

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

   2. pow3 95.8%

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

7. Applied egg-rr 95.8%

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

8. Taylor expanded in uy around 0 81.1%

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

9. Final simplification 81.1%

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

Alternative 13: 51.3% accurate, 41.9× speedup

Math

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

FPCore

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

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (maxCos * ((1.0f - ux) * (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 * ((1.0e0 - ux) * (ux * zi)))
end function

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (maxCos * ((single(1.0) - ux) * (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%

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

5. Taylor expanded in uy around 0 55.4%

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

   1. associate-*r* 55.4%

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

   2. unpow2 55.4%

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

   3. unpow2 55.4%

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

   4. swap-sqr 55.4%

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

   5. unpow2 55.4%

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

   6. swap-sqr 55.4%

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

   7. *-commutative 55.4%

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

   8. *-commutative 55.4%

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

   9. *-commutative 55.4%

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

   10. *-commutative 55.4%

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

   11. unpow2 55.4%

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

   12. *-commutative 55.4%

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

7. Simplified 55.4%

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

8. Taylor expanded in maxCos around 0 55.3%

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

   1. +-commutative 55.3%

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

   2. associate-*r* 55.3%

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

10. Simplified 55.3%

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

11. Final simplification 55.3%

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

Alternative 14: 51.3% accurate, 41.9× speedup

Math

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

FPCore

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

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (maxCos * (ux * ((1.0f - 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 * ((1.0e0 - ux) * zi)))
end function

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (maxCos * (ux * ((single(1.0) - 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%

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

5. Taylor expanded in uy around 0 55.4%

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

   1. associate-*r* 55.4%

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

   2. unpow2 55.4%

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

   3. unpow2 55.4%

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

   4. swap-sqr 55.4%

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

   5. unpow2 55.4%

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

   6. swap-sqr 55.4%

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

   7. *-commutative 55.4%

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

   8. *-commutative 55.4%

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

   9. *-commutative 55.4%

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

   10. *-commutative 55.4%

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

   11. unpow2 55.4%

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

   12. *-commutative 55.4%

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

7. Simplified 55.4%

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

8. Taylor expanded in maxCos around 0 55.3%

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

9. Final simplification 55.3%

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

Alternative 15: 49.1% accurate, 65.9× 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 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

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in ux around 0 96.1%

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

   1. add-cube-cbrt 95.8%

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

   2. pow3 95.8%

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

7. Applied egg-rr 95.8%

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

8. Taylor expanded in uy around 0 53.6%

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

Alternative 16: 44.9% accurate, 461.0× speedup

Math

\[\begin{array}{l} \\ xi \end{array} \]

FPCore

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

C

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

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
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in uy around 0 55.4%

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

   1. associate-*r* 55.4%

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

   2. unpow2 55.4%

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

   3. unpow2 55.4%

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

   4. swap-sqr 55.4%

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

   5. unpow2 55.4%

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

   6. swap-sqr 55.4%

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

   7. *-commutative 55.4%

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

   8. *-commutative 55.4%

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

   9. *-commutative 55.4%

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

   10. *-commutative 55.4%

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

   11. unpow2 55.4%

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

   12. *-commutative 55.4%

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

7. Simplified 55.4%

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

8. Taylor expanded in ux around 0 50.5%

   \[\leadsto \color{blue}{xi} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(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)))