UniformSampleCone, x

Percentage Accurate: 57.1% → 99.0%

Time: 13.4s

Alternatives: 13

Speedup: 2.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in ux around 0 98.7%

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

4. Taylor expanded in maxCos around 0 98.7%

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

5. Taylor expanded in ux around 0 98.7%

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

   1. neg-mul-1 98.7%

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

   2. unsub-neg 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.8%

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

   1. associate-*l* 56.8%

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

   2. sub-neg 56.8%

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

   3. +-commutative 56.8%

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

   4. distribute-rgt-neg-in 56.8%

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

   5. fma-define 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in ux around inf 98.4%

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

   1. expm1-log1p-u 97.0%

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

   2. expm1-undefine 97.0%

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

   3. mul-1-neg 97.0%

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

   4. sub-neg 97.0%

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

   5. metadata-eval 97.0%

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

   6. +-commutative 97.0%

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

7. Applied egg-rr 97.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(-\frac{-1 + maxCos}{ux}\right)} - 1\right)} + \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right) + \frac{1}{ux}\right)\right) - \frac{maxCos}{ux}\right)} \]
8. Step-by-step derivation

   1. expm1-define 97.0%

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

   2. distribute-neg-frac 97.0%

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

   3. mul-1-neg 97.0%

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

   4. distribute-lft-in 97.0%

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

   5. metadata-eval 97.0%

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

   6. neg-mul-1 97.0%

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

   7. sub-neg 97.0%

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

9. Simplified 97.0%

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

10. Taylor expanded in uy around inf 98.2%

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

    1. associate-*l* 98.2%

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

    2. associate-*r/ 98.2%

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

    3. metadata-eval 98.2%

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

    4. +-commutative 98.2%

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

    5. metadata-eval 98.2%

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

    6. associate-*r/ 98.2%

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

    7. associate--l+ 98.2%

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

    8. sub-neg 98.2%

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

    9. metadata-eval 98.2%

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

    10. +-commutative 98.2%

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

    11. associate-*r/ 98.2%

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

    12. metadata-eval 98.2%

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

    13. associate-*r/ 98.2%

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

12. Simplified 98.2%

    \[\leadsto \color{blue}{ux \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right) + \frac{2 + maxCos \cdot -2}{ux}}\right)} \]
13. Add Preprocessing

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in ux around 0 98.7%

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

4. Taylor expanded in ux around inf 98.5%

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

5. Taylor expanded in maxCos around 0 97.4%

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

   1. sub-neg 97.4%

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

   2. associate-*r/ 97.4%

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

   3. metadata-eval 97.4%

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

   4. metadata-eval 97.4%

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

   5. +-commutative 97.4%

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

7. Simplified 97.4%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in ux around 0 98.7%

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

4. Taylor expanded in maxCos around 0 98.2%

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

5. Final simplification 98.2%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 + \left(maxCos \cdot \left(2 \cdot ux - 2\right) - ux\right)\right)} \]
6. Add Preprocessing

Alternative 5: 93.1% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (cos (* (* uy 2.0) PI)) (sqrt (* ux (- 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	return cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (2.0f - ux)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt((ux * (single(2.0) - ux)));
end

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in ux around 0 98.7%

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

4. Taylor expanded in maxCos around 0 98.7%

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

5. Taylor expanded in maxCos around 0 94.6%

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

   1. neg-mul-1 94.6%

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

   2. unsub-neg 94.6%

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

7. Simplified 94.6%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]
8. Add Preprocessing

Alternative 6: 80.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ ux \cdot \sqrt{\left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right) + 2 \cdot \frac{1}{ux}\right) - 2 \cdot \frac{maxCos}{ux}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  ux
  (sqrt
   (-
    (+ (* (- 1.0 maxCos) (+ -1.0 maxCos)) (* 2.0 (/ 1.0 ux)))
    (* 2.0 (/ maxCos ux))))))

C

float code(float ux, float uy, float maxCos) {
	return ux * sqrtf(((((1.0f - maxCos) * (-1.0f + maxCos)) + (2.0f * (1.0f / ux))) - (2.0f * (maxCos / ux))));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ux * sqrt(((((1.0e0 - maxcos) * ((-1.0e0) + maxcos)) + (2.0e0 * (1.0e0 / ux))) - (2.0e0 * (maxcos / ux))))
end function

Julia

function code(ux, uy, maxCos)
	return Float32(ux * sqrt(Float32(Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(-1.0) + maxCos)) + Float32(Float32(2.0) * Float32(Float32(1.0) / ux))) - Float32(Float32(2.0) * Float32(maxCos / ux)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = ux * sqrt(((((single(1.0) - maxCos) * (single(-1.0) + maxCos)) + (single(2.0) * (single(1.0) / ux))) - (single(2.0) * (maxCos / ux))));
end

Derivation

1. Initial program 56.8%

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

   1. associate-*l* 56.8%

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

   2. sub-neg 56.8%

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

   3. +-commutative 56.8%

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

   4. distribute-rgt-neg-in 56.8%

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

   5. fma-define 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in ux around inf 98.4%

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

   1. expm1-log1p-u 97.0%

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

   2. expm1-undefine 97.0%

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

   3. mul-1-neg 97.0%

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

   4. sub-neg 97.0%

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

   5. metadata-eval 97.0%

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

   6. +-commutative 97.0%

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

7. Applied egg-rr 97.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(-\frac{-1 + maxCos}{ux}\right)} - 1\right)} + \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right) + \frac{1}{ux}\right)\right) - \frac{maxCos}{ux}\right)} \]
8. Step-by-step derivation

   1. expm1-define 97.0%

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

   2. distribute-neg-frac 97.0%

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

   3. mul-1-neg 97.0%

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

   4. distribute-lft-in 97.0%

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

   5. metadata-eval 97.0%

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

   6. neg-mul-1 97.0%

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

   7. sub-neg 97.0%

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

9. Simplified 97.0%

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

10. Taylor expanded in uy around 0 79.0%

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

11. Final simplification 79.0%

    \[\leadsto ux \cdot \sqrt{\left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right) + 2 \cdot \frac{1}{ux}\right) - 2 \cdot \frac{maxCos}{ux}} \]
12. Add Preprocessing

Alternative 7: 80.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ ux \cdot \sqrt{\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right) + \frac{2 + maxCos \cdot -2}{ux}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  ux
  (sqrt
   (+ (* (- 1.0 maxCos) (+ -1.0 maxCos)) (/ (+ 2.0 (* maxCos -2.0)) ux)))))

C

float code(float ux, float uy, float maxCos) {
	return ux * sqrtf((((1.0f - maxCos) * (-1.0f + maxCos)) + ((2.0f + (maxCos * -2.0f)) / ux)));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ux * sqrt((((1.0e0 - maxcos) * ((-1.0e0) + maxcos)) + ((2.0e0 + (maxcos * (-2.0e0))) / ux)))
end function

Julia

function code(ux, uy, maxCos)
	return Float32(ux * sqrt(Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(-1.0) + maxCos)) + Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))) / ux))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = ux * sqrt((((single(1.0) - maxCos) * (single(-1.0) + maxCos)) + ((single(2.0) + (maxCos * single(-2.0))) / ux)));
end

Derivation

1. Initial program 56.8%

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

   1. associate-*l* 56.8%

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

   2. sub-neg 56.8%

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

   3. +-commutative 56.8%

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

   4. distribute-rgt-neg-in 56.8%

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

   5. fma-define 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in ux around inf 98.4%

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

   1. expm1-log1p-u 97.0%

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

   2. expm1-undefine 97.0%

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

   3. mul-1-neg 97.0%

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

   4. sub-neg 97.0%

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

   5. metadata-eval 97.0%

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

   6. +-commutative 97.0%

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

7. Applied egg-rr 97.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(-\frac{-1 + maxCos}{ux}\right)} - 1\right)} + \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right) + \frac{1}{ux}\right)\right) - \frac{maxCos}{ux}\right)} \]
8. Step-by-step derivation

   1. expm1-define 97.0%

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

   2. distribute-neg-frac 97.0%

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

   3. mul-1-neg 97.0%

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

   4. distribute-lft-in 97.0%

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

   5. metadata-eval 97.0%

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

   6. neg-mul-1 97.0%

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

   7. sub-neg 97.0%

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

9. Simplified 97.0%

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

10. Taylor expanded in uy around 0 79.0%

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

    1. associate-*r/ 79.0%

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

    2. metadata-eval 79.0%

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

    3. +-commutative 79.0%

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

    4. metadata-eval 79.0%

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

    5. associate-*r/ 79.0%

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

    6. associate--l+ 79.0%

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

    7. sub-neg 79.0%

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

    8. metadata-eval 79.0%

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

    9. +-commutative 79.0%

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

    10. associate-*r/ 79.0%

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

    11. metadata-eval 79.0%

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

    12. associate-*r/ 79.0%

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

    13. div-sub 79.0%

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

    14. cancel-sign-sub-inv 79.0%

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

    15. metadata-eval 79.0%

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

    16. *-commutative 79.0%

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

12. Simplified 79.0%

    \[\leadsto \color{blue}{ux \cdot \sqrt{\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right) + \frac{2 + maxCos \cdot -2}{ux}}} \]
13. Add Preprocessing

Alternative 8: 79.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(2 + \left(maxCos \cdot \left(2 \cdot ux - 2\right) - ux\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt (* ux (+ 2.0 (- (* maxCos (- (* 2.0 ux) 2.0)) ux)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * (2.0f + ((maxCos * ((2.0f * ux) - 2.0f)) - ux))));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (2.0e0 + ((maxcos * ((2.0e0 * ux) - 2.0e0)) - ux))))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(maxCos * Float32(Float32(Float32(2.0) * ux) - Float32(2.0))) - ux))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * (single(2.0) + ((maxCos * ((single(2.0) * ux) - single(2.0))) - ux))));
end

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in ux around 0 98.7%

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

4. Taylor expanded in uy around 0 79.4%

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

5. Taylor expanded in maxCos around 0 79.0%

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

6. Final simplification 79.0%

   \[\leadsto \sqrt{ux \cdot \left(2 + \left(maxCos \cdot \left(2 \cdot ux - 2\right) - ux\right)\right)} \]
7. Add Preprocessing

Alternative 9: 79.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(\left(2 - ux\right) - 2 \cdot maxCos\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt (* ux (- (- 2.0 ux) (* 2.0 maxCos)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * ((2.0f - ux) - (2.0f * maxCos))));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * ((2.0e0 - ux) - (2.0e0 * maxcos))))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(Float32(Float32(2.0) - ux) - Float32(Float32(2.0) * maxCos))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * ((single(2.0) - ux) - (single(2.0) * maxCos))));
end

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in ux around 0 98.7%

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

4. Taylor expanded in uy around 0 79.4%

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

5. Taylor expanded in maxCos around 0 78.6%

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

   1. neg-mul-1 78.6%

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

   2. unsub-neg 78.6%

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(2 - ux\right)} - 2 \cdot maxCos\right)} \]

7. Simplified 78.6%

   \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(2 - ux\right)} - 2 \cdot maxCos\right)} \]
8. Add Preprocessing

Alternative 10: 76.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(2 - ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (* ux (- 2.0 ux))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * (2.0f - ux)));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (2.0e0 - ux)))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(Float32(2.0) - ux)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * (single(2.0) - ux)));
end

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in ux around 0 98.7%

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

4. Taylor expanded in uy around 0 79.4%

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

5. Taylor expanded in maxCos around 0 76.2%

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

   1. neg-mul-1 76.2%

      \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(-ux\right)}\right)} \]

7. Simplified 76.2%

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + \left(-ux\right)\right)}} \]

8. Final simplification 76.2%

   \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \]
9. Add Preprocessing

Alternative 11: 14.7% accurate, 74.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (/ 1.0 ux))

C

float code(float ux, float uy, float maxCos) {
	return 1.0f / ux;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0 / ux
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) / ux)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0) / ux;
end

Derivation

1. Initial program 56.8%

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

   1. associate-*l* 56.8%

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

   2. sub-neg 56.8%

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

   3. +-commutative 56.8%

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

   4. distribute-rgt-neg-in 56.8%

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

   5. fma-define 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in uy around 0 47.7%

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

7. Simplified 47.8%

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

8. Taylor expanded in ux around -inf -0.0%

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

   1. associate-*r* -0.0%

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

   2. neg-mul-1 -0.0%

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

   3. *-commutative -0.0%

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

   4. sub-neg -0.0%

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

   5. metadata-eval -0.0%

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

   6. +-commutative -0.0%

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

   7. neg-mul-1 -0.0%

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

   8. sub-neg -0.0%

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

10. Simplified -0.0%

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

11. Taylor expanded in maxCos around 0 -0.0%

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

    1. associate-*r* -0.0%

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

    2. neg-mul-1 -0.0%

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

13. Simplified -0.0%

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

15. Applied egg-rr 14.8%

    \[\leadsto \color{blue}{{ux}^{-1}} \]
16. Step-by-step derivation

    1. unpow-1 14.8%

       \[\leadsto \color{blue}{\frac{1}{ux}} \]

17. Simplified 14.8%

    \[\leadsto \color{blue}{\frac{1}{ux}} \]
18. Add Preprocessing

Alternative 12: 4.4% accurate, 111.5× speedup

Math

\[\begin{array}{l} \\ -ux \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (- ux))

C

float code(float ux, float uy, float maxCos) {
	return -ux;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = -ux
end function

Julia

function code(ux, uy, maxCos)
	return Float32(-ux)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = -ux;
end

Derivation

1. Initial program 56.8%

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

   1. associate-*l* 56.8%

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

   2. sub-neg 56.8%

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

   3. +-commutative 56.8%

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

   4. distribute-rgt-neg-in 56.8%

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

   5. fma-define 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in uy around 0 47.7%

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

7. Simplified 47.8%

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

8. Taylor expanded in ux around -inf -0.0%

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

   1. associate-*r* -0.0%

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

   2. neg-mul-1 -0.0%

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

   3. *-commutative -0.0%

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

   4. sub-neg -0.0%

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

   5. metadata-eval -0.0%

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

   6. +-commutative -0.0%

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

   7. neg-mul-1 -0.0%

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

   8. sub-neg -0.0%

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

10. Simplified -0.0%

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

11. Taylor expanded in maxCos around 0 -0.0%

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

    1. associate-*r* -0.0%

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

    2. neg-mul-1 -0.0%

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

13. Simplified -0.0%

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

15. Applied egg-rr 4.4%

    \[\leadsto \color{blue}{0 - ux} \]

16. Final simplification 4.4%

    \[\leadsto -ux \]
17. Add Preprocessing

Alternative 13: 18.9% accurate, 223.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos) :precision binary32 ux)

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ux
end function

Julia

function code(ux, uy, maxCos)
	return ux
end

MATLAB

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

Derivation

1. Initial program 56.8%

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

   1. associate-*l* 56.8%

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

   2. sub-neg 56.8%

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

   3. +-commutative 56.8%

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

   4. distribute-rgt-neg-in 56.8%

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

   5. fma-define 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in uy around 0 47.7%

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

7. Simplified 47.8%

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

8. Taylor expanded in ux around -inf -0.0%

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

   1. associate-*r* -0.0%

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

   2. neg-mul-1 -0.0%

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

   3. *-commutative -0.0%

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

   4. sub-neg -0.0%

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

   5. metadata-eval -0.0%

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

   6. +-commutative -0.0%

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

   7. neg-mul-1 -0.0%

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

   8. sub-neg -0.0%

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

10. Simplified -0.0%

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

11. Taylor expanded in maxCos around 0 -0.0%

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

    1. associate-*r* -0.0%

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

    2. neg-mul-1 -0.0%

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

13. Simplified -0.0%

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

15. Applied egg-rr 19.0%

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

16. Final simplification 19.0%

    \[\leadsto ux \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary32
  :pre (and (and (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) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))