UniformSampleCone, x

Percentage Accurate: 57.0% → 98.9%

Time: 13.7s

Alternatives: 10

Speedup: 2.2×

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: 57.0% 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: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos \left(uy \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(\left(1 - maxCos\right) + ux \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right)\right)\right) - maxCos\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* uy (expm1 (log1p (* 2.0 PI)))))
  (sqrt
   (*
    ux
    (-
     (+ 1.0 (+ (- 1.0 maxCos) (* ux (* (+ -1.0 maxCos) (- 1.0 maxCos)))))
     maxCos)))))

C

float code(float ux, float uy, float maxCos) {
	return cosf((uy * expm1f(log1pf((2.0f * ((float) M_PI)))))) * sqrtf((ux * ((1.0f + ((1.0f - maxCos) + (ux * ((-1.0f + maxCos) * (1.0f - maxCos))))) - maxCos)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(uy * expm1(log1p(Float32(Float32(2.0) * Float32(pi)))))) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - maxCos) + Float32(ux * Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(1.0) - maxCos))))) - maxCos))))
end

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-define 56.2%

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

3. Simplified 56.5%

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

5. Taylor expanded in ux around inf 98.5%

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

6. Taylor expanded in ux around 0 98.8%

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

   1. add-cube-cbrt 98.9%

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

   2. pow3 98.9%

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

8. Applied egg-rr 98.9%

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

   1. rem-cube-cbrt 98.8%

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

   2. expm1-log1p-u 98.9%

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

   3. expm1-undefine 98.9%

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

10. Applied egg-rr 98.9%

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

    1. expm1-define 98.9%

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

    \[\leadsto \cos \left(uy \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(\left(1 - maxCos\right) + ux \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right)\right)\right) - maxCos\right)} \]
14. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-define 56.2%

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

3. Simplified 56.5%

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

5. Taylor expanded in ux around inf 98.5%

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

6. Taylor expanded in ux around 0 98.8%

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

7. Taylor expanded in maxCos around 0 98.8%

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

   1. neg-mul-1 98.8%

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

   2. sub-neg 98.8%

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

9. Simplified 98.8%

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

10. Final simplification 98.8%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-define 56.2%

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

3. Simplified 56.5%

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

5. Taylor expanded in ux around inf 98.5%

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

6. Taylor expanded in ux around 0 98.8%

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

   1. add-cbrt-cube 98.8%

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

   2. pow1/3 95.3%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left(\left(\sqrt{ux \cdot \left(\left(1 + \left(-1 \cdot \left(maxCos - 1\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - maxCos\right)} \cdot \sqrt{ux \cdot \left(\left(1 + \left(-1 \cdot \left(maxCos - 1\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - maxCos\right)}\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(-1 \cdot \left(maxCos - 1\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - maxCos\right)}\right)}^{0.3333333333333333}} \]

8. Applied egg-rr 95.5%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left({\left(ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(-1 + maxCos\right) \cdot \left(1 - maxCos\right), -\left(-1 + maxCos\right)\right) - maxCos\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

10. Simplified 98.7%

    \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\sqrt[3]{{\left(ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), 1 - maxCos\right) - maxCos\right)\right)\right)}^{1.5}}} \]

11. Taylor expanded in uy around inf 98.8%

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

    1. *-commutative 98.8%

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

    2. associate-*r* 98.8%

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

    3. *-commutative 98.8%

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

    4. associate-*r* 98.8%

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

    5. associate--l+ 98.8%

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

    6. *-commutative 98.8%

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

    7. sub-neg 98.8%

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

    8. metadata-eval 98.8%

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

    9. +-commutative 98.8%

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

    10. *-commutative 98.8%

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

13. Simplified 98.8%

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

14. Final simplification 98.8%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

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(uy * Float32(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 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-define 56.2%

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

3. Simplified 56.5%

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

5. Taylor expanded in ux around inf 98.5%

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

6. Taylor expanded in ux around 0 98.8%

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

7. Taylor expanded in maxCos around 0 97.6%

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

8. Final simplification 97.6%

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

Alternative 5: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 + \left(\left(1 - maxCos\right) + ux \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right)\right)\right) - maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.9999999949504854e-6)
   (* (cos (* PI (* uy 2.0))) (sqrt (* ux (- 2.0 ux))))
   (sqrt
    (*
     ux
     (-
      (+ 1.0 (+ (- 1.0 maxCos) (* ux (* (+ -1.0 maxCos) (- 1.0 maxCos)))))
      maxCos)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.9999999949504854e-6f) {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf((ux * ((1.0f + ((1.0f - maxCos) + (ux * ((-1.0f + maxCos) * (1.0f - maxCos))))) - maxCos)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.9999999949504854e-6))
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - maxCos) + Float32(ux * Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(1.0) - maxCos))))) - maxCos)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(1.9999999949504854e-6))
		tmp = cos((single(pi) * (uy * single(2.0)))) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = sqrt((ux * ((single(1.0) + ((single(1.0) - maxCos) + (ux * ((single(-1.0) + maxCos) * (single(1.0) - maxCos))))) - maxCos)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999999e-6

   1. Initial program 55.0%

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

   3. Taylor expanded in ux around 0 98.8%

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

      1. cancel-sign-sub-inv 98.8%

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

      2. associate-*r* 98.8%

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

      3. mul-1-neg 98.8%

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

      4. sub-neg 98.8%

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

      5. metadata-eval 98.8%

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

      6. +-commutative 98.8%

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

      7. metadata-eval 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in maxCos around 0 98.6%

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

      1. neg-mul-1 98.6%

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

      2. unsub-neg 98.6%

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

   8. Simplified 98.6%

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

   if 1.99999999e-6 < maxCos

   1. Initial program 61.6%

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

      1. associate-*l* 61.6%

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

      2. sub-neg 61.6%

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

      3. +-commutative 61.6%

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

      4. distribute-rgt-neg-in 61.6%

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

      5. fma-define 62.1%

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

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), -1 - ux \cdot \left(maxCos + -1\right), 1\right)}} \]
   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. Taylor expanded in ux around 0 98.7%

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

   7. Taylor expanded in uy around 0 80.3%

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(\left(1 + \left(-1 \cdot \left(maxCos - 1\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - maxCos\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 + \left(\left(1 - maxCos\right) + ux \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right)\right)\right) - maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * ((1.0f + ((1.0f - maxCos) + (ux * ((-1.0f + maxCos) * (1.0f - maxCos))))) - 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 * ((1.0e0 + ((1.0e0 - maxcos) + (ux * (((-1.0e0) + maxcos) * (1.0e0 - maxcos))))) - maxcos)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-define 56.2%

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

3. Simplified 56.5%

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

5. Taylor expanded in ux around inf 98.5%

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

6. Taylor expanded in ux around 0 98.8%

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

7. Taylor expanded in uy around 0 78.9%

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

8. Final simplification 78.9%

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

Alternative 7: 80.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * ((2.0f + (ux * ((-1.0f + maxCos) * (1.0f - maxCos)))) - (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 * (((-1.0e0) + maxcos) * (1.0e0 - maxcos)))) - (2.0e0 * maxcos))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-define 56.2%

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

3. Simplified 56.5%

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

5. Taylor expanded in ux around inf 98.5%

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

6. Taylor expanded in ux around 0 98.8%

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

   1. add-cbrt-cube 98.8%

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

   2. pow1/3 95.3%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left(\left(\sqrt{ux \cdot \left(\left(1 + \left(-1 \cdot \left(maxCos - 1\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - maxCos\right)} \cdot \sqrt{ux \cdot \left(\left(1 + \left(-1 \cdot \left(maxCos - 1\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - maxCos\right)}\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(-1 \cdot \left(maxCos - 1\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - maxCos\right)}\right)}^{0.3333333333333333}} \]

8. Applied egg-rr 95.5%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left({\left(ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(-1 + maxCos\right) \cdot \left(1 - maxCos\right), -\left(-1 + maxCos\right)\right) - maxCos\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

10. Simplified 98.7%

    \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\sqrt[3]{{\left(ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), 1 - maxCos\right) - maxCos\right)\right)\right)}^{1.5}}} \]

11. Taylor expanded in uy around 0 78.9%

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

12. Final simplification 78.9%

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

Alternative 8: 79.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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)) - ((ux * maxcos) * (2.0e0 + (ux * (-2.0e0))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-define 56.2%

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

3. Simplified 56.5%

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

5. Taylor expanded in uy around 0 46.6%

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

7. Simplified 46.7%

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

8. Taylor expanded in ux around 0 49.3%

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

9. Taylor expanded in maxCos around 0 78.1%

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

    1. mul-1-neg 78.1%

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

    2. associate-*r* 78.1%

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

    3. *-commutative 78.1%

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

    4. *-commutative 78.1%

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

    5. sub-neg 78.1%

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

    6. metadata-eval 78.1%

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

11. Simplified 78.1%

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

12. Final simplification 78.1%

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

Alternative 9: 75.9% accurate, 2.1× speedup

Math

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

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(-Float32(-2.0)) - ux)))
end

MATLAB

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-define 56.2%

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

3. Simplified 56.5%

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

5. Taylor expanded in uy around 0 46.6%

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

7. Simplified 46.7%

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

8. Taylor expanded in ux around 0 49.3%

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

9. Taylor expanded in maxCos around 0 73.5%

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

    1. associate-*r* 73.5%

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

    2. mul-1-neg 73.5%

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

    3. sub-neg 73.5%

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

    4. metadata-eval 73.5%

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

11. Simplified 73.5%

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

12. Final simplification 73.5%

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

Alternative 10: 62.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot ux} \end{array} \]

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((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((2.0e0 * ux))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-define 56.2%

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

3. Simplified 56.5%

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

5. Taylor expanded in uy around 0 46.6%

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

7. Simplified 46.7%

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

8. Taylor expanded in ux around 0 65.1%

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

9. Taylor expanded in maxCos around 0 61.7%

   \[\leadsto \sqrt{\color{blue}{2 \cdot ux}} \]
10. Step-by-step derivation

    1. *-commutative 61.7%

       \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

11. Simplified 61.7%

    \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

12. Final simplification 61.7%

    \[\leadsto \sqrt{2 \cdot ux} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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)))))))