UniformSampleCone, x

Percentage Accurate: 56.9% → 99.0%

Time: 14.0s

Alternatives: 9

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: 56.9% 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

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (((maxCos * ((2.0f * ux) - (ux * maxCos))) - ux) + (2.0f - (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(Float32(maxCos * Float32(Float32(Float32(2.0) * ux) - Float32(ux * maxCos))) - ux) + Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))))))
end

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.9%

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

3. Simplified 57.9%

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

5. Taylor expanded in ux around 0 98.8%

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

6. Taylor expanded in maxCos around 0 98.8%

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

7. Final simplification 98.8%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* uy (* 2.0 PI)))
  (sqrt
   (*
    ux
    (+ (- 2.0 (* 2.0 maxCos)) (* ux (+ -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 - (2.0f * maxCos)) + (ux * (-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(Float32(2.0) - Float32(Float32(2.0) * maxCos)) + Float32(ux * Float32(Float32(-1.0) + Float32(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) - (single(2.0) * maxCos)) + (ux * (single(-1.0) + (maxCos * (single(2.0) - maxCos)))))));
end

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.9%

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

3. Simplified 57.9%

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

5. Taylor expanded in ux around 0 98.8%

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

6. Taylor expanded in maxCos around 0 98.8%

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

7. Taylor expanded in ux around 0 98.8%

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

   1. distribute-lft-in 98.8%

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

   2. *-commutative 98.8%

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

   3. mul-1-neg 98.8%

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

9. Applied egg-rr 98.8%

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

    1. *-commutative 98.8%

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

    2. distribute-lft-in 98.8%

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

    3. unsub-neg 98.8%

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

11. Simplified 98.8%

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

12. Final simplification 98.8%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. associate--l+ 98.8%

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

   2. associate-*r* 98.8%

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

   3. mul-1-neg 98.8%

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

   4. fma-neg 98.8%

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

   5. sub-neg 98.8%

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

   6. metadata-eval 98.8%

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

   7. +-commutative 98.8%

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

   8. distribute-lft-neg-in 98.8%

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

   9. metadata-eval 98.8%

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

   10. *-commutative 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in maxCos around 0 97.3%

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

7. Final simplification 97.3%

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

Alternative 4: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00011999999696854502:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(maxCos \cdot \left(2 \cdot ux - ux \cdot maxCos\right) - ux\right) + \left(2 - 2 \cdot maxCos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.00011999999696854502)
   (sqrt
    (*
     ux
     (+
      (- (* maxCos (- (* 2.0 ux) (* ux maxCos))) ux)
      (- 2.0 (* 2.0 maxCos)))))
   (* (cos (* PI (* uy 2.0))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.00011999999696854502f) {
		tmp = sqrtf((ux * (((maxCos * ((2.0f * ux) - (ux * maxCos))) - ux) + (2.0f - (2.0f * maxCos)))));
	} else {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.00011999999696854502))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(maxCos * Float32(Float32(Float32(2.0) * ux) - Float32(ux * maxCos))) - ux) + Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.00011999999696854502))
		tmp = sqrt((ux * (((maxCos * ((single(2.0) * ux) - (ux * maxCos))) - ux) + (single(2.0) - (single(2.0) * maxCos)))));
	else
		tmp = cos((single(pi) * (uy * single(2.0)))) * sqrt((ux * (single(2.0) - ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 1.19999997e-4

   1. Initial program 59.2%

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

      1. associate-*l* 59.2%

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

      2. sub-neg 59.2%

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

      3. +-commutative 59.2%

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

      4. distribute-rgt-neg-in 59.2%

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

      5. fma-define 59.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in ux around 0 99.6%

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

   6. Taylor expanded in maxCos around 0 99.6%

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

   7. Taylor expanded in uy around 0 99.6%

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

   if 1.19999997e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.1%

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

      1. associate-*l* 55.1%

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

      2. sub-neg 55.1%

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

      3. +-commutative 55.1%

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

      4. distribute-rgt-neg-in 55.1%

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

      5. fma-define 55.6%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in ux around 0 97.7%

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

   6. Taylor expanded in maxCos around 0 97.7%

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

   7. Taylor expanded in maxCos around 0 92.4%

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

      1. *-commutative 92.4%

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

      2. associate-*r* 92.4%

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

      3. neg-mul-1 92.4%

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

      4. unsub-neg 92.4%

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

   9. Simplified 92.4%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00011999999696854502:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(maxCos \cdot \left(2 \cdot ux - ux \cdot maxCos\right) - ux\right) + \left(2 - 2 \cdot maxCos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - 2 \cdot maxCos\\ \mathbf{if}\;uy \leq 0.0004199999966658652:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(maxCos \cdot \left(2 \cdot ux - ux \cdot maxCos\right) - ux\right) + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot t\_0} \cdot \left(1 + -2 \cdot \left(\left(uy \cdot \pi\right) \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- 2.0 (* 2.0 maxCos))))
   (if (<= uy 0.0004199999966658652)
     (sqrt (* ux (+ (- (* maxCos (- (* 2.0 ux) (* ux maxCos))) ux) t_0)))
     (* (sqrt (* ux t_0)) (+ 1.0 (* -2.0 (* (* uy PI) (* uy PI))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = 2.0f - (2.0f * maxCos);
	float tmp;
	if (uy <= 0.0004199999966658652f) {
		tmp = sqrtf((ux * (((maxCos * ((2.0f * ux) - (ux * maxCos))) - ux) + t_0)));
	} else {
		tmp = sqrtf((ux * t_0)) * (1.0f + (-2.0f * ((uy * ((float) M_PI)) * (uy * ((float) M_PI)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))
	tmp = Float32(0.0)
	if (uy <= Float32(0.0004199999966658652))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(maxCos * Float32(Float32(Float32(2.0) * ux) - Float32(ux * maxCos))) - ux) + t_0)));
	else
		tmp = Float32(sqrt(Float32(ux * t_0)) * Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * Float32(pi)) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(2.0) - (single(2.0) * maxCos);
	tmp = single(0.0);
	if (uy <= single(0.0004199999966658652))
		tmp = sqrt((ux * (((maxCos * ((single(2.0) * ux) - (ux * maxCos))) - ux) + t_0)));
	else
		tmp = sqrt((ux * t_0)) * (single(1.0) + (single(-2.0) * ((uy * single(pi)) * (uy * single(pi)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 4.19999997e-4

   1. Initial program 58.6%

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

      1. associate-*l* 58.6%

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

      2. sub-neg 58.6%

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

      3. +-commutative 58.6%

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

      4. distribute-rgt-neg-in 58.6%

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

      5. fma-define 58.9%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in ux around 0 99.5%

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

   6. Taylor expanded in maxCos around 0 99.5%

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

   7. Taylor expanded in uy around 0 98.7%

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

   if 4.19999997e-4 < uy

   1. Initial program 55.6%

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

   3. Taylor expanded in ux around 0 77.9%

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

   4. Taylor expanded in uy around 0 58.4%

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

      1. unpow2 58.4%

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

      2. unpow2 58.4%

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

      3. swap-sqr 58.4%

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

      4. unpow1 58.4%

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

      5. pow-plus 58.4%

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

      6. metadata-eval 58.4%

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

   6. Simplified 58.4%

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

      1. unpow2 58.4%

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

   8. Applied egg-rr 58.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0004199999966658652:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(maxCos \cdot \left(2 \cdot ux - ux \cdot maxCos\right) - ux\right) + \left(2 - 2 \cdot maxCos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left(1 + -2 \cdot \left(\left(uy \cdot \pi\right) \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - 2 \cdot maxCos\\ \mathbf{if}\;uy \leq 0.0004199999966658652:\\ \;\;\;\;\sqrt{ux \cdot \left(t\_0 + ux \cdot \left(-1 + maxCos \cdot \left(2 - maxCos\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot t\_0} \cdot \left(1 + -2 \cdot \left(\left(uy \cdot \pi\right) \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- 2.0 (* 2.0 maxCos))))
   (if (<= uy 0.0004199999966658652)
     (sqrt (* ux (+ t_0 (* ux (+ -1.0 (* maxCos (- 2.0 maxCos)))))))
     (* (sqrt (* ux t_0)) (+ 1.0 (* -2.0 (* (* uy PI) (* uy PI))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = 2.0f - (2.0f * maxCos);
	float tmp;
	if (uy <= 0.0004199999966658652f) {
		tmp = sqrtf((ux * (t_0 + (ux * (-1.0f + (maxCos * (2.0f - maxCos)))))));
	} else {
		tmp = sqrtf((ux * t_0)) * (1.0f + (-2.0f * ((uy * ((float) M_PI)) * (uy * ((float) M_PI)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))
	tmp = Float32(0.0)
	if (uy <= Float32(0.0004199999966658652))
		tmp = sqrt(Float32(ux * Float32(t_0 + Float32(ux * Float32(Float32(-1.0) + Float32(maxCos * Float32(Float32(2.0) - maxCos)))))));
	else
		tmp = Float32(sqrt(Float32(ux * t_0)) * Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * Float32(pi)) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(2.0) - (single(2.0) * maxCos);
	tmp = single(0.0);
	if (uy <= single(0.0004199999966658652))
		tmp = sqrt((ux * (t_0 + (ux * (single(-1.0) + (maxCos * (single(2.0) - maxCos)))))));
	else
		tmp = sqrt((ux * t_0)) * (single(1.0) + (single(-2.0) * ((uy * single(pi)) * (uy * single(pi)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 4.19999997e-4

   1. Initial program 58.6%

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

      1. associate-*l* 58.6%

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

      2. sub-neg 58.6%

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

      3. +-commutative 58.6%

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

      4. distribute-rgt-neg-in 58.6%

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

      5. fma-define 58.9%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in ux around 0 99.5%

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

   6. Taylor expanded in maxCos around 0 99.5%

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

   7. Taylor expanded in ux around 0 99.5%

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

   8. Taylor expanded in uy around 0 98.7%

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

   if 4.19999997e-4 < uy

   1. Initial program 55.6%

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

   3. Taylor expanded in ux around 0 77.9%

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

   4. Taylor expanded in uy around 0 58.4%

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

      1. unpow2 58.4%

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

      2. unpow2 58.4%

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

      3. swap-sqr 58.4%

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

      4. unpow1 58.4%

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

      5. pow-plus 58.4%

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

      6. metadata-eval 58.4%

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

   6. Simplified 58.4%

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

      1. unpow2 58.4%

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

   8. Applied egg-rr 58.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0004199999966658652:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 - 2 \cdot maxCos\right) + ux \cdot \left(-1 + maxCos \cdot \left(2 - maxCos\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left(1 + -2 \cdot \left(\left(uy \cdot \pi\right) \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left(1 + -2 \cdot \left(\left(uy \cdot \pi\right) \cdot \left(uy \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 1.9999999494757503e-5)
   (*
    (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
    (+ 1.0 (* -2.0 (* (* uy PI) (* uy PI)))))
   (sqrt (* ux (- 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 1.9999999494757503e-5f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos)))) * (1.0f + (-2.0f * ((uy * ((float) M_PI)) * (uy * ((float) M_PI)))));
	} else {
		tmp = sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(1.9999999494757503e-5))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))) * Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * Float32(pi)) * Float32(uy * Float32(pi))))));
	else
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(1.9999999494757503e-5))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos)))) * (single(1.0) + (single(-2.0) * ((uy * single(pi)) * (uy * single(pi)))));
	else
		tmp = sqrt((ux * (single(2.0) - ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-5

   1. Initial program 31.1%

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

   3. Taylor expanded in ux around 0 95.9%

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

   4. Taylor expanded in uy around 0 86.3%

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

      1. unpow2 86.3%

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

      2. unpow2 86.3%

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

      3. swap-sqr 86.3%

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

      4. unpow1 86.3%

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

      5. pow-plus 86.3%

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

      6. metadata-eval 86.3%

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

   6. Simplified 86.3%

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

      1. unpow2 86.3%

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

   8. Applied egg-rr 86.3%

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

   if 1.99999995e-5 < ux

   1. Initial program 84.4%

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

      1. associate-*l* 84.4%

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

      2. sub-neg 84.4%

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

      3. +-commutative 84.4%

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

      4. distribute-rgt-neg-in 84.4%

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

      5. fma-define 85.1%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in ux around 0 98.8%

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

      1. add-log-exp 97.1%

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

   7. Applied egg-rr 97.1%

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

   8. Taylor expanded in maxCos around 0 91.8%

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

      1. neg-mul-1 91.8%

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

   10. Simplified 91.8%

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

   11. Taylor expanded in uy around 0 75.9%

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

Alternative 8: 75.2% 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 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.9%

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

3. Simplified 57.9%

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

5. Taylor expanded in ux around 0 98.8%

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

   1. add-log-exp 97.2%

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

7. Applied egg-rr 97.2%

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

8. Taylor expanded in maxCos around 0 91.5%

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

   1. neg-mul-1 91.5%

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

10. Simplified 91.5%

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

11. Taylor expanded in uy around 0 75.6%

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

Alternative 9: 6.6% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 0.0)

C

float code(float ux, float uy, float maxCos) {
	return 0.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 0.0e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(0.0)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(0.0);
end

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.9%

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

3. Simplified 57.9%

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

5. Taylor expanded in uy around 0 49.5%

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

7. Simplified 49.5%

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

8. Taylor expanded in ux around 0 6.6%

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

   1. pow1/2 6.6%

      \[\leadsto \color{blue}{{\left(1 + \left(-1\right)\right)}^{0.5}} \]

   2. metadata-eval 6.6%

      \[\leadsto {\left(1 + \color{blue}{-1}\right)}^{0.5} \]

   3. metadata-eval 6.6%

      \[\leadsto {\color{blue}{0}}^{0.5} \]

   4. metadata-eval 6.6%

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

10. Applied egg-rr 6.6%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024117 
(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)))))))