UniformSampleCone, x

Percentage Accurate: 58.1% → 98.9%

Time: 14.6s

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: 58.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.0%

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

3. Taylor expanded in ux around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in maxCos around 0 98.9%

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

   1. expm1-log1p-u 98.9%

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

   2. neg-mul-1 98.9%

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

   3. log1p-define 99.0%

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

   4. sub-neg 99.0%

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

   5. expm1-undefine 99.0%

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

   6. add-exp-log 99.0%

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 60.0%

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

   1. associate-*l* 60.0%

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

   2. sub-neg 60.0%

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

   3. +-commutative 60.0%

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

   4. distribute-rgt-neg-in 60.0%

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

   5. fma-define 60.0%

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

3. Simplified 60.1%

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

5. Taylor expanded in uy around inf 60.0%

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

7. Simplified 60.0%

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

8. Taylor expanded in ux around -inf 59.9%

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

   1. fma-define 59.9%

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

   2. mul-1-neg 59.9%

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

   3. fmm-undef 59.9%

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

   4. mul-1-neg 59.9%

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

   5. *-commutative 59.9%

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

   6. sub-neg 59.9%

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

   7. metadata-eval 59.9%

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

   8. neg-mul-1 59.9%

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

10. Simplified 59.2%

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

11. Taylor expanded in ux around 0 98.9%

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

    1. associate--l+ 98.9%

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

    2. *-commutative 98.9%

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

    3. sub-neg 98.9%

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

    4. metadata-eval 98.9%

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

    5. +-commutative 98.9%

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

13. Simplified 98.9%

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

14. Final simplification 98.9%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.0%

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

3. Taylor expanded in ux around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in maxCos around 0 97.7%

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

7. Final simplification 97.7%

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

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.0%

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

3. Taylor expanded in ux around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in maxCos around 0 98.9%

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

   1. expm1-log1p-u 98.9%

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

   2. neg-mul-1 98.9%

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

   3. log1p-define 99.0%

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

   4. sub-neg 99.0%

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

   5. expm1-undefine 99.0%

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

   6. add-exp-log 99.0%

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

8. Applied egg-rr 99.0%

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

9. Taylor expanded in ux around 0 96.8%

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

    1. *-commutative 96.8%

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

11. Simplified 96.8%

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

12. Final simplification 96.8%

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

Alternative 5: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0004600000102072954:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) - 2 \cdot maxCos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(uy \cdot \left(\pi \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0004600000102072954)
   (sqrt
    (*
     ux
     (+ 2.0 (- (* ux (* (+ maxCos -1.0) (- 1.0 maxCos))) (* 2.0 maxCos)))))
   (* (sqrt (* ux (- 2.0 ux))) (cos (* uy (* PI -2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0004600000102072954f) {
		tmp = sqrtf((ux * (2.0f + ((ux * ((maxCos + -1.0f) * (1.0f - maxCos))) - (2.0f * maxCos)))));
	} else {
		tmp = sqrtf((ux * (2.0f - ux))) * cosf((uy * (((float) M_PI) * -2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0004600000102072954))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(ux * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos))) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - ux))) * cos(Float32(uy * Float32(Float32(pi) * Float32(-2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0004600000102072954))
		tmp = sqrt((ux * (single(2.0) + ((ux * ((maxCos + single(-1.0)) * (single(1.0) - maxCos))) - (single(2.0) * maxCos)))));
	else
		tmp = sqrt((ux * (single(2.0) - ux))) * cos((uy * (single(pi) * single(-2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 4.6000001e-4

   1. Initial program 60.1%

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

      1. associate-*l* 60.1%

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

      2. sub-neg 60.1%

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

      3. +-commutative 60.1%

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

      4. distribute-rgt-neg-in 60.1%

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

      5. fma-define 60.1%

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

   3. Simplified 60.2%

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

   5. Taylor expanded in uy around inf 60.0%

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

   7. Simplified 60.1%

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

   8. Taylor expanded in ux around -inf 60.3%

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

      1. fma-define 60.3%

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

      2. mul-1-neg 60.3%

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

      3. fmm-undef 60.3%

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

      4. mul-1-neg 60.3%

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

      5. *-commutative 60.3%

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

      6. sub-neg 60.3%

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

      7. metadata-eval 60.3%

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

      8. neg-mul-1 60.3%

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

   10. Simplified 59.2%

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

   11. Taylor expanded in ux around 0 99.4%

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

       1. associate--l+ 99.4%

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

       2. *-commutative 99.4%

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

       3. sub-neg 99.4%

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

       4. metadata-eval 99.4%

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

       5. +-commutative 99.4%

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

   13. Simplified 99.4%

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

   14. Taylor expanded in uy around 0 98.7%

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

       1. associate--l+ 98.8%

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

       2. sub-neg 98.8%

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

       3. metadata-eval 98.8%

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

       4. +-commutative 98.8%

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

   16. Simplified 98.8%

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

   if 4.6000001e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 59.8%

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

   3. Taylor expanded in ux around 0 98.1%

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

      1. associate--l+ 98.1%

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

      2. associate-*r* 98.1%

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

      3. mul-1-neg 98.1%

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

      4. sub-neg 98.1%

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

      5. metadata-eval 98.1%

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

      6. +-commutative 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in maxCos around 0 98.1%

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

      1. associate-*r* 98.1%

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

      2. add-sqr-sqrt 97.5%

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

      3. pow2 97.5%

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

      4. associate-*r* 97.5%

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

      5. *-commutative 97.5%

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

   8. Applied egg-rr 97.5%

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

   9. Taylor expanded in maxCos around 0 89.4%

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

       1. neg-mul-1 89.4%

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

       2. sub-neg 89.4%

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

       3. associate-*r* 89.0%

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

       4. unpow2 89.0%

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

       5. rem-square-sqrt 89.9%

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

       6. *-commutative 89.9%

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

       7. *-commutative 89.9%

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

       8. metadata-eval 89.9%

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

       9. distribute-rgt-neg-in 89.9%

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

       10. rem-square-sqrt -0.0%

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

       11. unpow2 -0.0%

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

       12. associate-*r* -0.0%

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

       13. cos-neg -0.0%

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

       14. *-commutative -0.0%

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

       15. unpow2 -0.0%

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

       16. rem-square-sqrt 89.9%

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

   11. Simplified 89.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0004600000102072954:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) - 2 \cdot maxCos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(uy \cdot \left(\pi \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

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))));
}

Julia

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

Derivation

1. Initial program 60.0%

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

3. Taylor expanded in ux around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in maxCos around 0 98.9%

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

7. Taylor expanded in ux around 0 96.8%

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

   1. *-commutative 96.8%

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

9. Simplified 96.8%

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

10. Final simplification 96.8%

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

Alternative 7: 80.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.0%

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

   1. associate-*l* 60.0%

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

   2. sub-neg 60.0%

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

   3. +-commutative 60.0%

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

   4. distribute-rgt-neg-in 60.0%

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

   5. fma-define 60.0%

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

3. Simplified 60.1%

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

5. Taylor expanded in uy around inf 60.0%

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

7. Simplified 60.0%

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

8. Taylor expanded in ux around -inf 59.9%

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

   1. fma-define 59.9%

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

   2. mul-1-neg 59.9%

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

   3. fmm-undef 59.9%

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

   4. mul-1-neg 59.9%

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

   5. *-commutative 59.9%

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

   6. sub-neg 59.9%

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

   7. metadata-eval 59.9%

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

   8. neg-mul-1 59.9%

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

10. Simplified 59.2%

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

11. Taylor expanded in ux around 0 98.9%

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

    1. associate--l+ 98.9%

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

    2. *-commutative 98.9%

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

    3. sub-neg 98.9%

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

    4. metadata-eval 98.9%

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

    5. +-commutative 98.9%

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

13. Simplified 98.9%

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

14. Taylor expanded in uy around 0 79.3%

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

    1. associate--l+ 79.3%

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

    2. sub-neg 79.3%

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

    3. metadata-eval 79.3%

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

    4. +-commutative 79.3%

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

16. Simplified 79.3%

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

17. Final simplification 79.3%

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

Alternative 8: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.0%

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

3. Taylor expanded in ux around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in uy around 0 79.3%

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

7. Taylor expanded in maxCos around 0 79.3%

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

8. Taylor expanded in maxCos around 0 78.5%

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

9. Final simplification 78.5%

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

Alternative 9: 75.9% 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 60.0%

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

3. Taylor expanded in ux around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in uy around 0 79.3%

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

7. Taylor expanded in maxCos around 0 79.3%

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

8. Taylor expanded in maxCos around 0 74.3%

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

   1. neg-mul-1 74.3%

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

   2. sub-neg 74.3%

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

10. Simplified 74.3%

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

Alternative 10: 61.8% 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 60.0%

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

   1. associate-*l* 60.0%

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

   2. sub-neg 60.0%

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

   3. +-commutative 60.0%

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

   4. distribute-rgt-neg-in 60.0%

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

   5. fma-define 60.0%

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

3. Simplified 60.1%

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

5. Taylor expanded in uy around 0 50.8%

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

   1. mul-1-neg 50.8%

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

   2. unsub-neg 50.8%

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

   3. sub-neg 50.8%

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

   4. metadata-eval 50.8%

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

   5. distribute-lft-in 50.8%

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

   6. *-commutative 50.8%

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

   7. mul-1-neg 50.8%

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

   8. sub-neg 50.8%

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

   9. *-commutative 50.8%

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

   10. associate--l+ 50.6%

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

   11. unpow2 50.6%

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

   12. sub-neg 50.6%

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

7. Simplified 50.8%

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

8. Taylor expanded in maxCos around 0 48.7%

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

9. Taylor expanded in ux around 0 60.4%

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

    1. *-commutative 60.4%

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

11. Simplified 60.4%

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

12. Final simplification 60.4%

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

Reproduce

herbie shell --seed 2024149 
(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)))))))