UniformSampleCone, x

Percentage Accurate: 57.3% → 98.9%

Time: 13.9s

Alternatives: 14

Speedup: 2.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-define 55.8%

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

3. Simplified 55.9%

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

5. Taylor expanded in ux around 0 99.1%

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

   1. add-log-exp 99.1%

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

   2. mul-1-neg 99.1%

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

   3. sub-neg 99.1%

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

   4. metadata-eval 99.1%

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

   5. +-commutative 99.1%

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

7. Applied egg-rr 99.1%

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

8. Taylor expanded in ux around inf 99.1%

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

9. Final simplification 99.1%

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

Alternative 2: 95.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.999998927116394:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot {\left(ux \cdot \left(2 - ux\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \sqrt{\left(\left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right) + \frac{1}{ux}\right) - \frac{-1 + maxCos}{ux}\right) - \frac{maxCos}{ux}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* PI (* uy 2.0))) 0.999998927116394)
   (* (cos (* 2.0 (* uy PI))) (pow (* ux (- 2.0 ux)) 0.5))
   (*
    ux
    (sqrt
     (-
      (-
       (+ (* (+ -1.0 maxCos) (- 1.0 maxCos)) (/ 1.0 ux))
       (/ (+ -1.0 maxCos) ux))
      (/ maxCos ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf((((float) M_PI) * (uy * 2.0f))) <= 0.999998927116394f) {
		tmp = cosf((2.0f * (uy * ((float) M_PI)))) * powf((ux * (2.0f - ux)), 0.5f);
	} else {
		tmp = ux * sqrtf((((((-1.0f + maxCos) * (1.0f - maxCos)) + (1.0f / ux)) - ((-1.0f + maxCos) / ux)) - (maxCos / ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) <= Float32(0.999998927116394))
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * (Float32(ux * Float32(Float32(2.0) - ux)) ^ Float32(0.5)));
	else
		tmp = Float32(ux * sqrt(Float32(Float32(Float32(Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(1.0) - maxCos)) + Float32(Float32(1.0) / ux)) - Float32(Float32(Float32(-1.0) + maxCos) / ux)) - Float32(maxCos / ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (cos((single(pi) * (uy * single(2.0)))) <= single(0.999998927116394))
		tmp = cos((single(2.0) * (uy * single(pi)))) * ((ux * (single(2.0) - ux)) ^ single(0.5));
	else
		tmp = ux * sqrt((((((single(-1.0) + maxCos) * (single(1.0) - maxCos)) + (single(1.0) / ux)) - ((single(-1.0) + maxCos) / ux)) - (maxCos / ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999998927

   1. Initial program 56.8%

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

      1. associate-*l* 56.8%

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

      2. sub-neg 56.8%

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

      3. +-commutative 56.8%

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

      4. distribute-rgt-neg-in 56.8%

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

      5. fma-define 57.2%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in ux around 0 98.2%

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

      1. add-log-exp 98.3%

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

      2. mul-1-neg 98.3%

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

      3. sub-neg 98.3%

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

      4. metadata-eval 98.3%

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

      5. +-commutative 98.3%

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

   7. Applied egg-rr 98.3%

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

   8. Taylor expanded in maxCos around 0 91.9%

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

      1. pow1/2 91.9%

         \[\leadsto \color{blue}{{\left(ux \cdot \left(2 - ux\right)\right)}^{0.5}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   10. Applied egg-rr 91.9%

       \[\leadsto \color{blue}{{\left(ux \cdot \left(2 - ux\right)\right)}^{0.5}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   if 0.999998927 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 55.0%

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

      1. associate-*l* 55.0%

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

      2. sub-neg 55.0%

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

      3. +-commutative 55.0%

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

      4. distribute-rgt-neg-in 55.0%

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

      5. fma-define 54.9%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in ux around inf 99.2%

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

   6. Taylor expanded in uy around 0 98.4%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.999998927116394:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot {\left(ux \cdot \left(2 - ux\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \sqrt{\left(\left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right) + \frac{1}{ux}\right) - \frac{-1 + maxCos}{ux}\right) - \frac{maxCos}{ux}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.999998927116394:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \sqrt{\left(\left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right) + \frac{1}{ux}\right) - \frac{-1 + maxCos}{ux}\right) - \frac{maxCos}{ux}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* PI (* uy 2.0))) 0.999998927116394)
   (* (cos (* 2.0 (* uy PI))) (sqrt (* ux (- 2.0 ux))))
   (*
    ux
    (sqrt
     (-
      (-
       (+ (* (+ -1.0 maxCos) (- 1.0 maxCos)) (/ 1.0 ux))
       (/ (+ -1.0 maxCos) ux))
      (/ maxCos ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf((((float) M_PI) * (uy * 2.0f))) <= 0.999998927116394f) {
		tmp = cosf((2.0f * (uy * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = ux * sqrtf((((((-1.0f + maxCos) * (1.0f - maxCos)) + (1.0f / ux)) - ((-1.0f + maxCos) / ux)) - (maxCos / ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) <= Float32(0.999998927116394))
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = Float32(ux * sqrt(Float32(Float32(Float32(Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(1.0) - maxCos)) + Float32(Float32(1.0) / ux)) - Float32(Float32(Float32(-1.0) + maxCos) / ux)) - Float32(maxCos / ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (cos((single(pi) * (uy * single(2.0)))) <= single(0.999998927116394))
		tmp = cos((single(2.0) * (uy * single(pi)))) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = ux * sqrt((((((single(-1.0) + maxCos) * (single(1.0) - maxCos)) + (single(1.0) / ux)) - ((single(-1.0) + maxCos) / ux)) - (maxCos / ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999998927

   1. Initial program 56.8%

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

      1. associate-*l* 56.8%

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

      2. sub-neg 56.8%

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

      3. +-commutative 56.8%

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

      4. distribute-rgt-neg-in 56.8%

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

      5. fma-define 57.2%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in ux around 0 98.2%

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

      1. add-log-exp 98.3%

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

      2. mul-1-neg 98.3%

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

      3. sub-neg 98.3%

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

      4. metadata-eval 98.3%

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

      5. +-commutative 98.3%

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

   7. Applied egg-rr 98.3%

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

   8. Taylor expanded in maxCos around 0 91.9%

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

   if 0.999998927 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 55.0%

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

      1. associate-*l* 55.0%

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

      2. sub-neg 55.0%

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

      3. +-commutative 55.0%

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

      4. distribute-rgt-neg-in 55.0%

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

      5. fma-define 54.9%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in ux around inf 99.2%

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

   6. Taylor expanded in uy around 0 98.4%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.999998927116394:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \sqrt{\left(\left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right) + \frac{1}{ux}\right) - \frac{-1 + maxCos}{ux}\right) - \frac{maxCos}{ux}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999960064888:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \sqrt{\left(\left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right) + \frac{1}{ux}\right) - \frac{-1 + maxCos}{ux}\right) - \frac{maxCos}{ux}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* PI (* uy 2.0))) 0.9999960064888)
   (* (cos (* 2.0 (* uy PI))) (sqrt (* 2.0 ux)))
   (*
    ux
    (sqrt
     (-
      (-
       (+ (* (+ -1.0 maxCos) (- 1.0 maxCos)) (/ 1.0 ux))
       (/ (+ -1.0 maxCos) ux))
      (/ maxCos ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf((((float) M_PI) * (uy * 2.0f))) <= 0.9999960064888f) {
		tmp = cosf((2.0f * (uy * ((float) M_PI)))) * sqrtf((2.0f * ux));
	} else {
		tmp = ux * sqrtf((((((-1.0f + maxCos) * (1.0f - maxCos)) + (1.0f / ux)) - ((-1.0f + maxCos) / ux)) - (maxCos / ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) <= Float32(0.9999960064888))
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	else
		tmp = Float32(ux * sqrt(Float32(Float32(Float32(Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(1.0) - maxCos)) + Float32(Float32(1.0) / ux)) - Float32(Float32(Float32(-1.0) + maxCos) / ux)) - Float32(maxCos / ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (cos((single(pi) * (uy * single(2.0)))) <= single(0.9999960064888))
		tmp = cos((single(2.0) * (uy * single(pi)))) * sqrt((single(2.0) * ux));
	else
		tmp = ux * sqrt((((((single(-1.0) + maxCos) * (single(1.0) - maxCos)) + (single(1.0) / ux)) - ((single(-1.0) + maxCos) / ux)) - (maxCos / ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999996006

   1. Initial program 57.1%

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

      1. associate-*l* 57.1%

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

      2. sub-neg 57.1%

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

      3. +-commutative 57.1%

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

      4. distribute-rgt-neg-in 57.1%

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

      5. fma-define 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in ux around 0 98.2%

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

      1. add-log-exp 98.2%

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

      2. mul-1-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. metadata-eval 98.2%

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

      5. +-commutative 98.2%

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

   7. Applied egg-rr 98.2%

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

   8. Taylor expanded in maxCos around 0 91.4%

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

   9. Taylor expanded in ux around 0 71.8%

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

       1. *-commutative 71.8%

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

   11. Simplified 71.8%

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

   if 0.999996006 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 55.0%

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

      1. associate-*l* 55.0%

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

      2. sub-neg 55.0%

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

      3. +-commutative 55.0%

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

      4. distribute-rgt-neg-in 55.0%

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

      5. fma-define 54.8%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in ux around inf 99.2%

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

   6. Taylor expanded in uy around 0 97.5%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999960064888:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \sqrt{\left(\left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right) + \frac{1}{ux}\right) - \frac{-1 + maxCos}{ux}\right) - \frac{maxCos}{ux}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(ux + Float32(maxCos * Float32(Float32(ux * maxCos) + Float32(Float32(2.0) + Float32(ux * Float32(-2.0))))))))))
end

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-define 55.8%

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

3. Simplified 55.9%

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

5. Taylor expanded in ux around 0 99.1%

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

   1. add-log-exp 99.1%

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

   2. mul-1-neg 99.1%

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

   3. sub-neg 99.1%

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

   4. metadata-eval 99.1%

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

   5. +-commutative 99.1%

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

7. Applied egg-rr 99.1%

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

8. Taylor expanded in maxCos around 0 99.0%

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

   1. associate--l+ 99.0%

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

   2. mul-1-neg 99.0%

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

   3. *-commutative 99.0%

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

   4. *-commutative 99.0%

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 6: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.4048999901206116 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(ux \cdot \left(-1 + \frac{2}{ux}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.4048999901206116e-6)
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux (* ux (+ -1.0 (/ 2.0 ux))))))
   (* (sqrt (* ux (- 2.0 (* 2.0 maxCos)))) (cos (* 2.0 (* uy PI))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.4048999901206116e-6f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (ux * (-1.0f + (2.0f / ux)))));
	} else {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos)))) * cosf((2.0f * (uy * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.4048999901206116e-6))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(ux * Float32(Float32(-1.0) + Float32(Float32(2.0) / ux))))));
	else
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))) * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(1.4048999901206116e-6))
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((ux * (ux * (single(-1.0) + (single(2.0) / ux)))));
	else
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos)))) * cos((single(2.0) * (uy * single(pi))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.40489999e-6

   1. Initial program 57.2%

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

      1. associate-*l* 57.2%

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

      2. sub-neg 57.2%

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

      3. +-commutative 57.2%

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

      4. distribute-rgt-neg-in 57.2%

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

      5. fma-define 57.3%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in ux around 0 99.0%

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

      1. add-log-exp 99.1%

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

      2. mul-1-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. +-commutative 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in ux around inf 99.0%

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

   9. Taylor expanded in maxCos around 0 98.8%

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

       1. sub-neg 98.8%

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

       2. metadata-eval 98.8%

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

       3. +-commutative 98.8%

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

       4. associate-*r/ 98.8%

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

       5. metadata-eval 98.8%

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

   11. Simplified 98.8%

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

   if 1.40489999e-6 < maxCos

   1. Initial program 47.7%

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

   3. Taylor expanded in ux around 0 85.0%

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

Alternative 7: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(ux + Float32(maxCos * Float32(Float32(2.0) + Float32(ux * Float32(-2.0)))))))))
end

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-define 55.8%

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

3. Simplified 55.9%

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

5. Taylor expanded in ux around 0 99.1%

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

   1. add-log-exp 99.1%

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

   2. mul-1-neg 99.1%

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

   3. sub-neg 99.1%

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

   4. metadata-eval 99.1%

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

   5. +-commutative 99.1%

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

7. Applied egg-rr 99.1%

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

8. Taylor expanded in maxCos around 0 98.6%

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

   1. associate--l+ 98.6%

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

   2. mul-1-neg 98.6%

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

   3. *-commutative 98.6%

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

10. Simplified 98.6%

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

11. Final simplification 98.6%

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

Alternative 8: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{if}\;maxCos \leq 1.4048999901206116 \cdot 10^{-6}:\\ \;\;\;\;t\_0 \cdot {\left(ux \cdot \left(2 - ux\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* 2.0 (* uy PI)))))
   (if (<= maxCos 1.4048999901206116e-6)
     (* t_0 (pow (* ux (- 2.0 ux)) 0.5))
     (* (sqrt (* ux (- 2.0 (* 2.0 maxCos)))) t_0))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf((2.0f * (uy * ((float) M_PI))));
	float tmp;
	if (maxCos <= 1.4048999901206116e-6f) {
		tmp = t_0 * powf((ux * (2.0f - ux)), 0.5f);
	} else {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos)))) * t_0;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(2.0) * Float32(uy * Float32(pi))))
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.4048999901206116e-6))
		tmp = Float32(t_0 * (Float32(ux * Float32(Float32(2.0) - ux)) ^ Float32(0.5)));
	else
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = cos((single(2.0) * (uy * single(pi))));
	tmp = single(0.0);
	if (maxCos <= single(1.4048999901206116e-6))
		tmp = t_0 * ((ux * (single(2.0) - ux)) ^ single(0.5));
	else
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos)))) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.40489999e-6

   1. Initial program 57.2%

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

      1. associate-*l* 57.2%

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

      2. sub-neg 57.2%

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

      3. +-commutative 57.2%

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

      4. distribute-rgt-neg-in 57.2%

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

      5. fma-define 57.3%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in ux around 0 99.0%

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

      1. add-log-exp 99.1%

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

      2. mul-1-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. +-commutative 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in maxCos around 0 98.8%

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

      1. pow1/2 98.8%

         \[\leadsto \color{blue}{{\left(ux \cdot \left(2 - ux\right)\right)}^{0.5}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   10. Applied egg-rr 98.8%

       \[\leadsto \color{blue}{{\left(ux \cdot \left(2 - ux\right)\right)}^{0.5}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   if 1.40489999e-6 < maxCos

   1. Initial program 47.7%

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

   3. Taylor expanded in ux around 0 85.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.4048999901206116 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot {\left(ux \cdot \left(2 - ux\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

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)));
}

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)) - ux))))
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)));
end

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-define 55.8%

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

3. Simplified 55.9%

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

5. Taylor expanded in ux around 0 99.1%

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

6. Taylor expanded in maxCos around 0 98.1%

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

   1. neg-mul-1 98.1%

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

8. Simplified 98.1%

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

9. Final simplification 98.1%

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

Alternative 10: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) - ux) + Float32(maxCos * Float32(-2.0))))))
end

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-define 55.8%

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

3. Simplified 55.9%

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

5. Taylor expanded in ux around 0 99.1%

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

6. Taylor expanded in maxCos around 0 98.1%

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

   1. neg-mul-1 98.1%

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

8. Simplified 98.1%

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

9. Taylor expanded in maxCos around 0 98.0%

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

    1. +-commutative 98.0%

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

    2. *-commutative 98.0%

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

    3. associate-*r* 98.0%

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

    4. distribute-rgt-out 98.0%

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

11. Simplified 98.0%

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

12. Final simplification 98.0%

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

Alternative 11: 80.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-define 55.8%

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

3. Simplified 55.9%

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

5. Taylor expanded in ux around inf 98.8%

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

6. Taylor expanded in uy around 0 78.6%

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

7. Final simplification 78.6%

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

Alternative 12: 79.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-define 55.8%

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

3. Simplified 55.9%

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

5. Taylor expanded in ux around 0 99.1%

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

6. Taylor expanded in maxCos around 0 98.1%

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

   1. neg-mul-1 98.1%

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

8. Simplified 98.1%

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

9. Taylor expanded in uy around 0 78.4%

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

10. Final simplification 78.4%

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

Alternative 13: 76.1% 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 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-define 55.8%

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

3. Simplified 55.9%

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

5. Taylor expanded in ux around 0 99.1%

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

   1. add-log-exp 99.1%

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

   2. mul-1-neg 99.1%

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

   3. sub-neg 99.1%

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

   4. metadata-eval 99.1%

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

   5. +-commutative 99.1%

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

7. Applied egg-rr 99.1%

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

8. Taylor expanded in maxCos around 0 92.4%

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

9. Taylor expanded in uy around 0 74.5%

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

Alternative 14: 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 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-define 55.8%

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

3. Simplified 55.9%

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

5. Taylor expanded in uy around 0 46.4%

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

   1. mul-1-neg 46.4%

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

   2. unsub-neg 46.4%

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

   3. sub-neg 46.4%

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

   4. metadata-eval 46.4%

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

   5. distribute-lft-in 46.4%

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

   6. *-commutative 46.4%

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

   7. mul-1-neg 46.4%

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

   8. sub-neg 46.4%

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

   9. *-commutative 46.4%

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

   10. associate--l+ 46.2%

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

   11. unpow2 46.2%

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

   12. sub-neg 46.2%

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

7. Simplified 46.4%

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

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 2024191 
(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)))))))