Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.8% → 99.1%

Time: 12.3s

Alternatives: 8

Speedup: 4.0×

Specification

\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

Initial Program: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

Alternative 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (cos (* 2.0 (* PI u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * cosf((2.0f * (((float) M_PI) * u2)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(Float32(Float32(2.0) * Float32(Float32(pi) * u2))))
end

Derivation

1. Initial program 60.4%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 60.4%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 99.1%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. associate-*l* 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

4. Final simplification 99.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

Alternative 2: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999977350234985:\\ \;\;\;\;t_0 \cdot \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* 2.0 PI)))))
   (if (<= t_0 0.9999977350234985)
     (* t_0 (sqrt (- u1 (* u1 (* u1 -0.5)))))
     (sqrt (- (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * (2.0f * ((float) M_PI))));
	float tmp;
	if (t_0 <= 0.9999977350234985f) {
		tmp = t_0 * sqrtf((u1 - (u1 * (u1 * -0.5f))));
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999977350234985))
		tmp = Float32(t_0 * sqrt(Float32(u1 - Float32(u1 * Float32(u1 * Float32(-0.5))))));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2)) < 0.999997735

   1. Initial program 52.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0 88.0%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. +-commutative 49.4%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot 1 \]

      2. mul-1-neg 49.4%

         \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot 1 \]

      3. unsub-neg 49.4%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot 1 \]

      4. unpow2 49.4%

         \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot 1 \]

      5. associate-*r* 49.4%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot 1 \]

   4. Simplified 88.0%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.999997735 < (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2))

   1. Initial program 64.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. sub-neg 64.4%

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. log1p-def 99.6%

         \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. associate-*l* 99.6%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 99.6%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot \color{blue}{\log \left(e^{u2}\right)}\right)\right) \]

      2. log-pow 99.6%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \color{blue}{\log \left({\left(e^{u2}\right)}^{\pi}\right)}\right) \]

   5. Applied egg-rr 99.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \color{blue}{\log \left({\left(e^{u2}\right)}^{\pi}\right)}\right) \]

   6. Taylor expanded in u2 around 0 98.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \log \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999977350234985:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

Alternative 3: 87.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999935030937195:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right) + u1 \cdot \left(u1 \cdot u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* u2 (* 2.0 PI))) 0.9999935030937195)
   (* (cos (* PI (* 2.0 u2))) (sqrt u1))
   (sqrt (+ (log1p (fma u1 u1 u1)) (* u1 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((u2 * (2.0f * ((float) M_PI)))) <= 0.9999935030937195f) {
		tmp = cosf((((float) M_PI) * (2.0f * u2))) * sqrtf(u1);
	} else {
		tmp = sqrtf((log1pf(fmaf(u1, u1, u1)) + (u1 * (u1 * u1))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))) <= Float32(0.9999935030937195))
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) * sqrt(u1));
	else
		tmp = sqrt(Float32(log1p(fma(u1, u1, u1)) + Float32(u1 * Float32(u1 * u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2)) < 0.999993503

   1. Initial program 52.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. add-cube-cbrt 52.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right) \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}} \]

      2. pow3 52.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)}^{3}} \]

   3. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3}} \]

   4. Taylor expanded in u1 around 0 78.7%

      \[\leadsto \color{blue}{{1}^{0.16666666666666666} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1}\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 78.7%

         \[\leadsto \color{blue}{1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1}\right) \]

      2. associate-*r* 78.7%

         \[\leadsto \color{blue}{\left(1 \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \sqrt{u1}} \]

      3. *-lft-identity 78.7%

         \[\leadsto \color{blue}{\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)} \cdot \sqrt{u1} \]

      4. *-commutative 78.7%

         \[\leadsto \cos \left(2 \cdot \color{blue}{\left(\pi \cdot u2\right)}\right) \cdot \sqrt{u1} \]

      5. *-commutative 78.7%

         \[\leadsto \cos \color{blue}{\left(\left(\pi \cdot u2\right) \cdot 2\right)} \cdot \sqrt{u1} \]

      6. associate-*r* 78.7%

         \[\leadsto \cos \color{blue}{\left(\pi \cdot \left(u2 \cdot 2\right)\right)} \cdot \sqrt{u1} \]

      7. *-commutative 78.7%

         \[\leadsto \cos \left(\pi \cdot \color{blue}{\left(2 \cdot u2\right)}\right) \cdot \sqrt{u1} \]

   6. Simplified 78.7%

      \[\leadsto \color{blue}{\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1}} \]

   if 0.999993503 < (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2))

   1. Initial program 63.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 63.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]
   3. Step-by-step derivation

      1. flip3-- 60.4%

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)}} \cdot 1 \]

      2. div-inv 60.4%

         \[\leadsto \sqrt{-\log \color{blue}{\left(\left({1}^{3} - {u1}^{3}\right) \cdot \frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)}} \cdot 1 \]

      3. log-prod 60.3%

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left({1}^{3} - {u1}^{3}\right) + \log \left(\frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)}} \cdot 1 \]

      4. metadata-eval 60.3%

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - {u1}^{3}\right) + \log \left(\frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot 1 \]

      5. pow3 60.3%

         \[\leadsto \sqrt{-\left(\log \left(1 - \color{blue}{\left(u1 \cdot u1\right) \cdot u1}\right) + \log \left(\frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot 1 \]

      6. sub-neg 60.3%

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(-\left(u1 \cdot u1\right) \cdot u1\right)\right)} + \log \left(\frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot 1 \]

      7. distribute-rgt-neg-out 60.3%

         \[\leadsto \sqrt{-\left(\log \left(1 + \color{blue}{\left(u1 \cdot u1\right) \cdot \left(-u1\right)}\right) + \log \left(\frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot 1 \]

      8. add-sqr-sqrt -0.0%

         \[\leadsto \sqrt{-\left(\log \left(1 + \left(u1 \cdot u1\right) \cdot \color{blue}{\left(\sqrt{-u1} \cdot \sqrt{-u1}\right)}\right) + \log \left(\frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot 1 \]

      9. sqrt-unprod 46.3%

         \[\leadsto \sqrt{-\left(\log \left(1 + \left(u1 \cdot u1\right) \cdot \color{blue}{\sqrt{\left(-u1\right) \cdot \left(-u1\right)}}\right) + \log \left(\frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot 1 \]

      10. sqr-neg 46.3%

          \[\leadsto \sqrt{-\left(\log \left(1 + \left(u1 \cdot u1\right) \cdot \sqrt{\color{blue}{u1 \cdot u1}}\right) + \log \left(\frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot 1 \]

      11. sqrt-unprod 46.3%

          \[\leadsto \sqrt{-\left(\log \left(1 + \left(u1 \cdot u1\right) \cdot \color{blue}{\left(\sqrt{u1} \cdot \sqrt{u1}\right)}\right) + \log \left(\frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot 1 \]

      12. add-sqr-sqrt 46.3%

          \[\leadsto \sqrt{-\left(\log \left(1 + \left(u1 \cdot u1\right) \cdot \color{blue}{u1}\right) + \log \left(\frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot 1 \]

      13. log1p-udef 44.9%

          \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(u1 \cdot u1\right) \cdot u1\right)} + \log \left(\frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot 1 \]

      14. pow3 44.9%

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{{u1}^{3}}\right) + \log \left(\frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot 1 \]

      15. metadata-eval 44.9%

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({u1}^{3}\right) + \log \left(\frac{1}{\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot 1 \]

      16. *-un-lft-identity 44.9%

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({u1}^{3}\right) + \log \left(\frac{1}{1 + \left(u1 \cdot u1 + \color{blue}{u1}\right)}\right)\right)} \cdot 1 \]

      17. fma-def 44.9%

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({u1}^{3}\right) + \log \left(\frac{1}{1 + \color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}}\right)\right)} \cdot 1 \]

   4. Applied egg-rr 44.9%

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left({u1}^{3}\right) + \log \left(\frac{1}{1 + \mathsf{fma}\left(u1, u1, u1\right)}\right)\right)}} \cdot 1 \]
   5. Step-by-step derivation

      1. log-div 45.2%

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({u1}^{3}\right) + \color{blue}{\left(\log 1 - \log \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)\right)}\right)} \cdot 1 \]

      2. metadata-eval 45.2%

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({u1}^{3}\right) + \left(\color{blue}{0} - \log \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)\right)\right)} \cdot 1 \]

      3. log1p-def 81.0%

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({u1}^{3}\right) + \left(0 - \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}\right)\right)} \cdot 1 \]

      4. neg-sub0 81.0%

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({u1}^{3}\right) + \color{blue}{\left(-\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)}\right)} \cdot 1 \]

      5. sub-neg 81.0%

         \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left({u1}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)}} \cdot 1 \]

   6. Simplified 81.0%

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left({u1}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)}} \cdot 1 \]

   7. Taylor expanded in u1 around 0 80.9%

      \[\leadsto \sqrt{-\left(\color{blue}{{u1}^{3}} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]
   8. Step-by-step derivation

      1. unpow3 80.9%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot u1\right) \cdot u1} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]

      2. add-sqr-sqrt 80.9%

         \[\leadsto \sqrt{-\left(\left(u1 \cdot u1\right) \cdot \color{blue}{\left(\sqrt{u1} \cdot \sqrt{u1}\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]

      3. sqrt-unprod 80.9%

         \[\leadsto \sqrt{-\left(\left(u1 \cdot u1\right) \cdot \color{blue}{\sqrt{u1 \cdot u1}} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]

      4. sqr-neg 80.9%

         \[\leadsto \sqrt{-\left(\left(u1 \cdot u1\right) \cdot \sqrt{\color{blue}{\left(-u1\right) \cdot \left(-u1\right)}} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]

      5. sqrt-unprod -0.0%

         \[\leadsto \sqrt{-\left(\left(u1 \cdot u1\right) \cdot \color{blue}{\left(\sqrt{-u1} \cdot \sqrt{-u1}\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]

      6. add-sqr-sqrt 92.1%

         \[\leadsto \sqrt{-\left(\left(u1 \cdot u1\right) \cdot \color{blue}{\left(-u1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]

      7. neg-mul-1 92.1%

         \[\leadsto \sqrt{-\left(\left(u1 \cdot u1\right) \cdot \color{blue}{\left(-1 \cdot u1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]

      8. associate-*r* 92.1%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(\left(u1 \cdot u1\right) \cdot -1\right) \cdot u1} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]

   9. Applied egg-rr 92.1%

      \[\leadsto \sqrt{-\left(\color{blue}{\left(\left(u1 \cdot u1\right) \cdot -1\right) \cdot u1} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]
   10. Step-by-step derivation

       1. *-commutative 92.1%

          \[\leadsto \sqrt{-\left(\color{blue}{u1 \cdot \left(\left(u1 \cdot u1\right) \cdot -1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]

       2. associate-*l* 92.1%

          \[\leadsto \sqrt{-\left(u1 \cdot \color{blue}{\left(u1 \cdot \left(u1 \cdot -1\right)\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]

       3. *-commutative 92.1%

          \[\leadsto \sqrt{-\left(u1 \cdot \left(u1 \cdot \color{blue}{\left(-1 \cdot u1\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]

       4. mul-1-neg 92.1%

          \[\leadsto \sqrt{-\left(u1 \cdot \left(u1 \cdot \color{blue}{\left(-u1\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]

   11. Simplified 92.1%

       \[\leadsto \sqrt{-\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(-u1\right)\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999935030937195:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right) + u1 \cdot \left(u1 \cdot u1\right)}\\ \end{array} \]

Alternative 4: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.996999979019165:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.996999979019165)
   (sqrt (- (log (- 1.0 u1))))
   (* (cos (* u2 (* 2.0 PI))) (sqrt (- u1 (* u1 (* u1 -0.5)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.996999979019165f) {
		tmp = sqrtf(-logf((1.0f - u1)));
	} else {
		tmp = cosf((u2 * (2.0f * ((float) M_PI)))) * sqrtf((u1 - (u1 * (u1 * -0.5f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.996999979019165))
		tmp = sqrt(Float32(-log(Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(u1 - Float32(u1 * Float32(u1 * Float32(-0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.996999979019165))
		tmp = sqrt(-log((single(1.0) - u1)));
	else
		tmp = cos((u2 * (single(2.0) * single(pi)))) * sqrt((u1 - (u1 * (u1 * single(-0.5)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.996999979

   1. Initial program 95.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 83.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]

   if 0.996999979 < (-.f32 1 u1)

   1. Initial program 44.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0 98.6%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. +-commutative 80.6%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot 1 \]

      2. mul-1-neg 80.6%

         \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot 1 \]

      3. unsub-neg 80.6%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot 1 \]

      4. unpow2 80.6%

         \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot 1 \]

      5. associate-*r* 80.6%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot 1 \]

   4. Simplified 98.6%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.996999979019165:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \end{array} \]

Alternative 5: 85.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9980599880218506:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9980599880218506)
   (sqrt (- (log (- 1.0 u1))))
   (* (cos (* PI (* 2.0 u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9980599880218506f) {
		tmp = sqrtf(-logf((1.0f - u1)));
	} else {
		tmp = cosf((((float) M_PI) * (2.0f * u2))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9980599880218506))
		tmp = sqrt(Float32(-log(Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.9980599880218506))
		tmp = sqrt(-log((single(1.0) - u1)));
	else
		tmp = cos((single(pi) * (single(2.0) * u2))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.998059988

   1. Initial program 94.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 83.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]

   if 0.998059988 < (-.f32 1 u1)

   1. Initial program 43.5%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. add-cube-cbrt 43.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right) \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}} \]

      2. pow3 43.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)}^{3}} \]

   3. Applied egg-rr 86.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3}} \]

   4. Taylor expanded in u1 around 0 88.9%

      \[\leadsto \color{blue}{{1}^{0.16666666666666666} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1}\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 88.9%

         \[\leadsto \color{blue}{1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1}\right) \]

      2. associate-*r* 88.9%

         \[\leadsto \color{blue}{\left(1 \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \sqrt{u1}} \]

      3. *-lft-identity 88.9%

         \[\leadsto \color{blue}{\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)} \cdot \sqrt{u1} \]

      4. *-commutative 88.9%

         \[\leadsto \cos \left(2 \cdot \color{blue}{\left(\pi \cdot u2\right)}\right) \cdot \sqrt{u1} \]

      5. *-commutative 88.9%

         \[\leadsto \cos \color{blue}{\left(\left(\pi \cdot u2\right) \cdot 2\right)} \cdot \sqrt{u1} \]

      6. associate-*r* 88.9%

         \[\leadsto \cos \color{blue}{\left(\pi \cdot \left(u2 \cdot 2\right)\right)} \cdot \sqrt{u1} \]

      7. *-commutative 88.9%

         \[\leadsto \cos \left(\pi \cdot \color{blue}{\left(2 \cdot u2\right)}\right) \cdot \sqrt{u1} \]

   6. Simplified 88.9%

      \[\leadsto \color{blue}{\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9980599880218506:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 6: 79.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.996999979019165:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.996999979019165)
   (sqrt (- (log (- 1.0 u1))))
   (sqrt (- u1 (* u1 (* u1 -0.5))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.996999979019165f) {
		tmp = sqrtf(-logf((1.0f - u1)));
	} else {
		tmp = sqrtf((u1 - (u1 * (u1 * -0.5f))));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((1.0e0 - u1) <= 0.996999979019165e0) then
        tmp = sqrt(-log((1.0e0 - u1)))
    else
        tmp = sqrt((u1 - (u1 * (u1 * (-0.5e0)))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.996999979019165))
		tmp = sqrt(Float32(-log(Float32(Float32(1.0) - u1))));
	else
		tmp = sqrt(Float32(u1 - Float32(u1 * Float32(u1 * Float32(-0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.996999979019165))
		tmp = sqrt(-log((single(1.0) - u1)));
	else
		tmp = sqrt((u1 - (u1 * (u1 * single(-0.5)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.996999979

   1. Initial program 95.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 83.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]

   if 0.996999979 < (-.f32 1 u1)

   1. Initial program 44.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 40.6%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]

   3. Taylor expanded in u1 around 0 80.6%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot 1 \]
   4. Step-by-step derivation

      1. +-commutative 80.6%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot 1 \]

      2. mul-1-neg 80.6%

         \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot 1 \]

      3. unsub-neg 80.6%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot 1 \]

      4. unpow2 80.6%

         \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot 1 \]

      5. associate-*r* 80.6%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot 1 \]

   5. Simplified 80.6%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.996999979019165:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \end{array} \]

Alternative 7: 72.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (- u1 (* u1 (* u1 -0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 - (u1 * (u1 * -0.5f))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 - (u1 * (u1 * (-0.5e0)))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 - Float32(u1 * Float32(u1 * Float32(-0.5)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 - (u1 * (u1 * single(-0.5)))));
end

Derivation

1. Initial program 60.4%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Taylor expanded in u2 around 0 54.1%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]

3. Taylor expanded in u1 around 0 72.9%

   \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot 1 \]
4. Step-by-step derivation

   1. +-commutative 72.9%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot 1 \]

   2. mul-1-neg 72.9%

      \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot 1 \]

   3. unsub-neg 72.9%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot 1 \]

   4. unpow2 72.9%

      \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot 1 \]

   5. associate-*r* 72.9%

      \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot 1 \]

5. Simplified 72.9%

   \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot 1 \]

6. Final simplification 72.9%

   \[\leadsto \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \]

Alternative 8: 64.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

Derivation

1. Initial program 60.4%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. add-cube-cbrt 60.3%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right) \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}} \]

   2. pow3 60.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)}^{3}} \]

3. Applied egg-rr 71.9%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3}} \]

4. Taylor expanded in u2 around 0 38.1%

   \[\leadsto {\color{blue}{\left({\log \left(1 + u1\right)}^{0.16666666666666666}\right)}}^{3} \]

5. Taylor expanded in u1 around 0 64.0%

   \[\leadsto \color{blue}{\sqrt{u1}} \]

6. Final simplification 64.0%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023207 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))