Beckmann Sample, near normal, slope_x

Percentage Accurate: 58.1% → 99.6%

Time: 20.9s

Alternatives: 9

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: 58.1% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \langle \left( \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (* (sqrt (- (log1p (- u1)))) (cos (* 2.0 (* PI u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	double tmp = sqrt(-log1p(-((double) u1))) * cos((2.0 * (((double) M_PI) * ((double) u2))));
	return (float) tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float64(sqrt(Float64(-log1p(Float64(-Float64(u1))))) * cos(Float64(2.0 * Float64(pi * Float64(u2)))))
	return Float32(tmp)
end

Derivation

1. Initial program 99.5%

   \[\langle \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \rangle_{\text{binary64}} \]

2. Final simplification 99.5%

   \[\leadsto \langle \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \rangle_{\text{binary64}} \]

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \sqrt[3]{\left(\left(u2 \cdot u2\right) \cdot {\pi}^{2}\right) \cdot \left(\pi \cdot u2\right)}\right) \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * cosf((2.0f * cbrtf((((u2 * u2) * powf(((float) M_PI), 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) * cbrt(Float32(Float32(Float32(u2 * u2) * (Float32(pi) ^ Float32(2.0))) * Float32(Float32(pi) * u2))))))
end

Derivation

1. Initial program 53.3%

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

   1. sub-neg 53.3%

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

   2. log1p-def 99.0%

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

   3. associate-*l* 99.0%

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

3. Simplified 99.0%

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

   1. add-log-exp_binary32 99.0%

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

5. Applied rewrite-once 99.0%

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

   1. add-cbrt-cube_binary32 99.0%

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

7. Applied rewrite-once 99.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \color{blue}{\sqrt[3]{\left(\log \left(e^{\pi \cdot u2}\right) \cdot \log \left(e^{\pi \cdot u2}\right)\right) \cdot \log \left(e^{\pi \cdot u2}\right)}}\right) \]
8. Step-by-step derivation

   1. rem-log-exp 99.0%

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

   2. rem-log-exp 99.0%

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

   3. *-commutative 99.0%

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

   4. *-commutative 99.0%

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

   5. swap-sqr 99.0%

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

   6. unpow2 99.0%

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

   7. rem-log-exp 99.1%

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

   8. *-commutative 99.1%

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

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

Alternative 3: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \sqrt[3]{{u2}^{3} \cdot {\pi}^{3}}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (cos (* 2.0 (cbrt (* (pow u2 3.0) (pow PI 3.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * cosf((2.0f * cbrtf((powf(u2, 3.0f) * powf(((float) M_PI), 3.0f)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(Float32(Float32(2.0) * cbrt(Float32((u2 ^ Float32(3.0)) * (Float32(pi) ^ Float32(3.0)))))))
end

Derivation

1. Initial program 53.3%

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

   1. sub-neg 53.3%

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

   2. log1p-def 99.0%

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

   3. associate-*l* 99.0%

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

3. Simplified 99.0%

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

   1. add-log-exp_binary32 99.0%

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

5. Applied rewrite-once 99.0%

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

   1. add-cbrt-cube_binary32 99.0%

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

7. Applied rewrite-once 99.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \color{blue}{\sqrt[3]{\left(\log \left(e^{\pi \cdot u2}\right) \cdot \log \left(e^{\pi \cdot u2}\right)\right) \cdot \log \left(e^{\pi \cdot u2}\right)}}\right) \]
8. Step-by-step derivation

   1. rem-log-exp 99.0%

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

   2. rem-log-exp 99.0%

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

   3. *-commutative 99.0%

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

   4. *-commutative 99.0%

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

   5. swap-sqr 99.0%

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

   6. unpow2 99.0%

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

   7. rem-log-exp 99.1%

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

   8. *-commutative 99.1%

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

9. Simplified 99.1%

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

10. Taylor expanded in u2 around 0 99.0%

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

11. Final simplification 99.0%

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

Alternative 4: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999979734420776:\\ \;\;\;\;t_0 \cdot \sqrt{u1 + u1 \cdot \left(u1 \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* 2.0 PI)))))
   (if (<= t_0 0.9999979734420776)
     (* 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 <= 0.9999979734420776f) {
		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(0.9999979734420776))
		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)) < 0.999997973

   1. Initial program 53.0%

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

   2. Taylor expanded in u1 around 0 89.9%

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

      1. +-commutative 89.9%

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

      2. neg-mul-1 89.9%

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

      3. fma-def 89.9%

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

      4. unpow2 89.9%

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

   4. Simplified 89.9%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. pow1/2 89.9%

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

      2. metadata-eval 89.9%

         \[\leadsto {\left(-\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}^{\color{blue}{\left(0.25 \cdot 2\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. sqr-pow 89.6%

         \[\leadsto \color{blue}{\left({\left(-\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)} \cdot {\left(-\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. neg-sub0 89.6%

         \[\leadsto \left({\color{blue}{\left(0 - \mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}}^{\left(\frac{0.25 \cdot 2}{2}\right)} \cdot {\left(-\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. fma-neg 89.6%

         \[\leadsto \left({\left(0 - \color{blue}{\left(-0.5 \cdot \left(u1 \cdot u1\right) - u1\right)}\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)} \cdot {\left(-\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. associate--r- 89.6%

         \[\leadsto \left({\color{blue}{\left(\left(0 - -0.5 \cdot \left(u1 \cdot u1\right)\right) + u1\right)}}^{\left(\frac{0.25 \cdot 2}{2}\right)} \cdot {\left(-\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. neg-sub0 89.6%

         \[\leadsto \left({\left(\color{blue}{\left(--0.5 \cdot \left(u1 \cdot u1\right)\right)} + u1\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)} \cdot {\left(-\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. +-commutative 89.6%

         \[\leadsto \left({\color{blue}{\left(u1 + \left(--0.5 \cdot \left(u1 \cdot u1\right)\right)\right)}}^{\left(\frac{0.25 \cdot 2}{2}\right)} \cdot {\left(-\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. *-commutative 89.6%

         \[\leadsto \left({\left(u1 + \left(-\color{blue}{\left(u1 \cdot u1\right) \cdot -0.5}\right)\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)} \cdot {\left(-\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. distribute-rgt-neg-in 89.6%

          \[\leadsto \left({\left(u1 + \color{blue}{\left(u1 \cdot u1\right) \cdot \left(--0.5\right)}\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)} \cdot {\left(-\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. metadata-eval 89.6%

          \[\leadsto \left({\left(u1 + \left(u1 \cdot u1\right) \cdot \color{blue}{0.5}\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)} \cdot {\left(-\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      12. metadata-eval 89.6%

          \[\leadsto \left({\left(u1 + \left(u1 \cdot u1\right) \cdot 0.5\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)} \cdot {\left(-\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      13. metadata-eval 89.6%

          \[\leadsto \left({\left(u1 + \left(u1 \cdot u1\right) \cdot 0.5\right)}^{\color{blue}{0.25}} \cdot {\left(-\mathsf{fma}\left(-0.5, u1 \cdot u1, -u1\right)\right)}^{\left(\frac{0.25 \cdot 2}{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\left({\left(u1 + \left(u1 \cdot u1\right) \cdot 0.5\right)}^{0.25} \cdot {\left(u1 + \left(u1 \cdot u1\right) \cdot 0.5\right)}^{0.25}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   7. Step-by-step derivation

      1. pow-sqr 89.9%

         \[\leadsto \color{blue}{{\left(u1 + \left(u1 \cdot u1\right) \cdot 0.5\right)}^{\left(2 \cdot 0.25\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. metadata-eval 89.9%

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

      3. unpow1/2 89.9%

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

      4. associate-*l* 89.9%

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

   8. Simplified 89.9%

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

   if 0.999997973 < (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2))

   1. Initial program 53.5%

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

      1. sub-neg 53.5%

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

      2. log1p-def 99.7%

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

      3. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. add-log-exp_binary32 99.7%

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

   5. Applied rewrite-once 99.7%

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

   6. Taylor expanded in u2 around 0 98.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999979734420776:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 + u1 \cdot \left(u1 \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

Alternative 5: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.001979999942705035:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.001979999942705035)
   (sqrt (- (log1p (- u1))))
   (* (cos (* 2.0 (* PI u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.001979999942705035f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf((2.0f * (((float) M_PI) * u2))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.001979999942705035))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(Float32(pi) * u2))) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00197999994

   1. Initial program 53.5%

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

      1. sub-neg 53.5%

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

      2. log1p-def 99.7%

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

      3. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. add-log-exp_binary32 99.7%

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

   5. Applied rewrite-once 99.7%

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

   6. Taylor expanded in u2 around 0 98.9%

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

   if 0.00197999994 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 53.0%

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

      1. sub-neg 53.0%

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

      2. log1p-def 97.9%

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

      3. associate-*l* 97.9%

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

   3. Simplified 97.9%

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

      1. pow1/2 97.9%

         \[\leadsto \color{blue}{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.5}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

      2. log1p-udef 53.0%

         \[\leadsto {\left(-\color{blue}{\log \left(1 + \left(-u1\right)\right)}\right)}^{0.5} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

      3. sub-neg 53.0%

         \[\leadsto {\left(-\log \color{blue}{\left(1 - u1\right)}\right)}^{0.5} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

      4. sqr-pow 53.1%

         \[\leadsto \color{blue}{\left({\left(-\log \left(1 - u1\right)\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(-\log \left(1 - u1\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

      5. pow2 53.1%

         \[\leadsto \color{blue}{{\left({\left(-\log \left(1 - u1\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

      6. sub-neg 53.1%

         \[\leadsto {\left({\left(-\log \color{blue}{\left(1 + \left(-u1\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

      7. log1p-udef 97.6%

         \[\leadsto {\left({\left(-\color{blue}{\mathsf{log1p}\left(-u1\right)}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

      8. metadata-eval 97.6%

         \[\leadsto {\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

   5. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.25}\right)}^{2}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

   6. Taylor expanded in u1 around 0 79.7%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.001979999942705035:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 6: 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 53.3%

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

   1. sub-neg 53.3%

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

   2. log1p-def 99.0%

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

   3. associate-*l* 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 7: 80.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (- (log1p (- u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(-log1p(Float32(-u1))))
end

Derivation

1. Initial program 53.3%

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

   1. sub-neg 53.3%

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

   2. log1p-def 99.0%

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

   3. associate-*l* 99.0%

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

3. Simplified 99.0%

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

   1. add-log-exp_binary32 99.0%

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

5. Applied rewrite-once 99.0%

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

6. Taylor expanded in u2 around 0 80.0%

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

7. Final simplification 80.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]

Alternative 8: 72.6% 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 53.3%

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

2. Taylor expanded in u2 around 0 45.9%

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

   1. flip-- 43.5%

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

   2. div-inv 43.4%

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

   3. log-prod 43.4%

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

   4. metadata-eval 43.4%

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

   5. sub-neg 43.4%

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

   6. log1p-def 46.1%

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

   7. distribute-rgt-neg-in 46.1%

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

4. Applied egg-rr 46.1%

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

5. Taylor expanded in u1 around 0 73.9%

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

   1. +-commutative 73.9%

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

   2. neg-mul-1 73.9%

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

   3. sub-neg 73.9%

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

   4. *-commutative 73.9%

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

   5. unpow2 73.9%

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

   6. associate-*r* 73.9%

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

7. Simplified 73.9%

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

8. Final simplification 73.9%

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

Alternative 9: 64.5% 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 53.3%

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

2. Taylor expanded in u2 around 0 45.9%

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

3. Taylor expanded in u1 around 0 66.8%

   \[\leadsto \sqrt{-\color{blue}{-1 \cdot u1}} \cdot 1 \]
4. Step-by-step derivation

   1. neg-mul-1 66.8%

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

5. Simplified 66.8%

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

6. Final simplification 66.8%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023297 
(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))))