Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.4% → 98.4%

Time: 6.6s

Alternatives: 10

Speedup: 4.7×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 58.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\sin \left(\mathsf{fma}\left(-u2, \pi, \pi \cdot 0.5\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) \cdot 2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (* (* (sin (fma (- u2) PI (* PI 0.5))) (sin (* u2 PI))) 2.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * ((sinf(fmaf(-u2, ((float) M_PI), (((float) M_PI) * 0.5f))) * sinf((u2 * ((float) M_PI)))) * 2.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(sin(fma(Float32(-u2), Float32(pi), Float32(Float32(pi) * Float32(0.5)))) * sin(Float32(u2 * Float32(pi)))) * Float32(2.0)))
end

Derivation

1. Initial program 58.4%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.4

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

3. Applied rewrites 98.4%

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

   1. lift-sin.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. associate-*l* N/A

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

   7. sin-2 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-cos.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-sin.f32 N/A

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

   17. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(u2 \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sin \left(u2 \cdot \pi\right)\right) \cdot 2\right) \]

   4. lower-sin.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\color{blue}{\sin \left(\left(\mathsf{neg}\left(u2 \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sin \left(u2 \cdot \pi\right)\right) \cdot 2\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\sin \left(\left(\mathsf{neg}\left(\color{blue}{u2 \cdot \pi}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(u2 \cdot \pi\right)\right) \cdot 2\right) \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\sin \left(\color{blue}{\left(\mathsf{neg}\left(u2\right)\right) \cdot \pi} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(u2 \cdot \pi\right)\right) \cdot 2\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(u2\right), \pi, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \cdot \sin \left(u2 \cdot \pi\right)\right) \cdot 2\right) \]

   8. lower-neg.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\sin \left(\mathsf{fma}\left(\color{blue}{-u2}, \pi, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) \cdot 2\right) \]

   9. lift-PI.f32 N/A

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

   10. mult-flip N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\sin \left(\mathsf{fma}\left(-u2, \pi, \color{blue}{\pi \cdot \frac{1}{2}}\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) \cdot 2\right) \]

   11. metadata-eval N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\sin \left(\mathsf{fma}\left(-u2, \pi, \pi \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) \cdot 2\right) \]

   12. lower-*.f32 98.3

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\sin \left(\mathsf{fma}\left(-u2, \pi, \color{blue}{\pi \cdot 0.5}\right)\right) \cdot \sin \left(u2 \cdot \pi\right)\right) \cdot 2\right) \]

7. Applied rewrites 98.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\color{blue}{\sin \left(\mathsf{fma}\left(-u2, \pi, \pi \cdot 0.5\right)\right)} \cdot \sin \left(u2 \cdot \pi\right)\right) \cdot 2\right) \]
8. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (sin (* (+ PI PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
}

Julia

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

Derivation

1. Initial program 58.4%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.4

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

3. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-PI.f32 N/A

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

   6. add-cube-cbrt N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-cbrt.f32 N/A

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

   9. associate-*l* N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-cbrt.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-cbrt.f32 N/A

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

   14. lift-*.f32 N/A

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

5. Applied rewrites 98.4%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\pi + \pi\right) \cdot u2\right)} \]
6. Add Preprocessing

Alternative 3: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.0034000000450760126:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, 0.5, u1\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (+ PI PI) u2))))
   (if (<= t_0 -0.0034000000450760126)
     (* (sqrt (- t_0)) t_1)
     (* (sqrt (fma (* u1 u1) 0.5 u1)) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= -0.0034000000450760126f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf(fmaf((u1 * u1), 0.5f, u1)) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0034000000450760126))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(fma(Float32(u1 * u1), Float32(0.5), u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00340000005

   1. Initial program 58.4%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 58.4

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

   3. Applied rewrites 58.4%

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

   if -0.00340000005 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 58.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4

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

   3. Applied rewrites 98.4%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-cbrt.f32 N/A

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

      9. associate-*l* N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-cbrt.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-cbrt.f32 N/A

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

      14. lift-*.f32 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 87.8

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

   8. Applied rewrites 87.8%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

         \[\leadsto \sqrt{u1 \cdot \left(\frac{1}{2} \cdot u1\right) + u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \left(\frac{1}{2} \cdot u1\right) + u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \left(u1 \cdot \frac{1}{2}\right) + u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(u1 \cdot u1\right) \cdot \frac{1}{2} + u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{{u1}^{2} \cdot \frac{1}{2} + u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

      10. lift-pow.f32 N/A

          \[\leadsto \sqrt{{u1}^{2} \cdot \frac{1}{2} + u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

      11. lower-fma.f32 87.8

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

      12. lift-pow.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left({u1}^{2}, \frac{1}{2}, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

      13. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \frac{1}{2}, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

      14. lower-*.f32 87.8

          \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, 0.5, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

   10. Applied rewrites 87.8%

       \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{0.5}, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.00046999999904073775:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.00046999999904073775)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* u2 PI)))
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.00046999999904073775f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (u2 * ((float) M_PI)));
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.00046999999904073775))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 4.69999999e-4

   1. Initial program 58.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 80.9

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

   6. Applied rewrites 80.9%

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

   if 4.69999999e-4 < u2

   1. Initial program 58.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4

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

   3. Applied rewrites 98.4%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-cbrt.f32 N/A

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

      9. associate-*l* N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-cbrt.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-cbrt.f32 N/A

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

      14. lift-*.f32 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 87.8

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

   8. Applied rewrites 87.8%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 87.8

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

      4. lift-+.f32 N/A

         \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

      7. lower-fma.f32 87.8

         \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

   10. Applied rewrites 87.8%

       \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0006000000284984708:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0006000000284984708)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* u2 PI)))
   (* (sqrt u1) (sin (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0006000000284984708f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (u2 * ((float) M_PI)));
	} else {
		tmp = sqrtf(u1) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0006000000284984708))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 6.00000028e-4

   1. Initial program 58.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 80.9

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

   6. Applied rewrites 80.9%

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

   if 6.00000028e-4 < u2

   1. Initial program 58.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4

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

   3. Applied rewrites 98.4%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-cbrt.f32 N/A

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

      9. associate-*l* N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-cbrt.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-cbrt.f32 N/A

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

      14. lift-*.f32 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 76.1%

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

Alternative 6: 80.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.4%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.4

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

3. Applied rewrites 98.4%

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

4. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 80.9

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

6. Applied rewrites 80.9%

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

Alternative 7: 76.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.0001900000061141327:\\ \;\;\;\;\left(u2 \cdot \left(\pi + \pi\right)\right) \cdot \sqrt{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.0001900000061141327)
     (* (* u2 (+ PI PI)) (sqrt (- t_0)))
     (* (+ u2 u2) (* u1 (* PI (sqrt (/ 1.0 u1))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.0001900000061141327f) {
		tmp = (u2 * (((float) M_PI) + ((float) M_PI))) * sqrtf(-t_0);
	} else {
		tmp = (u2 + u2) * (u1 * (((float) M_PI) * sqrtf((1.0f / u1))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0001900000061141327))
		tmp = Float32(Float32(u2 * Float32(Float32(pi) + Float32(pi))) * sqrt(Float32(-t_0)));
	else
		tmp = Float32(Float32(u2 + u2) * Float32(u1 * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) / u1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = log((single(1.0) - u1));
	tmp = single(0.0);
	if (t_0 <= single(-0.0001900000061141327))
		tmp = (u2 * (single(pi) + single(pi))) * sqrt(-t_0);
	else
		tmp = (u2 + u2) * (u1 * (single(pi) * sqrt((single(1.0) / u1))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -1.90000006e-4

   1. Initial program 58.4%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}}\right)\right) \]

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lower--.f32 51.0

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

   4. Applied rewrites 51.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. lower-*.f32 51.0

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 51.0

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

      15. lift-*.f32 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 51.0

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

   6. Applied rewrites 51.0%

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

   if -1.90000006e-4 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 58.4%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}}\right)\right) \]

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lower--.f32 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)\right) \]

      2. lower-PI.f32 N/A

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

      3. lower-sqrt.f32 65.6

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

   7. Applied rewrites 65.6%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. count-2 N/A

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

      5. lift-+.f32 N/A

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

      6. lower-*.f32 65.6

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

      7. lift-*.f32 N/A

         \[\leadsto \left(u2 + u2\right) \cdot \left(\pi \cdot \sqrt{u1}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

      9. lower-*.f32 65.6

         \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

   9. Applied rewrites 65.6%

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

   10. Taylor expanded in u1 around inf

       \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{\frac{1}{u1}}}\right)\right) \]
   11. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

       2. lower-*.f32 N/A

          \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

       3. lower-PI.f32 N/A

          \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

       4. lower-sqrt.f32 N/A

          \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

       5. lower-/.f32 65.6

          \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

   12. Applied rewrites 65.6%

       \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \color{blue}{\sqrt{\frac{1}{u1}}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.0001900000061141327:\\ \;\;\;\;\left(\sqrt{-t\_0} \cdot \left(\pi + \pi\right)\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.0001900000061141327)
     (* (* (sqrt (- t_0)) (+ PI PI)) u2)
     (* (+ u2 u2) (* u1 (* PI (sqrt (/ 1.0 u1))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.0001900000061141327f) {
		tmp = (sqrtf(-t_0) * (((float) M_PI) + ((float) M_PI))) * u2;
	} else {
		tmp = (u2 + u2) * (u1 * (((float) M_PI) * sqrtf((1.0f / u1))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0001900000061141327))
		tmp = Float32(Float32(sqrt(Float32(-t_0)) * Float32(Float32(pi) + Float32(pi))) * u2);
	else
		tmp = Float32(Float32(u2 + u2) * Float32(u1 * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) / u1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = log((single(1.0) - u1));
	tmp = single(0.0);
	if (t_0 <= single(-0.0001900000061141327))
		tmp = (sqrt(-t_0) * (single(pi) + single(pi))) * u2;
	else
		tmp = (u2 + u2) * (u1 * (single(pi) * sqrt((single(1.0) / u1))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -1.90000006e-4

   1. Initial program 58.4%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}}\right)\right) \]

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lower--.f32 51.0

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

   4. Applied rewrites 51.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. distribute-rgt-out N/A

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

      10. count-2-rev N/A

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

      11. lift-*.f32 N/A

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

      12. lower-*.f32 51.0

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

      13. lift-*.f32 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f32 51.0

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

   6. Applied rewrites 51.0%

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

   if -1.90000006e-4 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 58.4%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}}\right)\right) \]

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lower--.f32 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)\right) \]

      2. lower-PI.f32 N/A

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

      3. lower-sqrt.f32 65.6

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

   7. Applied rewrites 65.6%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. count-2 N/A

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

      5. lift-+.f32 N/A

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

      6. lower-*.f32 65.6

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

      7. lift-*.f32 N/A

         \[\leadsto \left(u2 + u2\right) \cdot \left(\pi \cdot \sqrt{u1}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

      9. lower-*.f32 65.6

         \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

   9. Applied rewrites 65.6%

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

   10. Taylor expanded in u1 around inf

       \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{\frac{1}{u1}}}\right)\right) \]
   11. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

       2. lower-*.f32 N/A

          \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

       3. lower-PI.f32 N/A

          \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

       4. lower-sqrt.f32 N/A

          \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

       5. lower-/.f32 65.6

          \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

   12. Applied rewrites 65.6%

       \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \color{blue}{\sqrt{\frac{1}{u1}}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (+ u2 u2) (* u1 (* PI (sqrt (/ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 + u2) * (u1 * (((float) M_PI) * sqrtf((1.0f / u1))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u2 + u2) * Float32(u1 * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) / u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 + u2) * (u1 * (single(pi) * sqrt((single(1.0) / u1))));
end

Derivation

1. Initial program 58.4%

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

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto 2 \cdot \color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto 2 \cdot \left(u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}}\right)\right) \]

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower--.f32 51.0

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

4. Applied rewrites 51.0%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

      \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)\right) \]

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 65.6

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

7. Applied rewrites 65.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. count-2 N/A

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

   5. lift-+.f32 N/A

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

   6. lower-*.f32 65.6

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

   7. lift-*.f32 N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\pi \cdot \sqrt{u1}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

   9. lower-*.f32 65.6

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

9. Applied rewrites 65.6%

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

10. Taylor expanded in u1 around inf

    \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{\frac{1}{u1}}}\right)\right) \]
11. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

    2. lower-*.f32 N/A

       \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

    3. lower-PI.f32 N/A

       \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

    4. lower-sqrt.f32 N/A

       \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

    5. lower-/.f32 65.6

       \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

12. Applied rewrites 65.6%

    \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \color{blue}{\sqrt{\frac{1}{u1}}}\right)\right) \]
13. Add Preprocessing

Alternative 10: 65.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (+ u2 u2) (* (sqrt u1) PI)))

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u2 + u2) * Float32(sqrt(u1) * Float32(pi)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 + u2) * (sqrt(u1) * single(pi));
end

Derivation

1. Initial program 58.4%

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

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto 2 \cdot \color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto 2 \cdot \left(u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}}\right)\right) \]

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower--.f32 51.0

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

4. Applied rewrites 51.0%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

      \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)\right) \]

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 65.6

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

7. Applied rewrites 65.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. count-2 N/A

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

   5. lift-+.f32 N/A

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

   6. lower-*.f32 65.6

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

   7. lift-*.f32 N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\pi \cdot \sqrt{u1}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

   9. lower-*.f32 65.6

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

9. Applied rewrites 65.6%

   \[\leadsto \left(u2 + u2\right) \cdot \color{blue}{\left(\sqrt{u1} \cdot \pi\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025155 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :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)))) (sin (* (* 2.0 PI) u2))))