Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.7% → 98.2%

Time: 6.1s

Alternatives: 16

Speedup: 4.5×

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: 57.7% 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: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.03999999910593033:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1\right)}^{0.5} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(-\log \left(1 - u1\right)\right) \cdot 0.5} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* 2.0 PI) u2))))
   (if (<= u1 0.03999999910593033)
     (*
      (pow
       (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1)
       0.5)
      t_0)
     (* (exp (* (log (- (log (- 1.0 u1)))) 0.5)) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (u1 <= 0.03999999910593033f) {
		tmp = powf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1), 0.5f) * t_0;
	} else {
		tmp = expf((logf(-logf((1.0f - u1))) * 0.5f)) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03999999910593033))
		tmp = Float32((Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1) ^ Float32(0.5)) * t_0);
	else
		tmp = Float32(exp(Float32(log(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(0.5))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0399999991

   1. Initial program 57.7%

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

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1 + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot u1 + \frac{1}{3}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-fma.f32 93.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 93.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right), u1, 1\right) \cdot u1\right)}^{\frac{1}{2}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. lower-pow.f32 93.1

         \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1\right)}^{0.5}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. flip-- 93.1

         \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right)}, u1, 0.5\right), u1, 1\right) \cdot u1\right)}^{0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. metadata-eval 93.1

         \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1\right)}^{0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. diff-log 93.1

         \[\leadsto {\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right)}, u1, 1\right) \cdot u1\right)}^{0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. Applied rewrites 93.1%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1\right)}^{0.5}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.0399999991 < u1

   1. Initial program 57.7%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. pow1/2 N/A

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

      6. pow-to-exp N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-log.f32 N/A

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

      10. lift-log.f32 N/A

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

      11. lift--.f32 N/A

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

      12. lift-neg.f32 57.7

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

   3. Applied rewrites 57.7%

      \[\leadsto \color{blue}{e^{\log \left(-\log \left(1 - u1\right)\right) \cdot 0.5}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(-\log \left(1 - u1\right)\right) \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03999999910593033)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (+ PI PI) u2)))
   (* (exp (* (log (- (log (- 1.0 u1)))) 0.5)) (sin (* (* 2.0 PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.03999999910593033f) {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = expf((logf(-logf((1.0f - u1))) * 0.5f)) * sinf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03999999910593033))
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(exp(Float32(log(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(0.5))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0399999991

   1. Initial program 57.7%

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

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1 + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot u1 + \frac{1}{3}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-fma.f32 93.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 93.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. flip-- 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right)}, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. metadata-eval 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. diff-log 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{Rewrite=>}\left(lift-PI.f32, \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      5. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite=>}\left(lift-*.f32, \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      6. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite=>}\left(count-2-rev, \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      7. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite<=}\left(lift-+.f32, \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      8. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\mathsf{Rewrite<=}\left(lift-PI.f32, \pi\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \mathsf{Rewrite<=}\left(lift-PI.f32, \pi\right)\right) \cdot u2\right) \]

   6. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)} \]

   if 0.0399999991 < u1

   1. Initial program 57.7%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. pow1/2 N/A

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

      6. pow-to-exp N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-log.f32 N/A

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

      10. lift-log.f32 N/A

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

      11. lift--.f32 N/A

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

      12. lift-neg.f32 57.7

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

   3. Applied rewrites 57.7%

      \[\leadsto \color{blue}{e^{\log \left(-\log \left(1 - u1\right)\right) \cdot 0.5}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\log \left(1 - u1 \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03999999910593033)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (+ PI PI) u2)))
   (*
    (sqrt (- (- (log (- 1.0 (* u1 u1))) (log1p u1))))
    (sin (* (* 2.0 PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.03999999910593033f) {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-(logf((1.0f - (u1 * u1))) - log1pf(u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03999999910593033))
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-Float32(log(Float32(Float32(1.0) - Float32(u1 * u1))) - log1p(u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0399999991

   1. Initial program 57.7%

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

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1 + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot u1 + \frac{1}{3}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-fma.f32 93.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 93.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. flip-- 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right)}, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. metadata-eval 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. diff-log 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{Rewrite=>}\left(lift-PI.f32, \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      5. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite=>}\left(lift-*.f32, \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      6. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite=>}\left(count-2-rev, \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      7. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite<=}\left(lift-+.f32, \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      8. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\mathsf{Rewrite<=}\left(lift-PI.f32, \pi\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \mathsf{Rewrite<=}\left(lift-PI.f32, \pi\right)\right) \cdot u2\right) \]

   6. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)} \]

   if 0.0399999991 < u1

   1. Initial program 57.7%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. lower-log.f32 N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. lower--.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-log.f32 N/A

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

      13. lower-+.f32 54.9

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

   3. Applied rewrites 54.9%

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

      1. lift-+.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. lower-log1p.f32 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \sqrt{-\left(\log \left(1 - u1 \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.03999999910593033:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (+ PI PI) u2))))
   (if (<= t_0 -0.03999999910593033)
     (* (sqrt (- t_0)) t_1)
     (*
      (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) 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 <= -0.03999999910593033f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * 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(-0.03999999910593033))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0399999991

   1. Initial program 57.7%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 57.7

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

   3. Applied rewrites 57.7%

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

   if -0.0399999991 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.7%

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

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1 + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot u1 + \frac{1}{3}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-fma.f32 93.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 93.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. flip-- 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right)}, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. metadata-eval 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. diff-log 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{Rewrite=>}\left(lift-PI.f32, \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      5. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite=>}\left(lift-*.f32, \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      6. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite=>}\left(count-2-rev, \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      7. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite<=}\left(lift-+.f32, \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      8. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\mathsf{Rewrite<=}\left(lift-PI.f32, \pi\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \mathsf{Rewrite<=}\left(lift-PI.f32, \pi\right)\right) \cdot u2\right) \]

   6. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.017999999225139618:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (+ PI PI) u2))))
   (if (<= t_0 -0.017999999225139618)
     (* (sqrt (- t_0)) t_1)
     (* (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) 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 <= -0.017999999225139618f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) * 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(-0.017999999225139618))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0179999992

   1. Initial program 57.7%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 57.7

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

   3. Applied rewrites 57.7%

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

   if -0.0179999992 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.7%

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

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1 + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot u1 + \frac{1}{3}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-fma.f32 93.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 93.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. flip-- 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right)}, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. metadata-eval 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. diff-log 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{Rewrite=>}\left(lift-PI.f32, \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      5. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite=>}\left(lift-*.f32, \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      6. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite=>}\left(count-2-rev, \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      7. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite<=}\left(lift-+.f32, \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      8. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\mathsf{Rewrite<=}\left(lift-PI.f32, \pi\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \mathsf{Rewrite<=}\left(lift-PI.f32, \pi\right)\right) \cdot u2\right) \]

   6. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)} \]

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 91.3%

      \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.0026000000070780516:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (+ PI PI) u2))))
   (if (<= t_0 -0.0026000000070780516)
     (* (sqrt (- t_0)) t_1)
     (* (sqrt (* (fma 0.5 u1 1.0) 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 <= -0.0026000000070780516f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * 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(-0.0026000000070780516))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00260000001

   1. Initial program 57.7%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 57.7

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

   3. Applied rewrites 57.7%

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

   if -0.00260000001 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.7%

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

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1 + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot u1 + \frac{1}{3}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-fma.f32 93.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 93.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. flip-- 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right)}, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. metadata-eval 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. diff-log 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{Rewrite=>}\left(lift-PI.f32, \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      5. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite=>}\left(lift-*.f32, \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      6. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite=>}\left(count-2-rev, \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      7. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite<=}\left(lift-+.f32, \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      8. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\mathsf{Rewrite<=}\left(lift-PI.f32, \pi\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \mathsf{Rewrite<=}\left(lift-PI.f32, \pi\right)\right) \cdot u2\right) \]

   6. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)} \]

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 87.7%

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

Alternative 7: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.007000000216066837:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.007000000216066837)
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (+ PI PI) u2)))
   (*
    (sqrt (- (log (- 1.0 u1))))
    (*
     (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI))
     u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.007000000216066837f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * (fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.007000000216066837))
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00700000022

   1. Initial program 57.7%

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

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1 + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot u1 + \frac{1}{3}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-fma.f32 93.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 93.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. flip-- 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right)}, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. metadata-eval 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. diff-log 93.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{Rewrite=>}\left(lift-PI.f32, \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      5. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite=>}\left(lift-*.f32, \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      6. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite=>}\left(count-2-rev, \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      7. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{Rewrite<=}\left(lift-+.f32, \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot u2\right) \]

      8. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\mathsf{Rewrite<=}\left(lift-PI.f32, \pi\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. diff-log N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \mathsf{Rewrite<=}\left(lift-PI.f32, \pi\right)\right) \cdot u2\right) \]

   6. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)} \]

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 87.7%

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

   if 0.00700000022 < u1

   1. Initial program 57.7%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   4. Applied rewrites 53.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\\ \mathbf{if}\;u1 \leq 2.300000005561742 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{elif}\;u1 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0
         (*
          (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI))
          u2)))
   (if (<= u1 2.300000005561742e-6)
     (* (sqrt u1) (sin (* (+ PI PI) u2)))
     (if (<= u1 0.03999999910593033)
       (*
        (sqrt
         (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
        t_0)
       (* (sqrt (- (log (- 1.0 u1)))) t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2;
	float tmp;
	if (u1 <= 2.300000005561742e-6f) {
		tmp = sqrtf(u1) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else if (u1 <= 0.03999999910593033f) {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * t_0;
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(2.300000005561742e-6))
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	elseif (u1 <= Float32(0.03999999910593033))
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * t_0);
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if u1 < 2.30000001e-6

   1. Initial program 57.7%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-sin.f32 76.4

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

      10. lift-PI.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. lift-PI.f32 76.4

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

   4. Applied rewrites 76.4%

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

   if 2.30000001e-6 < u1 < 0.0399999991

   1. Initial program 57.7%

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

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot \color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1 + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot u1 + \frac{1}{3}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. lower-fma.f32 93.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 93.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u1, \frac{1}{3}\right), u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   7. Applied rewrites 84.8%

      \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]

   if 0.0399999991 < u1

   1. Initial program 57.7%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   4. Applied rewrites 53.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 88.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.00015999999595806003:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.00015999999595806003)
   (* (sqrt u1) (sin (* (+ PI PI) u2)))
   (*
    (sqrt (- (log (- 1.0 u1))))
    (*
     (fma (* (* u2 u2) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI))
     u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.00015999999595806003f) {
		tmp = sqrtf(u1) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * (fmaf(((u2 * u2) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.00015999999595806003))
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(fma(Float32(Float32(u2 * u2) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 1.59999996e-4

   1. Initial program 57.7%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-sin.f32 76.4

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

      10. lift-PI.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. lift-PI.f32 76.4

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

   4. Applied rewrites 76.4%

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

   if 1.59999996e-4 < u1

   1. Initial program 57.7%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   4. Applied rewrites 53.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 83.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot \pi\right) \cdot \pi\\ \mathbf{if}\;u1 \leq 0.00015999999595806003:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \sqrt{u1} \cdot t\_0, \left(\pi + \pi\right) \cdot \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot t\_0, -1.3333333333333333, \pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* PI PI) PI)))
   (if (<= u1 0.00015999999595806003)
     (*
      u2
      (fma
       (* -1.3333333333333333 (* u2 u2))
       (* (sqrt u1) t_0)
       (* (+ PI PI) (sqrt u1))))
     (*
      (sqrt (- (log (- 1.0 u1))))
      (* (fma (* (* u2 u2) t_0) -1.3333333333333333 (+ PI PI)) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (((float) M_PI) * ((float) M_PI)) * ((float) M_PI);
	float tmp;
	if (u1 <= 0.00015999999595806003f) {
		tmp = u2 * fmaf((-1.3333333333333333f * (u2 * u2)), (sqrtf(u1) * t_0), ((((float) M_PI) + ((float) M_PI)) * sqrtf(u1)));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * (fmaf(((u2 * u2) * t_0), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.00015999999595806003))
		tmp = Float32(u2 * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(sqrt(u1) * t_0), Float32(Float32(Float32(pi) + Float32(pi)) * sqrt(u1))));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(fma(Float32(Float32(u2 * u2) * t_0), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 1.59999996e-4

   1. Initial program 57.7%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-sin.f32 76.4

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

      10. lift-PI.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. lift-PI.f32 76.4

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto u2 \cdot \color{blue}{\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{u1}\right)\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{u1}\right)\right) + \color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto u2 \cdot \left(\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{u1}\right) + 2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \sqrt{u1}\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{3} \cdot \color{blue}{\sqrt{u1}}, 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)\right) \]

   7. Applied rewrites 70.5%

      \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \sqrt{u1} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \left(\pi + \pi\right) \cdot \sqrt{u1}\right)} \]

   if 1.59999996e-4 < u1

   1. Initial program 57.7%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

   4. Applied rewrites 53.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot u2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 81.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot u2\right) \cdot \left(\pi \cdot \sqrt{-\log \left(1 - u1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03999999910593033)
   (*
    (sqrt
     (-
      (*
       (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
       u1)))
    (* (+ PI PI) u2))
   (* (* 2.0 u2) (* PI (sqrt (- (log (- 1.0 u1))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.03999999910593033f) {
		tmp = sqrtf(-(((((((-0.25f * u1) - 0.3333333333333333f) * u1) - 0.5f) * u1) - 1.0f) * u1)) * ((((float) M_PI) + ((float) M_PI)) * u2);
	} else {
		tmp = (2.0f * u2) * (((float) M_PI) * sqrtf(-logf((1.0f - u1))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03999999910593033))
		tmp = Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u1) - Float32(0.3333333333333333)) * u1) - Float32(0.5)) * u1) - Float32(1.0)) * u1))) * Float32(Float32(Float32(pi) + Float32(pi)) * u2));
	else
		tmp = Float32(Float32(Float32(2.0) * u2) * Float32(Float32(pi) * sqrt(Float32(-log(Float32(Float32(1.0) - u1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.03999999910593033))
		tmp = sqrt(-(((((((single(-0.25) * u1) - single(0.3333333333333333)) * u1) - single(0.5)) * u1) - single(1.0)) * u1)) * ((single(pi) + single(pi)) * u2);
	else
		tmp = (single(2.0) * u2) * (single(pi) * sqrt(-log((single(1.0) - u1))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0399999991

   1. Initial program 57.7%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-PI.f32 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \sqrt{-u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{-u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      3. diff-log N/A

         \[\leadsto \sqrt{-\color{blue}{u1} \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{-\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      6. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{-\left(\left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      9. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(\left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{-\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      12. lower--.f32 N/A

          \[\leadsto \sqrt{-\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      13. lower-*.f32 78.1

          \[\leadsto \sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   7. Applied rewrites 78.1%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   if 0.0399999991 < u1

   1. Initial program 57.7%

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

   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. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-log.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. lift-sqrt.f32 51.1

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

   4. Applied rewrites 51.1%

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

Alternative 12: 81.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.017999999225139618:\\ \;\;\;\;\sqrt{-\left(\left(-0.3333333333333333 \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot u2\right) \cdot \left(\pi \cdot \sqrt{-\log \left(1 - u1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.017999999225139618)
   (*
    (sqrt (- (* (- (* (- (* -0.3333333333333333 u1) 0.5) u1) 1.0) u1)))
    (* (+ PI PI) u2))
   (* (* 2.0 u2) (* PI (sqrt (- (log (- 1.0 u1))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.017999999225139618f) {
		tmp = sqrtf(-(((((-0.3333333333333333f * u1) - 0.5f) * u1) - 1.0f) * u1)) * ((((float) M_PI) + ((float) M_PI)) * u2);
	} else {
		tmp = (2.0f * u2) * (((float) M_PI) * sqrtf(-logf((1.0f - u1))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.017999999225139618))
		tmp = Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(Float32(Float32(-0.3333333333333333) * u1) - Float32(0.5)) * u1) - Float32(1.0)) * u1))) * Float32(Float32(Float32(pi) + Float32(pi)) * u2));
	else
		tmp = Float32(Float32(Float32(2.0) * u2) * Float32(Float32(pi) * sqrt(Float32(-log(Float32(Float32(1.0) - u1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.017999999225139618))
		tmp = sqrt(-(((((single(-0.3333333333333333) * u1) - single(0.5)) * u1) - single(1.0)) * u1)) * ((single(pi) + single(pi)) * u2);
	else
		tmp = (single(2.0) * u2) * (single(pi) * sqrt(-log((single(1.0) - u1))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0179999992

   1. Initial program 57.7%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-PI.f32 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in u1 around 0

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

      1. flip-- N/A

         \[\leadsto \sqrt{-u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{-u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      3. diff-log N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{-\left(\left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      9. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(\left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

      10. lower-*.f32 76.9

          \[\leadsto \sqrt{-\left(\left(-0.3333333333333333 \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   7. Applied rewrites 76.9%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.3333333333333333 \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]

   if 0.0179999992 < u1

   1. Initial program 57.7%

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

   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. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-log.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. lift-sqrt.f32 51.1

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

   4. Applied rewrites 51.1%

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

Alternative 13: 81.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.0026000000070780516:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot u2\right) \cdot \left(\pi \cdot \sqrt{-\log \left(1 - u1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.0026000000070780516)
   (* (sqrt (- (* (- (* -0.5 u1) 1.0) u1))) (* (+ PI PI) u2))
   (* (* 2.0 u2) (* PI (sqrt (- (log (- 1.0 u1))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.0026000000070780516f) {
		tmp = sqrtf(-(((-0.5f * u1) - 1.0f) * u1)) * ((((float) M_PI) + ((float) M_PI)) * u2);
	} else {
		tmp = (2.0f * u2) * (((float) M_PI) * sqrtf(-logf((1.0f - u1))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.0026000000070780516))
		tmp = Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(-0.5) * u1) - Float32(1.0)) * u1))) * Float32(Float32(Float32(pi) + Float32(pi)) * u2));
	else
		tmp = Float32(Float32(Float32(2.0) * u2) * Float32(Float32(pi) * sqrt(Float32(-log(Float32(Float32(1.0) - u1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.0026000000070780516))
		tmp = sqrt(-(((single(-0.5) * u1) - single(1.0)) * u1)) * ((single(pi) + single(pi)) * u2);
	else
		tmp = (single(2.0) * u2) * (single(pi) * sqrt(-log((single(1.0) - u1))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00260000001

   1. Initial program 57.7%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-PI.f32 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in u1 around 0

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. diff-log N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 74.3

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

   7. Applied rewrites 74.3%

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

   if 0.00260000001 < u1

   1. Initial program 57.7%

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

   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. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-log.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. lift-sqrt.f32 51.1

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

   4. Applied rewrites 51.1%

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

Alternative 14: 81.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \left(\pi + \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq -0.0026000000070780516:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (* (+ PI PI) u2)))
   (if (<= t_0 -0.0026000000070780516)
     (* (sqrt (- t_0)) t_1)
     (* (sqrt (- (* (- (* -0.5 u1) 1.0) u1))) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = (((float) M_PI) + ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= -0.0026000000070780516f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf(-(((-0.5f * u1) - 1.0f) * u1)) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = Float32(Float32(Float32(pi) + Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0026000000070780516))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(-0.5) * u1) - Float32(1.0)) * u1))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = log((single(1.0) - u1));
	t_1 = (single(pi) + single(pi)) * u2;
	tmp = single(0.0);
	if (t_0 <= single(-0.0026000000070780516))
		tmp = sqrt(-t_0) * t_1;
	else
		tmp = sqrt(-(((single(-0.5) * u1) - single(1.0)) * u1)) * t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00260000001

   1. Initial program 57.7%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-PI.f32 51.1

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

   4. Applied rewrites 51.1%

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

   if -0.00260000001 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.7%

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

   2. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-PI.f32 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in u1 around 0

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. diff-log N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 74.3

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

   7. Applied rewrites 74.3%

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

Alternative 15: 74.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (* (- (* -0.5 u1) 1.0) u1))) (* (+ PI PI) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-(((-0.5f * u1) - 1.0f) * u1)) * ((((float) M_PI) + ((float) M_PI)) * u2);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(-0.5) * u1) - Float32(1.0)) * u1))) * Float32(Float32(Float32(pi) + Float32(pi)) * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-(((single(-0.5) * u1) - single(1.0)) * u1)) * ((single(pi) + single(pi)) * u2);
end

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f32 N/A

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

   4. count-2-rev N/A

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

   5. lower-+.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 51.1

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

4. Applied rewrites 51.1%

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

5. Taylor expanded in u1 around 0

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. diff-log N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-*.f32 74.3

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

7. Applied rewrites 74.3%

   \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot u1 - 1\right) \cdot u1}} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]
8. Add Preprocessing

Alternative 16: 66.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-sin.f32 76.4

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

   10. lift-PI.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. count-2-rev N/A

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

   13. lower-+.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-PI.f32 76.4

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-PI.f32 66.3

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

7. Applied rewrites 66.3%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-sqrt.f32 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-sqrt.f32 66.3

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

9. Applied rewrites 66.3%

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

Reproduce

herbie shell --seed 2025138 
(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))))