Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.8% → 98.2%

Time: 6.4s

Alternatives: 12

Speedup: 8.9×

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: 57.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.03500000014901161:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (+ PI PI) u2))))
   (if (<= t_0 -0.03500000014901161)
     (* (sqrt (- t_0)) t_1)
     (*
      (sqrt
       (fma
        u1
        1.0
        (* u1 (* (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1))))
      t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= -0.03500000014901161f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf(fmaf(u1, 1.0f, (u1 * (fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f) * u1)))) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.03500000014901161))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(fma(u1, Float32(1.0), Float32(u1 * Float32(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)) * u1)))) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0350000001

   1. Initial program 97.3%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 97.3

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

   3. Applied rewrites 97.3%

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

   if -0.0350000001 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 50.4%

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

   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 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 98.4

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

   6. Applied rewrites 98.4%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   8. Applied rewrites 98.4%

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

Alternative 2: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 57.8

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

3. Applied rewrites 57.8%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. pow2 N/A

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

   6. log-div N/A

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

   7. pow2 N/A

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

   8. lower--.f32 N/A

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

   9. fp-cancel-sub-sign-inv N/A

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

   10. lower-log1p.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-neg.f32 N/A

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

   13. lower-log1p.f32 98.3

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

5. Applied rewrites 98.3%

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

Alternative 3: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\left(\left(\left(\left(-0.25 \cdot \left(u1 \cdot u1\right) - 0.3333333333333333\right) \cdot \left(u1 \cdot u1\right) - 0.5\right) \cdot \left(u1 \cdot u1\right) - 1\right) \cdot \left(u1 \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt
   (-
    (-
     (*
      (-
       (*
        (- (* (- (* -0.25 (* u1 u1)) 0.3333333333333333) (* u1 u1)) 0.5)
        (* u1 u1))
       1.0)
      (* u1 u1))
     (log1p u1))))
  (sin (* (+ PI PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-((((((((-0.25f * (u1 * u1)) - 0.3333333333333333f) * (u1 * u1)) - 0.5f) * (u1 * u1)) - 1.0f) * (u1 * u1)) - log1pf(u1))) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * Float32(u1 * u1)) - Float32(0.3333333333333333)) * Float32(u1 * u1)) - Float32(0.5)) * Float32(u1 * u1)) - Float32(1.0)) * Float32(u1 * u1)) - log1p(u1)))) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)))
end

Derivation

1. Initial program 57.8%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 57.8

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

3. Applied rewrites 57.8%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. pow2 N/A

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

   6. log-div N/A

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

   7. pow2 N/A

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

   8. lower--.f32 N/A

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

   9. fp-cancel-sub-sign-inv N/A

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

   10. lower-log1p.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-neg.f32 N/A

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

   13. lower-log1p.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{-\left(\color{blue}{{u1}^{2} \cdot \left({u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{-\left(\left({u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{{u1}^{2}} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(\left({u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{{u1}^{2}} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

8. Applied rewrites 95.9%

   \[\leadsto \sqrt{-\left(\color{blue}{\left(\left(\left(-0.25 \cdot \left(u1 \cdot u1\right) - 0.3333333333333333\right) \cdot \left(u1 \cdot u1\right) - 0.5\right) \cdot \left(u1 \cdot u1\right) - 1\right) \cdot \left(u1 \cdot u1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]
9. Add Preprocessing

Alternative 4: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.01899999938905239:\\ \;\;\;\;\sqrt{-t\_0} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.01899999938905239)
     (* (sqrt (- t_0)) (* (* PI 2.0) u2))
     (*
      (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1))
      (sin (* (+ PI PI) u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.01899999938905239f) {
		tmp = sqrtf(-t_0) * ((((float) M_PI) * 2.0f) * u2);
	} else {
		tmp = sqrtf((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) * u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.01899999938905239))
		tmp = Float32(sqrt(Float32(-t_0)) * Float32(Float32(Float32(pi) * Float32(2.0)) * u2));
	else
		tmp = Float32(sqrt(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0189999994

   1. Initial program 96.8%

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

   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. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift-PI.f32 80.9

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

   4. Applied rewrites 80.9%

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

   if -0.0189999994 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 49.0%

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

   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 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 98.4

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

   6. Applied rewrites 98.4%

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

   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 98.2%

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

Alternative 5: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.01899999938905239:\\ \;\;\;\;\sqrt{-t\_0} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.01899999938905239)
     (* (sqrt (- t_0)) (* (* PI 2.0) u2))
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (+ PI PI) u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.01899999938905239f) {
		tmp = sqrtf(-t_0) * ((((float) M_PI) * 2.0f) * u2);
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.01899999938905239))
		tmp = Float32(sqrt(Float32(-t_0)) * Float32(Float32(Float32(pi) * Float32(2.0)) * u2));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0189999994

   1. Initial program 96.8%

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

   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. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift-PI.f32 80.9

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

   4. Applied rewrites 80.9%

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

   if -0.0189999994 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 49.0%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 95.9

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

   4. Applied rewrites 95.9%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 95.9

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

   6. Applied rewrites 95.9%

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

Alternative 6: 93.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\left(\left(\left(\left(-0.25 \cdot \left(u1 \cdot u1\right) - 0.3333333333333333\right) \cdot \left(u1 \cdot u1\right) - 0.5\right) \cdot \left(u1 \cdot u1\right) - 1\right) \cdot \left(u1 \cdot u1\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, u1, 0.3333333333333333\right) \cdot u1 - 0.5, u1, 1\right) \cdot u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt
   (-
    (-
     (*
      (-
       (*
        (- (* (- (* -0.25 (* u1 u1)) 0.3333333333333333) (* u1 u1)) 0.5)
        (* u1 u1))
       1.0)
      (* u1 u1))
     (* (fma (- (* (fma -0.25 u1 0.3333333333333333) u1) 0.5) u1 1.0) u1))))
  (sin (* (+ PI PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-((((((((-0.25f * (u1 * u1)) - 0.3333333333333333f) * (u1 * u1)) - 0.5f) * (u1 * u1)) - 1.0f) * (u1 * u1)) - (fmaf(((fmaf(-0.25f, u1, 0.3333333333333333f) * u1) - 0.5f), u1, 1.0f) * u1))) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * Float32(u1 * u1)) - Float32(0.3333333333333333)) * Float32(u1 * u1)) - Float32(0.5)) * Float32(u1 * u1)) - Float32(1.0)) * Float32(u1 * u1)) - Float32(fma(Float32(Float32(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)))
end

Derivation

1. Initial program 57.8%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 57.8

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

3. Applied rewrites 57.8%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. pow2 N/A

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

   6. log-div N/A

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

   7. pow2 N/A

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

   8. lower--.f32 N/A

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

   9. fp-cancel-sub-sign-inv N/A

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

   10. lower-log1p.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-neg.f32 N/A

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

   13. lower-log1p.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{-\left(\color{blue}{{u1}^{2} \cdot \left({u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{-\left(\left({u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{{u1}^{2}} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(\left({u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{{u1}^{2}} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

8. Applied rewrites 95.9%

   \[\leadsto \sqrt{-\left(\color{blue}{\left(\left(\left(-0.25 \cdot \left(u1 \cdot u1\right) - 0.3333333333333333\right) \cdot \left(u1 \cdot u1\right) - 0.5\right) \cdot \left(u1 \cdot u1\right) - 1\right) \cdot \left(u1 \cdot u1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]

9. Taylor expanded in u1 around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f32 N/A

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

    6. lower--.f32 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f32 N/A

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

    9. +-commutative N/A

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

    10. lower-fma.f32 93.4

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

11. Applied rewrites 93.4%

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

Alternative 7: 93.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt
   (fma u1 1.0 (* u1 (* (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1))))
  (sin (* (+ PI PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(fmaf(u1, 1.0f, (u1 * (fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f) * u1)))) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(fma(u1, Float32(1.0), Float32(u1 * Float32(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)) * u1)))) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)))
end

Derivation

1. Initial program 57.8%

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

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-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. lift-+.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 93.1

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

6. Applied rewrites 93.1%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. *-commutative N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. +-commutative N/A

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

8. Applied rewrites 93.2%

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

Alternative 8: 93.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
  (sin (* (+ PI PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
}

Julia

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

Derivation

1. Initial program 57.8%

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

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-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. lift-+.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 93.1

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

6. Applied rewrites 93.1%

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

Alternative 9: 87.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0010000000474974513:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0010000000474974513)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (* (* PI 2.0) u2))
   (* (sqrt u1) (sin (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0010000000474974513f) {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * ((((float) M_PI) * 2.0f) * u2);
	} else {
		tmp = sqrtf(u1) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0010000000474974513))
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * Float32(Float32(Float32(pi) * Float32(2.0)) * u2));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00100000005

   1. Initial program 57.6%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift-PI.f32 92.1

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

   7. Applied rewrites 92.1%

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

   if 0.00100000005 < u2

   1. Initial program 58.2%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 75.9%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 75.9

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

   6. Applied rewrites 75.9%

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

Alternative 10: 78.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
  (* (* PI 2.0) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * ((((float) M_PI) * 2.0f) * u2);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * Float32(Float32(Float32(pi) * Float32(2.0)) * u2))
end

Derivation

1. Initial program 57.8%

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

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

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 78.1

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

7. Applied rewrites 78.1%

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

Alternative 11: 74.3% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (* (fma 0.5 u1 1.0) u1)) (* (* PI 2.0) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * ((((float) M_PI) * 2.0f) * u2);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * Float32(Float32(Float32(pi) * Float32(2.0)) * u2))
end

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 87.7

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

4. Applied rewrites 87.7%

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

5. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 74.3

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

7. Applied rewrites 74.3%

   \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \color{blue}{\left(\left(\pi \cdot 2\right) \cdot u2\right)} \]
8. Add Preprocessing

Alternative 12: 66.4% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in u1 around 0

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 66.4

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

7. Applied rewrites 66.4%

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

Reproduce

herbie shell --seed 2025093 
(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))))