Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.9% → 98.4%

Time: 5.0s

Alternatives: 9

Speedup: 4.7×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.4

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

3. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. count-2-rev N/A

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

   5. rem-3cbrt-rft N/A

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

   6. cbrt-unprod N/A

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

   7. sqr-abs-rev N/A

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

   8. lift-fabs.f32 N/A

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

   9. lift-fabs.f32 N/A

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

   10. cbrt-unprod N/A

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

   11. lift-cbrt.f32 N/A

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

   12. lift-cbrt.f32 N/A

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

   13. associate-*l* N/A

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

   14. lift-cbrt.f32 N/A

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

   15. cbrt-prod N/A

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

   16. lift-*.f32 N/A

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

   17. lift-cbrt.f32 N/A

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

5. Applied rewrites 98.4%

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

Alternative 2: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.0024999999441206455:\\ \;\;\;\;\sqrt{-t\_0} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(u2 \cdot \left(\pi + \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.0024999999441206455)
     (* (sqrt (- t_0)) (sin (* (+ PI PI) u2)))
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* u2 (+ PI PI)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.0024999999441206455f) {
		tmp = sqrtf(-t_0) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * sinf((u2 * (((float) M_PI) + ((float) M_PI))));
	}
	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(-0.0024999999441206455))
		tmp = Float32(sqrt(Float32(-t_0)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * sin(Float32(u2 * Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00249999994

   1. Initial program 57.9%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.9

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

   3. Applied rewrites 57.9%

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

   if -0.00249999994 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.9%

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

   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. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 87.8

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

   4. Applied rewrites 87.8%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 87.8

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lower-fma.f32 87.8

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

      8. lower-fma.f32 N/A

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

      9. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \mathsf{Rewrite=>}\left(*-commutative, \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \mathsf{Rewrite=>}\left(lower-*.f32, \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right) \]

      11. lower-fma.f32 N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(u2 \cdot \mathsf{Rewrite=>}\left(lower-+.f32, \left(\pi + \pi\right)\right)\right) \]

   6. Applied rewrites 87.8%

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

Alternative 3: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.004999999888241291:\\ \;\;\;\;\sqrt{-t\_0} \cdot \mathsf{fma}\left(\pi, u2, \mathsf{fma}\left(u2, \pi, \left(\left(\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333\right) \cdot u2\right) \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(u2 \cdot \left(\pi + \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.004999999888241291)
     (*
      (sqrt (- t_0))
      (fma
       PI
       u2
       (fma
        u2
        PI
        (* (* (* (* (* u2 u2) -1.3333333333333333) u2) PI) (* PI PI)))))
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* u2 (+ PI PI)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.004999999888241291f) {
		tmp = sqrtf(-t_0) * fmaf(((float) M_PI), u2, fmaf(u2, ((float) M_PI), (((((u2 * u2) * -1.3333333333333333f) * u2) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI)))));
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * sinf((u2 * (((float) M_PI) + ((float) M_PI))));
	}
	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(-0.004999999888241291))
		tmp = Float32(sqrt(Float32(-t_0)) * fma(Float32(pi), u2, fma(u2, Float32(pi), Float32(Float32(Float32(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)) * u2) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))))));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * sin(Float32(u2 * Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00499999989

   1. Initial program 57.9%

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

   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. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-pow.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 53.6

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right) \]

   4. Applied rewrites 53.6%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, {u2}^{2} \cdot {\pi}^{3}, 2 \cdot \pi\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. associate-*r* N/A

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

      14. associate-*r* N/A

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

   6. Applied rewrites 53.6%

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

   if -0.00499999989 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.9%

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

   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. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 87.8

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

   4. Applied rewrites 87.8%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 87.8

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lower-fma.f32 87.8

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

      8. lower-fma.f32 N/A

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

      9. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \mathsf{Rewrite=>}\left(*-commutative, \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \mathsf{Rewrite=>}\left(lower-*.f32, \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right) \]

      11. lower-fma.f32 N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(u2 \cdot \mathsf{Rewrite=>}\left(lower-+.f32, \left(\pi + \pi\right)\right)\right) \]

   6. Applied rewrites 87.8%

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

Alternative 4: 90.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0024999999441206455)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* u2 PI)))
   (* (sqrt u1) (sin (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0024999999441206455f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (u2 * ((float) M_PI)));
	} else {
		tmp = sqrtf(u1) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0024999999441206455))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00249999994

   1. Initial program 57.9%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 81.6

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

   6. Applied rewrites 81.6%

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

   if 0.00249999994 < u2

   1. Initial program 57.9%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4

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

   3. Applied rewrites 98.4%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. rem-3cbrt-rft N/A

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

      6. cbrt-unprod N/A

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

      7. sqr-abs-rev N/A

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

      8. lift-fabs.f32 N/A

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

      9. lift-fabs.f32 N/A

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

      10. cbrt-unprod N/A

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

      11. lift-cbrt.f32 N/A

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

      12. lift-cbrt.f32 N/A

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

      13. associate-*l* N/A

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

      14. lift-cbrt.f32 N/A

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

      15. cbrt-prod N/A

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

      16. lift-*.f32 N/A

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

      17. lift-cbrt.f32 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 76.4%

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

Alternative 5: 81.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.4

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

3. Applied rewrites 98.4%

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

4. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 81.6

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

6. Applied rewrites 81.6%

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

Alternative 6: 80.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.002899999963119626:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(\pi + \pi\right)\right) \cdot u2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.002899999963119626)
   (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (* 2.0 (* u2 PI)))
   (* (* (sqrt (- (log (- 1.0 u1)))) (+ PI PI)) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.002899999963119626f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * (2.0f * (u2 * ((float) M_PI)));
	} else {
		tmp = (sqrtf(-logf((1.0f - u1))) * (((float) M_PI) + ((float) M_PI))) * u2;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.002899999963119626))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	else
		tmp = Float32(Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(Float32(pi) + Float32(pi))) * u2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.002899999963119626))
		tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * (single(2.0) * (u2 * single(pi)));
	else
		tmp = (sqrt(-log((single(1.0) - u1))) * (single(pi) + single(pi))) * u2;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00289999996

   1. Initial program 57.9%

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

   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. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 87.8

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 74.4

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

   7. Applied rewrites 74.4%

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

   if 0.00289999996 < u1

   1. Initial program 57.9%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lower--.f32 50.7

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

   4. Applied rewrites 50.7%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. distribute-rgt-out N/A

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

      10. count-2-rev N/A

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

      11. lift-*.f32 N/A

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

      12. lower-*.f32 50.7

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

      13. lift-*.f32 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f32 50.7

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

   6. Applied rewrites 50.7%

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

Alternative 7: 74.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * (single(2.0) * (u2 * single(pi)));
end

Derivation

1. Initial program 57.9%

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

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. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 87.8

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

4. Applied rewrites 87.8%

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

5. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 74.4

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

7. Applied rewrites 74.4%

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

Alternative 8: 66.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\sqrt{u1} \cdot \pi\right) \cdot u2\right) \cdot 18}{9} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/ (* (* (* (sqrt u1) PI) u2) 18.0) 9.0))

C

float code(float cosTheta_i, float u1, float u2) {
	return (((sqrtf(u1) * ((float) M_PI)) * u2) * 18.0f) / 9.0f;
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(Float32(sqrt(u1) * Float32(pi)) * u2) * Float32(18.0)) / Float32(9.0))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((sqrt(u1) * single(pi)) * u2) * single(18.0)) / single(9.0);
end

Derivation

1. Initial program 57.9%

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

2. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower--.f32 50.7

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 66.5

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

7. Applied rewrites 66.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. frac-add N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f32 N/A

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

9. Applied rewrites 66.5%

   \[\leadsto \frac{\left(\left(\sqrt{u1} \cdot \pi\right) \cdot u2\right) \cdot 18}{\color{blue}{9}} \]
10. Add Preprocessing

Alternative 9: 66.5% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

2. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower--.f32 50.7

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 66.5

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

7. Applied rewrites 66.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f32 66.5

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 66.5

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

9. Applied rewrites 66.5%

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

Reproduce

herbie shell --seed 2025156 
(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))))