Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.8% → 99.1%

Time: 12.2s

Alternatives: 19

Speedup: 14.4×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.9%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0 \leq 0.15000000596046448:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1 \cdot u1, u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* PI 2.0)))))
   (if (<= (* (sqrt (- (log (- 1.0 u1)))) t_0) 0.15000000596046448)
     (*
      (sqrt (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) (* u1 u1) u1))
      t_0)
     (* (fma (* -2.0 (* u2 u2)) (* PI PI) 1.0) (sqrt (- (log1p (- u1))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * (((float) M_PI) * 2.0f)));
	float tmp;
	if ((sqrtf(-logf((1.0f - u1))) * t_0) <= 0.15000000596046448f) {
		tmp = sqrtf(fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), (u1 * u1), u1)) * t_0;
	} else {
		tmp = fmaf((-2.0f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 1.0f) * sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(Float32(pi) * Float32(2.0))))
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0) <= Float32(0.15000000596046448))
		tmp = Float32(sqrt(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), Float32(u1 * u1), u1)) * t_0);
	else
		tmp = Float32(fma(Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)) * sqrt(Float32(-log1p(Float32(-u1)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.150000006

   1. Initial program 50.1%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f32 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.150000006 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 97.2%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{\left({u2}^{2} \cdot -2\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\mathsf{fma}\left({u2}^{2} \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(\color{blue}{{u2}^{2} \cdot -2}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(\color{blue}{\left(u2 \cdot u2\right)} \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(\color{blue}{\left(u2 \cdot u2\right)} \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-PI.f32 92.5

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

   7. Applied rewrites 92.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \leq 0.15000000596046448:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1 \cdot u1, u1\right)} \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024229 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))