Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.1% → 98.4%

Time: 13.5s

Alternatives: 5

Speedup: 2.9×

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: 57.1% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.0%

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

   1. sub-neg 54.0%

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

   2. log1p-define 98.4%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 90.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot u2\right)\\ \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.010999999940395355:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* 2.0 (* PI u2))))
   (if (<= (* (* 2.0 PI) u2) 0.010999999940395355)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (* (sqrt u1) (sin t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 2.0f * (((float) M_PI) * u2);
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.010999999940395355f) {
		tmp = sqrtf(-log1pf(-u1)) * t_0;
	} else {
		tmp = sqrtf(u1) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(2.0) * Float32(Float32(pi) * u2))
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.010999999940395355))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	else
		tmp = Float32(sqrt(u1) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0109999999

   1. Initial program 52.5%

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

      1. sub-neg 52.5%

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

      2. log1p-define 98.6%

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

   3. Simplified 98.6%

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

      1. add-exp-log 94.4%

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

      2. associate-*l* 94.4%

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

   6. Applied egg-rr 94.4%

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

   7. Taylor expanded in u2 around 0 96.6%

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

   if 0.0109999999 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 57.6%

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

   3. Taylor expanded in u1 around 0 75.7%

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

      1. mul-1-neg 75.7%

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

   5. Simplified 75.7%

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

   6. Taylor expanded in u2 around inf 75.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.010999999940395355:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in u1 around 0 78.0%

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

   1. mul-1-neg 78.0%

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

5. Simplified 78.0%

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

6. Taylor expanded in u2 around inf 78.0%

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

7. Final simplification 78.0%

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

Alternative 4: 66.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in u1 around 0 78.0%

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

   1. mul-1-neg 78.0%

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

5. Simplified 78.0%

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

6. Taylor expanded in u2 around 0 66.9%

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

8. Simplified 66.9%

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

9. Taylor expanded in u2 around 0 66.9%

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

    1. associate-*r* 66.9%

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

11. Simplified 66.9%

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

12. Final simplification 66.9%

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

Alternative 5: 66.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\pi \cdot \sqrt{u1 \cdot 4}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* u2 (* PI (sqrt (* u1 4.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (((float) M_PI) * sqrtf((u1 * 4.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(Float32(pi) * sqrt(Float32(u1 * Float32(4.0)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (single(pi) * sqrt((u1 * single(4.0))));
end

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in u1 around 0 78.0%

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

   1. mul-1-neg 78.0%

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

5. Simplified 78.0%

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

6. Taylor expanded in u2 around 0 66.9%

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

8. Simplified 66.9%

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

   1. add-cube-cbrt 66.6%

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

   2. pow3 66.6%

      \[\leadsto \left(2 \cdot \sqrt{u1}\right) \cdot \left(u2 \cdot \color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}}\right) \]

10. Applied egg-rr 66.6%

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

    1. pow1 66.6%

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

    2. *-commutative 66.6%

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

    3. rem-cube-cbrt 66.9%

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

    4. associate-*l* 66.9%

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

    5. add-sqr-sqrt 66.8%

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

    6. sqrt-unprod 66.9%

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

    7. *-commutative 66.9%

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

    8. *-commutative 66.9%

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

    9. swap-sqr 66.9%

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

    10. add-sqr-sqrt 66.9%

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

    11. metadata-eval 66.9%

        \[\leadsto {\left(u2 \cdot \left(\pi \cdot \sqrt{u1 \cdot \color{blue}{4}}\right)\right)}^{1} \]

12. Applied egg-rr 66.9%

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

    1. unpow1 66.9%

       \[\leadsto \color{blue}{u2 \cdot \left(\pi \cdot \sqrt{u1 \cdot 4}\right)} \]

14. Simplified 66.9%

    \[\leadsto \color{blue}{u2 \cdot \left(\pi \cdot \sqrt{u1 \cdot 4}\right)} \]

15. Final simplification 66.9%

    \[\leadsto u2 \cdot \left(\pi \cdot \sqrt{u1 \cdot 4}\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))))