Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.8% → 57.8%

Time: 2.9s

Alternatives: 7

Speedup: 1.0×

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

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

FPCore

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

Initial Program: 57.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 57.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.3%

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

Alternative 2: 53.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\\ t_2 := \cos t\_1\\ \mathbf{if}\;t\_0 \cdot t\_2 \leq 0:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1)))))
        (t_1 (* (* 2.0 (PI)) u2))
        (t_2 (cos t_1)))
   (if (<= (* t_0 t_2) 0.0) (* t_1 t_2) t_0)))

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))) < -0.0

   1. Initial program 12.6%

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

   3. Taylor expanded in u1 around 0

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

   5. Applied rewrites 19.0%

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

   if -0.0 < (*.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 71.0%

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

   3. Taylor expanded in u1 around 0

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

   5. Applied rewrites 62.5%

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

Alternative 3: 49.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (- (log (- 1.0 u1)))))

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(-log((1.0e0 - u1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
end

MATLAB

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in u1 around 0

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

5. Applied rewrites 47.4%

   \[\leadsto \color{blue}{\sqrt{-\log \left(1 - u1\right)}} \]
6. Add Preprocessing

Alternative 4: 20.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right) + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{6} \cdot \left(\sqrt{u1} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

5. Applied rewrites 17.2%

   \[\leadsto \color{blue}{-\log \left(1 - u1\right)} \]

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 19.8%

   \[\leadsto \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
9. Add Preprocessing

Alternative 5: 18.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2 \end{array} \]

FPCore

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right) + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{6} \cdot \left(\sqrt{u1} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

5. Applied rewrites 17.2%

   \[\leadsto \color{blue}{-\log \left(1 - u1\right)} \]

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 19.8%

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

9. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right) + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{u1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} + \frac{1}{16} \cdot \frac{1}{u1 \cdot {\left(\sqrt{-1}\right)}^{2}}\right)\right)\right) + \frac{-1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)} \]

11. Applied rewrites 19.2%

    \[\leadsto \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2} \]
12. Add Preprocessing

Alternative 6: 19.2% accurate, 57.8× speedup

Math

\[\begin{array}{l} \\ 1 - u1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (- 1.0 u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return 1.0f - u1;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 1.0e0 - u1
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(1.0) - u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(1.0) - u1;
end

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right) + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{u1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} + \frac{1}{16} \cdot \frac{1}{u1 \cdot {\left(\sqrt{-1}\right)}^{2}}\right)\right)\right) + \frac{-1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)} \]

5. Applied rewrites 4.8%

   \[\leadsto \color{blue}{\log \left(1 - u1\right)} \]

6. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right) + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{6} \cdot \left(\sqrt{u1} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

8. Applied rewrites 19.2%

   \[\leadsto \color{blue}{1 - u1} \]
9. Add Preprocessing

Alternative 7: 18.0% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{PI}\left(\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (PI))

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right) + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{6} \cdot \left(\sqrt{u1} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

5. Applied rewrites 17.2%

   \[\leadsto \color{blue}{-\log \left(1 - u1\right)} \]

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 19.8%

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

9. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right) + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{u1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} + \frac{1}{16} \cdot \frac{1}{u1 \cdot {\left(\sqrt{-1}\right)}^{2}}\right)\right)\right) + \frac{-1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)} \]

11. Applied rewrites 19.2%

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

12. Taylor expanded in u1 around 0

    \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right) + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{u1} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} + \frac{1}{16} \cdot \frac{1}{u1 \cdot {\left(\sqrt{-1}\right)}^{2}}\right)\right)\right) + \frac{-1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)} \]

14. Applied rewrites 18.0%

    \[\leadsto \color{blue}{\mathsf{PI}\left(\right)} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(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))))