Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.4% → 91.5%

Time: 9.4s

Alternatives: 6

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

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

FPCore

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

Initial Program: 57.4% 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: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{if}\;1 - u1 \leq 0.9998400211334229:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-\left(-u1\right)\right) - \left(-u1\right) \cdot u1} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* (PI) 2.0)))))
   (if (<= (- 1.0 u1) 0.9998400211334229)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (* (sqrt (- (- (- u1)) (* (- u1) u1))) t_0))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999840021

   1. Initial program 91.2%

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

   if 0.999840021 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 33.1%

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

   4. Applied rewrites 20.5%

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

   5. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 57.9

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

   7. Applied rewrites 57.9%

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

   8. Taylor expanded in u1 around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f32 93.1

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

   10. Applied rewrites 93.1%

       \[\leadsto \sqrt{-\left(\color{blue}{\left(-u1\right) \cdot u1} + \left(-u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998400211334229:\\ \;\;\;\;\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-\left(-u1\right)\right) - \left(-u1\right) \cdot u1} \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ t_1 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.02199999988079071:\\ \;\;\;\;\sqrt{\left(-\left(-u1\right)\right) - \left(-u1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* (PI) 2.0)))) (t_1 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 t_1) 0.02199999988079071)
     (* (sqrt (- (- (- u1)) (* (- u1) u1))) t_0)
     (* 1.0 t_1))))

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))) < 0.0219999999

   1. Initial program 38.5%

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

   4. Applied rewrites 20.6%

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

   5. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 55.9

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

   7. Applied rewrites 55.9%

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

   8. Taylor expanded in u1 around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f32 89.6

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

   10. Applied rewrites 89.6%

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

   if 0.0219999999 < (*.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 92.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 78.6%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.02199999988079071:\\ \;\;\;\;\sqrt{\left(-\left(-u1\right)\right) - \left(-u1\right) \cdot u1} \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ t_1 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.02199999988079071:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* (PI) 2.0)))) (t_1 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 t_1) 0.02199999988079071) (* (sqrt u1) t_0) (* 1.0 t_1))))

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))) < 0.0219999999

   1. Initial program 38.5%

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

   4. Applied rewrites 44.3%

      \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 89.5

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

   7. Applied rewrites 89.5%

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

   if 0.0219999999 < (*.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 92.4%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 78.6%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.02199999988079071:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot t\_0 \leq 0.013000000268220901:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* (cos (* u2 (* (PI) 2.0))) t_0) 0.013000000268220901)
     (* (pow (* u1 u1) 0.25) 1.0)
     (* 1.0 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))) < 0.0130000003

   1. Initial program 36.7%

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

   4. Applied rewrites 44.3%

      \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 90.6

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.6%

       \[\leadsto \sqrt{u1} \cdot \color{blue}{1} \]
   11. Step-by-step derivation

   12. Applied rewrites 69.6%

       \[\leadsto {\left(u1 \cdot u1\right)}^{\color{blue}{0.25}} \cdot 1 \]

   if 0.0130000003 < (*.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 90.9%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 77.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.013000000268220901:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(u1 \cdot u1\right)}^{0.25} \cdot 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (pow (* u1 u1) 0.25) 1.0))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf((u1 * u1), 0.25f) * 1.0f;
}

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 = ((u1 * u1) ** 0.25e0) * 1.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(u1 * u1) ^ Float32(0.25)) * Float32(1.0))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((u1 * u1) ^ single(0.25)) * single(1.0);
end

Derivation

1. Initial program 58.7%

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

4. Applied rewrites 39.2%

   \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 74.4

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

7. Applied rewrites 74.4%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{u1} \cdot \color{blue}{1} \]
9. Step-by-step derivation

10. Applied rewrites 60.9%

    \[\leadsto \sqrt{u1} \cdot \color{blue}{1} \]
11. Step-by-step derivation

12. Applied rewrites 60.9%

    \[\leadsto {\left(u1 \cdot u1\right)}^{\color{blue}{0.25}} \cdot 1 \]
13. Add Preprocessing

Alternative 6: 64.9% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \cdot 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (sqrt u1) 1.0))

C

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

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(u1) * 1.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.7%

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

4. Applied rewrites 39.2%

   \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 74.4

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

7. Applied rewrites 74.4%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{u1} \cdot \color{blue}{1} \]
9. Step-by-step derivation

10. Applied rewrites 60.9%

    \[\leadsto \sqrt{u1} \cdot \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024257 
(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))))