Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.4% → 97.0%

Time: 8.6s

Alternatives: 7

Speedup: 7.7×

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

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

FPCore

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

Initial Program: 57.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;1 - u1 \leq 0.9970999956130981:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\cos t\_0 \cdot \sin t\_0\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(-u1\right) \cdot \left(0.5 + \frac{1}{u1}\right)\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (PI))))
   (if (<= (- 1.0 u1) 0.9970999956130981)
     (* (sqrt (- (log (- 1.0 u1)))) (* (* (cos t_0) (sin t_0)) 2.0))
     (*
      (sqrt (- (* (* (- u1) (+ 0.5 (/ 1.0 u1))) u1)))
      (sin (* (* 2.0 (PI)) u2))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997099996

   1. Initial program 95.5%

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

      1. lift-sin.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. sin-2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-sin.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 95.5

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

   4. Applied rewrites 95.5%

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

   if 0.997099996 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 46.7%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 37.7

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in u1 around inf

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

   8. Applied rewrites 98.0%

      \[\leadsto \sqrt{-\left(\left(-u1\right) \cdot \left(0.5 + \frac{1}{u1}\right)\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9970999956130981:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(-u1\right) \cdot \left(0.5 + \frac{1}{u1}\right)\right) \cdot u1} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* 2.0 (PI)) u2))))
   (if (<= (- 1.0 u1) 0.9970999956130981)
     (* (sqrt (- (log (- 1.0 u1)))) t_0)
     (* (sqrt (- (* (* (- u1) (+ 0.5 (/ 1.0 u1))) u1))) t_0))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997099996

   1. Initial program 95.5%

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

   if 0.997099996 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 46.7%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 38.3

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

   5. Applied rewrites 37.5%

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

   6. Taylor expanded in u1 around inf

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

   8. Applied rewrites 98.0%

      \[\leadsto \sqrt{-\left(\left(-u1\right) \cdot \left(0.5 + \frac{1}{u1}\right)\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 88.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\left(\left(-u1\right) \cdot \left(0.5 + \frac{1}{u1}\right)\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 34.9

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

5. Applied rewrites 34.8%

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

6. Taylor expanded in u1 around inf

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

8. Applied rewrites 86.8%

   \[\leadsto \sqrt{-\left(\left(-u1\right) \cdot \left(0.5 + \frac{1}{u1}\right)\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
9. Add Preprocessing

Alternative 4: 88.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\left(-1 - 0.5 \cdot u1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 35.5

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

5. Applied rewrites 35.1%

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

7. Applied rewrites 86.7%

   \[\leadsto \sqrt{-\left(-1 - 0.5 \cdot u1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
8. Add Preprocessing

Alternative 5: 76.7% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 60.8%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lower-log1p.f32 N/A

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

   11. lower-neg.f32 N/A

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

   12. metadata-eval 12.7

       \[\leadsto {\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 13.3%

   \[\leadsto \color{blue}{{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 75.1

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

7. Applied rewrites 75.1%

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

Alternative 6: 66.3% accurate, 7.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 75.1

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

5. Applied rewrites 75.1%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 6.5

       \[\leadsto \sqrt{-\left(-u1\right)} \cdot \left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), {\mathsf{PI}\left(\right)}^{3}, \color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot u2\right) \]

8. Applied rewrites 6.5%

   \[\leadsto \sqrt{-\left(-u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)} \]

9. Taylor expanded in u2 around 0

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

11. Applied rewrites 66.8%

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

Alternative 7: 7.1% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 0.0)

C

float code(float cosTheta_i, float u1, float u2) {
	return 0.0f;
}

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 = 0.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(0.0)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
end

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 75.1

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

5. Applied rewrites 75.1%

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

7. Applied rewrites -0.0%

   \[\leadsto \color{blue}{\sqrt{-u1} \cdot 0} \]

8. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{0} \]
9. Step-by-step derivation

10. Applied rewrites 7.1%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024305 
(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))))