Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.8% → 99.0%

Time: 10.3s

Alternatives: 13

Speedup: 10.5×

Specification

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))

C

float code(float alpha, float u0) {
	return (-alpha * alpha) * logf((1.0f - u0));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (-alpha * alpha) * log((1.0e0 - u0))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (-alpha * alpha) * log((single(1.0) - u0));
end

Initial Program: 55.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))

C

float code(float alpha, float u0) {
	return (-alpha * alpha) * logf((1.0f - u0));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (-alpha * alpha) * log((1.0e0 - u0))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (-alpha * alpha) * log((single(1.0) - u0));
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-\alpha\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* alpha (* (log1p (- u0)) (- alpha))))

C

float code(float alpha, float u0) {
	return alpha * (log1pf(-u0) * -alpha);
}

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(log1p(Float32(-u0)) * Float32(-alpha)))
end

Derivation

1. Initial program 58.1%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\log \left(1 - u0\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \log \color{blue}{\left(1 - u0\right)} \]

   3. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)} \]

   4. lower-log1p.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)} \]

   5. lower-neg.f32 98.9

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\color{blue}{-u0}\right) \]

4. Applied rewrites 98.9%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)} \]
5. Step-by-step derivation

   1. lift-neg.f32 N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \alpha\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \alpha\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   4. lft-mult-inverse N/A

      \[\leadsto \left(\left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\alpha} \cdot \alpha}\right)\right) \cdot \alpha\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   5. lift-/.f32 N/A

      \[\leadsto \left(\left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\alpha}} \cdot \alpha\right)\right) \cdot \alpha\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\left(\left(\mathsf{neg}\left(\color{blue}{\alpha \cdot \frac{1}{\alpha}}\right)\right) \cdot \alpha\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   7. distribute-lft-neg-out N/A

      \[\leadsto \left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \frac{1}{\alpha}\right)} \cdot \alpha\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   8. lift-neg.f32 N/A

      \[\leadsto \left(\left(\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \frac{1}{\alpha}\right) \cdot \alpha\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{\alpha} \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \left(\color{blue}{\left(\frac{1}{\alpha} \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \left(\left(\frac{1}{\alpha} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)}\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \frac{1}{\alpha}\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   13. lift-/.f32 N/A

       \[\leadsto \left(\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\frac{1}{\alpha}}\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   14. div-inv N/A

       \[\leadsto \left(\color{blue}{\frac{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}{\alpha}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   15. clear-num N/A

       \[\leadsto \left(\color{blue}{\frac{1}{\frac{\alpha}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   16. lower-/.f32 N/A

       \[\leadsto \left(\color{blue}{\frac{1}{\frac{\alpha}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   17. clear-num N/A

       \[\leadsto \left(\frac{1}{\color{blue}{\frac{1}{\frac{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}{\alpha}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}{\alpha}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   19. div-inv N/A

       \[\leadsto \left(\frac{1}{\frac{\mathsf{neg}\left(-1\right)}{\color{blue}{\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \frac{1}{\alpha}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   20. lift-/.f32 N/A

       \[\leadsto \left(\frac{1}{\frac{\mathsf{neg}\left(-1\right)}{\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\frac{1}{\alpha}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   21. *-commutative N/A

       \[\leadsto \left(\frac{1}{\frac{\mathsf{neg}\left(-1\right)}{\color{blue}{\frac{1}{\alpha} \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   22. lift-*.f32 N/A

       \[\leadsto \left(\frac{1}{\frac{\mathsf{neg}\left(-1\right)}{\frac{1}{\alpha} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   23. associate-*r* N/A

       \[\leadsto \left(\frac{1}{\frac{\mathsf{neg}\left(-1\right)}{\color{blue}{\left(\frac{1}{\alpha} \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   24. *-commutative N/A

       \[\leadsto \left(\frac{1}{\frac{\mathsf{neg}\left(-1\right)}{\color{blue}{\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \frac{1}{\alpha}\right)} \cdot \alpha}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

6. Applied rewrites 98.7%

   \[\leadsto \left(\color{blue}{\frac{1}{\frac{-1}{\alpha}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right)} \]

   3. lift-log1p.f32 N/A

      \[\leadsto \color{blue}{\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)} \cdot \left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right) \]

   4. lift-neg.f32 N/A

      \[\leadsto \log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u0\right)\right)}\right) \cdot \left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right) \]

   5. sub-neg N/A

      \[\leadsto \log \color{blue}{\left(1 - u0\right)} \cdot \left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right) \]

   6. lift--.f32 N/A

      \[\leadsto \log \color{blue}{\left(1 - u0\right)} \cdot \left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right) \]

   7. lift-log.f32 N/A

      \[\leadsto \color{blue}{\log \left(1 - u0\right)} \cdot \left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \log \left(1 - u0\right) \cdot \color{blue}{\left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \log \left(1 - u0\right) \cdot \left(\color{blue}{\frac{1}{\frac{-1}{\alpha}}} \cdot \alpha\right) \]

   10. lift-/.f32 N/A

       \[\leadsto \log \left(1 - u0\right) \cdot \left(\frac{1}{\color{blue}{\frac{-1}{\alpha}}} \cdot \alpha\right) \]

   11. associate-/r/ N/A

       \[\leadsto \log \left(1 - u0\right) \cdot \left(\color{blue}{\left(\frac{1}{-1} \cdot \alpha\right)} \cdot \alpha\right) \]

   12. metadata-eval N/A

       \[\leadsto \log \left(1 - u0\right) \cdot \left(\left(\color{blue}{-1} \cdot \alpha\right) \cdot \alpha\right) \]

   13. neg-mul-1 N/A

       \[\leadsto \log \left(1 - u0\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right) \]

   14. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\log \left(1 - u0\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha} \]

   15. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(\log \left(1 - u0\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha} \]

   16. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(\log \left(1 - u0\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha \]

   17. lift-log.f32 N/A

       \[\leadsto \left(\color{blue}{\log \left(1 - u0\right)} \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha \]

   18. lift--.f32 N/A

       \[\leadsto \left(\log \color{blue}{\left(1 - u0\right)} \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha \]

   19. sub-neg N/A

       \[\leadsto \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha \]

   20. lift-neg.f32 N/A

       \[\leadsto \left(\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u0\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha \]

   21. lift-log1p.f32 N/A

       \[\leadsto \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)} \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha \]

   22. lift-neg.f32 99.0

       \[\leadsto \left(\mathsf{log1p}\left(-u0\right) \cdot \color{blue}{\left(-\alpha\right)}\right) \cdot \alpha \]

8. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(-u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha} \]

9. Final simplification 99.0%

   \[\leadsto \alpha \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-\alpha\right)\right) \]
10. Add Preprocessing

Alternative 2: 93.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(u0 \cdot \alpha\right) \cdot \mathsf{fma}\left(\alpha, 0.5, u0 \cdot \left(\alpha \cdot \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  u0
  (fma
   alpha
   alpha
   (*
    (* u0 alpha)
    (fma alpha 0.5 (* u0 (* alpha (fma u0 0.25 0.3333333333333333))))))))

C

float code(float alpha, float u0) {
	return u0 * fmaf(alpha, alpha, ((u0 * alpha) * fmaf(alpha, 0.5f, (u0 * (alpha * fmaf(u0, 0.25f, 0.3333333333333333f))))));
}

Julia

function code(alpha, u0)
	return Float32(u0 * fma(alpha, alpha, Float32(Float32(u0 * alpha) * fma(alpha, Float32(0.5), Float32(u0 * Float32(alpha * fma(u0, Float32(0.25), Float32(0.3333333333333333))))))))
end

Derivation

1. Initial program 55.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right)} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right)} \]

   2. lower-fma.f32 N/A

      \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right), {\alpha}^{2}\right)} \]

5. Applied rewrites 93.5%

   \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(u0, \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0, u0 \cdot 0.25, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)\right), \alpha \cdot \alpha\right)} \]
6. Step-by-step derivation

7. Applied rewrites 93.7%

   \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \color{blue}{\alpha}, \left(\alpha \cdot u0\right) \cdot \left(\alpha \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right)\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 93.7%

   \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(\alpha \cdot u0\right) \cdot \mathsf{fma}\left(\alpha, 0.5, u0 \cdot \left(\mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right) \cdot \alpha\right)\right)\right) \]

10. Final simplification 93.7%

    \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(u0 \cdot \alpha\right) \cdot \mathsf{fma}\left(\alpha, 0.5, u0 \cdot \left(\alpha \cdot \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right)\right)\right)\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (alpha u0)
  :name "Beckmann Distribution sample, tan2theta, alphax == alphay"
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0)) (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* (* (- alpha) alpha) (log (- 1.0 u0))))