Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.4% → 99.0%

Time: 11.7s

Alternatives: 17

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: 55.4% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.4%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 99.0

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

4. Applied rewrites 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.4%

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

4. Applied rewrites 84.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. cube-unmult N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{{u0}^{3}}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, u0\right)\right)\right) \]

   4. sqr-pow N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{{u0}^{\left(\frac{3}{2}\right)} \cdot {u0}^{\left(\frac{3}{2}\right)}}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, u0\right)\right)\right) \]

   5. unpow-prod-down N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{{\left(u0 \cdot u0\right)}^{\left(\frac{3}{2}\right)}}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, u0\right)\right)\right) \]

   6. sqr-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\mathsf{log1p}\left({\color{blue}{\left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)}}^{\left(\frac{3}{2}\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, u0\right)\right)\right) \]

   7. lift-neg.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\mathsf{log1p}\left({\left(\color{blue}{\left(\mathsf{neg}\left(u0\right)\right)} \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, u0\right)\right)\right) \]

   8. lift-neg.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\mathsf{log1p}\left({\left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(u0\right)\right)}\right)}^{\left(\frac{3}{2}\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, u0\right)\right)\right) \]

   9. unpow-prod-down N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{{\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)}}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, u0\right)\right)\right) \]

   10. sqr-pow N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{{\left(\mathsf{neg}\left(u0\right)\right)}^{3}}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, u0\right)\right)\right) \]

   11. unpow3 N/A

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

   12. lift-neg.f32 N/A

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

   13. lift-neg.f32 N/A

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

   14. sqr-neg N/A

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

   15. lift-*.f32 N/A

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

   16. lower-*.f32 98.7

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

6. Applied rewrites 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

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%

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

Reproduce

herbie shell --seed 2024228 
(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))))