Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.0% → 99.0%

Time: 7.7s

Alternatives: 4

Speedup: 21.6×

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: 56.0% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ {\alpha}^{2} \cdot \left(-\mathsf{log1p}\left(-u0\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* (pow alpha 2.0) (- (log1p (- u0)))))

C

float code(float alpha, float u0) {
	return powf(alpha, 2.0f) * -log1pf(-u0);
}

Julia

function code(alpha, u0)
	return Float32((alpha ^ Float32(2.0)) * Float32(-log1p(Float32(-u0))))
end

Derivation

1. Initial program 55.0%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. log1p-def 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in alpha around 0 55.0%

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

   1. mul-1-neg 55.0%

      \[\leadsto \color{blue}{-{\alpha}^{2} \cdot \log \left(1 - u0\right)} \]

   2. distribute-rgt-neg-in 55.0%

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot \left(-\log \left(1 - u0\right)\right)} \]

   3. sub-neg 55.0%

      \[\leadsto {\alpha}^{2} \cdot \left(-\log \color{blue}{\left(1 + \left(-u0\right)\right)}\right) \]

   4. mul-1-neg 55.0%

      \[\leadsto {\alpha}^{2} \cdot \left(-\log \left(1 + \color{blue}{-1 \cdot u0}\right)\right) \]

   5. log1p-def 99.1%

      \[\leadsto {\alpha}^{2} \cdot \left(-\color{blue}{\mathsf{log1p}\left(-1 \cdot u0\right)}\right) \]

   6. mul-1-neg 99.1%

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

6. Simplified 99.1%

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

7. Final simplification 99.1%

   \[\leadsto {\alpha}^{2} \cdot \left(-\mathsf{log1p}\left(-u0\right)\right) \]

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

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(Float32(-alpha) * Float32(alpha * log1p(Float32(-u0))))
end

Derivation

1. Initial program 55.0%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. log1p-def 98.9%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 3: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\alpha}^{2} \cdot u0 \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* (pow alpha 2.0) u0))

C

float code(float alpha, float u0) {
	return powf(alpha, 2.0f) * u0;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (alpha ** 2.0e0) * u0
end function

Julia

function code(alpha, u0)
	return Float32((alpha ^ Float32(2.0)) * u0)
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (alpha ^ single(2.0)) * u0;
end

Derivation

1. Initial program 55.0%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. log1p-def 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in u0 around 0 75.5%

   \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]

5. Final simplification 75.5%

   \[\leadsto {\alpha}^{2} \cdot u0 \]

Alternative 4: 74.4% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (alpha * u0)
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * u0);
end

Derivation

1. Initial program 55.0%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. log1p-def 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in u0 around 0 75.4%

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

   1. mul-1-neg 75.4%

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

   2. distribute-rgt-neg-in 75.4%

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

6. Simplified 75.4%

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

   1. add-sqr-sqrt 75.1%

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

   2. sqrt-unprod 75.4%

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

   3. associate-*r* 75.5%

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

   4. associate-*r* 75.5%

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

   5. swap-sqr 75.4%

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

8. Applied egg-rr 75.5%

   \[\leadsto \color{blue}{\sqrt{{\alpha}^{4} \cdot {u0}^{2}}} \]
9. Step-by-step derivation

   1. *-commutative 75.5%

      \[\leadsto \sqrt{\color{blue}{{u0}^{2} \cdot {\alpha}^{4}}} \]

10. Simplified 75.5%

    \[\leadsto \color{blue}{\sqrt{{u0}^{2} \cdot {\alpha}^{4}}} \]
11. Step-by-step derivation

    1. *-commutative 75.5%

       \[\leadsto \sqrt{\color{blue}{{\alpha}^{4} \cdot {u0}^{2}}} \]

    2. metadata-eval 75.5%

       \[\leadsto \sqrt{{\alpha}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot {u0}^{2}} \]

    3. pow-sqr 75.4%

       \[\leadsto \sqrt{\color{blue}{\left({\alpha}^{2} \cdot {\alpha}^{2}\right)} \cdot {u0}^{2}} \]

    4. unpow2 75.4%

       \[\leadsto \sqrt{\left({\alpha}^{2} \cdot {\alpha}^{2}\right) \cdot \color{blue}{\left(u0 \cdot u0\right)}} \]

    5. swap-sqr 75.5%

       \[\leadsto \sqrt{\color{blue}{\left({\alpha}^{2} \cdot u0\right) \cdot \left({\alpha}^{2} \cdot u0\right)}} \]

    6. sqrt-unprod 75.1%

       \[\leadsto \color{blue}{\sqrt{{\alpha}^{2} \cdot u0} \cdot \sqrt{{\alpha}^{2} \cdot u0}} \]

    7. add-sqr-sqrt 75.5%

       \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]

    8. *-commutative 75.5%

       \[\leadsto \color{blue}{u0 \cdot {\alpha}^{2}} \]

    9. unpow2 75.5%

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

    10. associate-*r* 75.4%

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

    11. *-commutative 75.4%

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

12. Applied egg-rr 75.4%

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

13. Final simplification 75.4%

    \[\leadsto \alpha \cdot \left(\alpha \cdot u0\right) \]

Reproduce

herbie shell --seed 2023314 
(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))))