Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.0% → 99.0%

Time: 8.2s

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, 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 59.7%

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

   1. associate-*l* 59.7%

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

   2. sub-neg 59.7%

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

   3. log1p-def 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 87.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \left(-\alpha\right) \cdot \left(\alpha \cdot \left(u0 \cdot \left(-1 + u0 \cdot -0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (- alpha) (* alpha (* u0 (+ -1.0 (* u0 -0.5))))))

C

float code(float alpha, float u0) {
	return -alpha * (alpha * (u0 * (-1.0f + (u0 * -0.5f))));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = -alpha * (alpha * (u0 * ((-1.0e0) + (u0 * (-0.5e0)))))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(-alpha) * Float32(alpha * Float32(u0 * Float32(Float32(-1.0) + Float32(u0 * Float32(-0.5))))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = -alpha * (alpha * (u0 * (single(-1.0) + (u0 * single(-0.5)))));
end

Derivation

1. Initial program 59.7%

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

   1. associate-*l* 59.7%

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

   2. sub-neg 59.7%

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

   3. log1p-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in u0 around 0 84.8%

   \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(-1 \cdot \left(\alpha \cdot u0\right) + -0.5 \cdot \left(\alpha \cdot {u0}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. mul-1-neg 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\color{blue}{\left(-\alpha \cdot u0\right)} + -0.5 \cdot \left(\alpha \cdot {u0}^{2}\right)\right) \]

   2. distribute-rgt-neg-in 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\color{blue}{\alpha \cdot \left(-u0\right)} + -0.5 \cdot \left(\alpha \cdot {u0}^{2}\right)\right) \]

   3. mul-1-neg 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\left(-1 \cdot u0\right)} + -0.5 \cdot \left(\alpha \cdot {u0}^{2}\right)\right) \]

   4. *-commutative 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-1 \cdot u0\right) + \color{blue}{\left(\alpha \cdot {u0}^{2}\right) \cdot -0.5}\right) \]

   5. associate-*l* 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-1 \cdot u0\right) + \color{blue}{\alpha \cdot \left({u0}^{2} \cdot -0.5\right)}\right) \]

   6. *-commutative 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-1 \cdot u0\right) + \alpha \cdot \color{blue}{\left(-0.5 \cdot {u0}^{2}\right)}\right) \]

   7. distribute-lft-out 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(\alpha \cdot \left(-1 \cdot u0 + -0.5 \cdot {u0}^{2}\right)\right)} \]

   8. *-commutative 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(\color{blue}{u0 \cdot -1} + -0.5 \cdot {u0}^{2}\right)\right) \]

   9. *-commutative 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(u0 \cdot -1 + \color{blue}{{u0}^{2} \cdot -0.5}\right)\right) \]

   10. unpow2 84.8%

       \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(u0 \cdot -1 + \color{blue}{\left(u0 \cdot u0\right)} \cdot -0.5\right)\right) \]

   11. associate-*l* 84.8%

       \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(u0 \cdot -1 + \color{blue}{u0 \cdot \left(u0 \cdot -0.5\right)}\right)\right) \]

   12. distribute-lft-out 84.6%

       \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\left(u0 \cdot \left(-1 + u0 \cdot -0.5\right)\right)}\right) \]

6. Simplified 84.6%

   \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(\alpha \cdot \left(u0 \cdot \left(-1 + u0 \cdot -0.5\right)\right)\right)} \]

7. Final simplification 84.6%

   \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(u0 \cdot \left(-1 + u0 \cdot -0.5\right)\right)\right) \]

Alternative 3: 87.4% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot \left(\alpha \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* alpha (* alpha (- u0 (* u0 (* u0 -0.5))))))

C

float code(float alpha, float u0) {
	return alpha * (alpha * (u0 - (u0 * (u0 * -0.5f))));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (alpha * (u0 - (u0 * (u0 * (-0.5e0)))))
end function

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(alpha * Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5))))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * (u0 - (u0 * (u0 * single(-0.5)))));
end

Derivation

1. Initial program 59.7%

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

   1. associate-*l* 59.7%

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

   2. sub-neg 59.7%

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

   3. log1p-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in u0 around 0 84.8%

   \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(-1 \cdot \left(\alpha \cdot u0\right) + -0.5 \cdot \left(\alpha \cdot {u0}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. mul-1-neg 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\color{blue}{\left(-\alpha \cdot u0\right)} + -0.5 \cdot \left(\alpha \cdot {u0}^{2}\right)\right) \]

   2. distribute-rgt-neg-in 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\color{blue}{\alpha \cdot \left(-u0\right)} + -0.5 \cdot \left(\alpha \cdot {u0}^{2}\right)\right) \]

   3. mul-1-neg 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\left(-1 \cdot u0\right)} + -0.5 \cdot \left(\alpha \cdot {u0}^{2}\right)\right) \]

   4. *-commutative 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-1 \cdot u0\right) + \color{blue}{\left(\alpha \cdot {u0}^{2}\right) \cdot -0.5}\right) \]

   5. associate-*l* 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-1 \cdot u0\right) + \color{blue}{\alpha \cdot \left({u0}^{2} \cdot -0.5\right)}\right) \]

   6. *-commutative 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-1 \cdot u0\right) + \alpha \cdot \color{blue}{\left(-0.5 \cdot {u0}^{2}\right)}\right) \]

   7. distribute-lft-out 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(\alpha \cdot \left(-1 \cdot u0 + -0.5 \cdot {u0}^{2}\right)\right)} \]

   8. *-commutative 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(\color{blue}{u0 \cdot -1} + -0.5 \cdot {u0}^{2}\right)\right) \]

   9. *-commutative 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(u0 \cdot -1 + \color{blue}{{u0}^{2} \cdot -0.5}\right)\right) \]

   10. unpow2 84.8%

       \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(u0 \cdot -1 + \color{blue}{\left(u0 \cdot u0\right)} \cdot -0.5\right)\right) \]

   11. associate-*l* 84.8%

       \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(u0 \cdot -1 + \color{blue}{u0 \cdot \left(u0 \cdot -0.5\right)}\right)\right) \]

   12. distribute-lft-out 84.6%

       \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\left(u0 \cdot \left(-1 + u0 \cdot -0.5\right)\right)}\right) \]

6. Simplified 84.6%

   \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(\alpha \cdot \left(u0 \cdot \left(-1 + u0 \cdot -0.5\right)\right)\right)} \]
7. Step-by-step derivation

   1. distribute-lft-in 84.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\left(u0 \cdot -1 + u0 \cdot \left(u0 \cdot -0.5\right)\right)}\right) \]

8. Applied egg-rr 84.8%

   \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\left(u0 \cdot -1 + u0 \cdot \left(u0 \cdot -0.5\right)\right)}\right) \]

9. Final simplification 84.8%

   \[\leadsto \alpha \cdot \left(\alpha \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)\right) \]

Alternative 4: 74.6% 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 59.7%

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

   1. associate-*l* 59.7%

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

   2. sub-neg 59.7%

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

   3. log1p-def 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* 99.0%

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

   2. log1p-udef 59.7%

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

   3. sub-neg 59.7%

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

   4. add-sqr-sqrt 59.6%

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

   5. pow2 59.6%

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

5. Applied egg-rr 69.6%

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

6. Taylor expanded in u0 around 0 71.7%

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

   1. unpow2 71.7%

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

   2. *-commutative 71.7%

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

   3. associate-*r* 71.8%

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

   4. associate-*r* 71.7%

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

   5. add-sqr-sqrt 72.0%

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

8. Applied egg-rr 72.0%

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

9. Final simplification 72.0%

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

Reproduce

herbie shell --seed 2023308 
(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))))