Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.2% → 98.9%

Time: 7.7s

Alternatives: 9

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: 56.2% 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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.4%

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

   1. *-commutative 56.4%

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

   2. sub-neg 56.4%

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

   3. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 93.4% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (u0 * (alpha + (u0 * ((alpha * 0.5e0) + (u0 * ((0.25e0 * (alpha * u0)) + (alpha * 0.3333333333333333e0)))))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (u0 * (alpha + (u0 * ((alpha * single(0.5)) + (u0 * ((single(0.25) * (alpha * u0)) + (alpha * single(0.3333333333333333))))))));
end

Derivation

1. Initial program 56.4%

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

   1. *-commutative 56.4%

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

   2. sub-neg 56.4%

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

   3. log1p-define 99.0%

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

3. Simplified 99.0%

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

   1. log1p-undefine 56.4%

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

   2. flip-+ 53.2%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 5.0%

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

   5. sqr-neg 5.0%

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

   6. sqrt-unprod 5.0%

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

   7. add-sqr-sqrt 5.0%

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

   8. unsub-neg 5.0%

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

   9. rem-exp-log 5.0%

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

   10. log1p-undefine 5.0%

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

   11. log-div 5.0%

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

   12. metadata-eval 5.0%

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

   13. sqr-neg 5.0%

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

   14. pow2 5.0%

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

   15. add-log-exp 4.4%

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

   16. add-sqr-sqrt -0.0%

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

   17. sqrt-unprod 89.3%

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

   18. sqr-neg 89.3%

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

   19. sqrt-unprod 89.1%

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

6. Applied egg-rr 89.3%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \color{blue}{\left(\log \left(1 - {u0}^{2}\right) - \mathsf{log1p}\left(u0\right)\right)} \]
7. Step-by-step derivation

   1. sub-neg 89.3%

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

   2. log1p-define 98.8%

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

8. Simplified 98.8%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(-{u0}^{2}\right) - \mathsf{log1p}\left(u0\right)\right)} \]
9. Step-by-step derivation

   1. add-cube-cbrt 97.4%

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

   2. pow3 97.5%

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

10. Applied egg-rr 97.5%

    \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(\mathsf{log1p}\left(-{u0}^{2}\right) - \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(u0\right)}\right)}^{3}}\right) \]
11. Step-by-step derivation

    1. associate-*l* 97.4%

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

    2. rem-cube-cbrt 98.8%

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

12. Applied egg-rr 98.8%

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

13. Taylor expanded in u0 around 0 93.5%

    \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + u0 \cdot \left(0.25 \cdot \left(\alpha \cdot u0\right) + 0.3333333333333333 \cdot \alpha\right)\right)\right)\right)} \]

14. Final simplification 93.5%

    \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + u0 \cdot \left(\alpha \cdot 0.5 + u0 \cdot \left(0.25 \cdot \left(\alpha \cdot u0\right) + \alpha \cdot 0.3333333333333333\right)\right)\right)\right) \]
15. Add Preprocessing

Alternative 3: 93.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 - u0 \cdot \left(u0 \cdot -0.25 - 0.3333333333333333\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (* alpha alpha)
  (* u0 (+ 1.0 (* u0 (- 0.5 (* u0 (- (* u0 -0.25) 0.3333333333333333))))))))

C

float code(float alpha, float u0) {
	return (alpha * alpha) * (u0 * (1.0f + (u0 * (0.5f - (u0 * ((u0 * -0.25f) - 0.3333333333333333f))))));
}

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 - (u0 * ((u0 * (-0.25e0)) - 0.3333333333333333e0))))))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * alpha) * Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) - Float32(u0 * Float32(Float32(u0 * Float32(-0.25)) - Float32(0.3333333333333333))))))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (alpha * alpha) * (u0 * (single(1.0) + (u0 * (single(0.5) - (u0 * ((u0 * single(-0.25)) - single(0.3333333333333333)))))));
end

Derivation

1. Initial program 56.4%

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

   1. *-commutative 56.4%

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

   2. sub-neg 56.4%

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

   3. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u0 around 0 93.3%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} \]

6. Final simplification 93.3%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 - u0 \cdot \left(u0 \cdot -0.25 - 0.3333333333333333\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 4: 93.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 - u0 \cdot \left(u0 \cdot -0.25 - 0.3333333333333333\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  alpha
  (*
   alpha
   (* u0 (+ 1.0 (* u0 (- 0.5 (* u0 (- (* u0 -0.25) 0.3333333333333333)))))))))

C

float code(float alpha, float u0) {
	return alpha * (alpha * (u0 * (1.0f + (u0 * (0.5f - (u0 * ((u0 * -0.25f) - 0.3333333333333333f)))))));
}

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 - (u0 * ((u0 * (-0.25e0)) - 0.3333333333333333e0)))))))
end function

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(alpha * Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) - Float32(u0 * Float32(Float32(u0 * Float32(-0.25)) - Float32(0.3333333333333333)))))))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * (u0 * (single(1.0) + (u0 * (single(0.5) - (u0 * ((u0 * single(-0.25)) - single(0.3333333333333333))))))));
end

Derivation

1. Initial program 56.4%

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

   1. *-commutative 56.4%

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

   2. sub-neg 56.4%

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

   3. log1p-define 99.0%

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

3. Simplified 99.0%

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

   1. log1p-undefine 56.4%

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

   2. flip-+ 53.2%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 5.0%

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

   5. sqr-neg 5.0%

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

   6. sqrt-unprod 5.0%

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

   7. add-sqr-sqrt 5.0%

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

   8. unsub-neg 5.0%

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

   9. rem-exp-log 5.0%

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

   10. log1p-undefine 5.0%

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

   11. log-div 5.0%

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

   12. metadata-eval 5.0%

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

   13. sqr-neg 5.0%

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

   14. pow2 5.0%

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

   15. add-log-exp 4.4%

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

   16. add-sqr-sqrt -0.0%

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

   17. sqrt-unprod 89.3%

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

   18. sqr-neg 89.3%

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

   19. sqrt-unprod 89.1%

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

6. Applied egg-rr 89.3%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \color{blue}{\left(\log \left(1 - {u0}^{2}\right) - \mathsf{log1p}\left(u0\right)\right)} \]
7. Step-by-step derivation

   1. sub-neg 89.3%

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

   2. log1p-define 98.8%

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

8. Simplified 98.8%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(-{u0}^{2}\right) - \mathsf{log1p}\left(u0\right)\right)} \]
9. Step-by-step derivation

   1. add-cube-cbrt 97.4%

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

   2. pow3 97.5%

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

10. Applied egg-rr 97.5%

    \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(\mathsf{log1p}\left(-{u0}^{2}\right) - \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(u0\right)}\right)}^{3}}\right) \]
11. Step-by-step derivation

    1. associate-*l* 97.4%

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

    2. rem-cube-cbrt 98.8%

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

12. Applied egg-rr 98.8%

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

13. Taylor expanded in u0 around 0 93.2%

    \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}\right) \]

14. Final simplification 93.2%

    \[\leadsto \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 - u0 \cdot \left(u0 \cdot -0.25 - 0.3333333333333333\right)\right)\right)\right)\right) \]
15. Add Preprocessing

Alternative 5: 91.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot \left(u0 \cdot \left(\alpha + u0 \cdot \left(\alpha \cdot 0.5 + \left(\alpha \cdot u0\right) \cdot 0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  alpha
  (*
   u0
   (+ alpha (* u0 (+ (* alpha 0.5) (* (* alpha u0) 0.3333333333333333)))))))

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (u0 * (alpha + (u0 * ((alpha * 0.5e0) + ((alpha * u0) * 0.3333333333333333e0)))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (u0 * (alpha + (u0 * ((alpha * single(0.5)) + ((alpha * u0) * single(0.3333333333333333))))));
end

Derivation

1. Initial program 56.4%

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

   1. *-commutative 56.4%

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

   2. sub-neg 56.4%

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

   3. log1p-define 99.0%

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

3. Simplified 99.0%

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

   1. log1p-undefine 56.4%

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

   2. flip-+ 53.2%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 5.0%

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

   5. sqr-neg 5.0%

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

   6. sqrt-unprod 5.0%

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

   7. add-sqr-sqrt 5.0%

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

   8. unsub-neg 5.0%

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

   9. rem-exp-log 5.0%

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

   10. log1p-undefine 5.0%

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

   11. log-div 5.0%

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

   12. metadata-eval 5.0%

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

   13. sqr-neg 5.0%

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

   14. pow2 5.0%

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

   15. add-log-exp 4.4%

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

   16. add-sqr-sqrt -0.0%

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

   17. sqrt-unprod 89.3%

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

   18. sqr-neg 89.3%

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

   19. sqrt-unprod 89.1%

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

6. Applied egg-rr 89.3%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \color{blue}{\left(\log \left(1 - {u0}^{2}\right) - \mathsf{log1p}\left(u0\right)\right)} \]
7. Step-by-step derivation

   1. sub-neg 89.3%

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

   2. log1p-define 98.8%

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

8. Simplified 98.8%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(-{u0}^{2}\right) - \mathsf{log1p}\left(u0\right)\right)} \]
9. Step-by-step derivation

   1. add-cube-cbrt 97.4%

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

   2. pow3 97.5%

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

10. Applied egg-rr 97.5%

    \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(\mathsf{log1p}\left(-{u0}^{2}\right) - \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(u0\right)}\right)}^{3}}\right) \]
11. Step-by-step derivation

    1. associate-*l* 97.4%

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

    2. rem-cube-cbrt 98.8%

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

12. Applied egg-rr 98.8%

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

13. Taylor expanded in u0 around 0 91.4%

    \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + u0 \cdot \left(0.3333333333333333 \cdot \left(\alpha \cdot u0\right) + 0.5 \cdot \alpha\right)\right)\right)} \]

14. Final simplification 91.4%

    \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + u0 \cdot \left(\alpha \cdot 0.5 + \left(\alpha \cdot u0\right) \cdot 0.3333333333333333\right)\right)\right) \]
15. Add Preprocessing

Alternative 6: 91.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 - u0 \cdot -0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* alpha alpha) (* u0 (+ 1.0 (* u0 (- 0.5 (* u0 -0.3333333333333333)))))))

C

float code(float alpha, float u0) {
	return (alpha * alpha) * (u0 * (1.0f + (u0 * (0.5f - (u0 * -0.3333333333333333f)))));
}

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 - (u0 * (-0.3333333333333333e0))))))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * alpha) * Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) - Float32(u0 * Float32(-0.3333333333333333)))))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (alpha * alpha) * (u0 * (single(1.0) + (u0 * (single(0.5) - (u0 * single(-0.3333333333333333))))));
end

Derivation

1. Initial program 56.4%

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

   1. *-commutative 56.4%

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

   2. sub-neg 56.4%

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

   3. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u0 around 0 91.2%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)} \]

6. Final simplification 91.2%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 - u0 \cdot -0.3333333333333333\right)\right)\right) \]
7. Add Preprocessing

Alternative 7: 91.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 - u0 \cdot -0.3333333333333333\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* alpha (* alpha (* u0 (+ 1.0 (* u0 (- 0.5 (* u0 -0.3333333333333333))))))))

C

float code(float alpha, float u0) {
	return alpha * (alpha * (u0 * (1.0f + (u0 * (0.5f - (u0 * -0.3333333333333333f))))));
}

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 - (u0 * (-0.3333333333333333e0)))))))
end function

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(alpha * Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) - Float32(u0 * Float32(-0.3333333333333333))))))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * (u0 * (single(1.0) + (u0 * (single(0.5) - (u0 * single(-0.3333333333333333)))))));
end

Derivation

1. Initial program 56.4%

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

   1. *-commutative 56.4%

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

   2. sub-neg 56.4%

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

   3. log1p-define 99.0%

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

3. Simplified 99.0%

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

   1. log1p-undefine 56.4%

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

   2. flip-+ 53.2%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 5.0%

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

   5. sqr-neg 5.0%

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

   6. sqrt-unprod 5.0%

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

   7. add-sqr-sqrt 5.0%

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

   8. unsub-neg 5.0%

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

   9. rem-exp-log 5.0%

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

   10. log1p-undefine 5.0%

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

   11. log-div 5.0%

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

   12. metadata-eval 5.0%

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

   13. sqr-neg 5.0%

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

   14. pow2 5.0%

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

   15. add-log-exp 4.4%

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

   16. add-sqr-sqrt -0.0%

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

   17. sqrt-unprod 89.3%

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

   18. sqr-neg 89.3%

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

   19. sqrt-unprod 89.1%

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

6. Applied egg-rr 89.3%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \color{blue}{\left(\log \left(1 - {u0}^{2}\right) - \mathsf{log1p}\left(u0\right)\right)} \]
7. Step-by-step derivation

   1. sub-neg 89.3%

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

   2. log1p-define 98.8%

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

8. Simplified 98.8%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(-{u0}^{2}\right) - \mathsf{log1p}\left(u0\right)\right)} \]
9. Step-by-step derivation

   1. add-cube-cbrt 97.4%

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

   2. pow3 97.5%

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

10. Applied egg-rr 97.5%

    \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(\mathsf{log1p}\left(-{u0}^{2}\right) - \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(u0\right)}\right)}^{3}}\right) \]
11. Step-by-step derivation

    1. associate-*l* 97.4%

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

    2. rem-cube-cbrt 98.8%

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

12. Applied egg-rr 98.8%

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

13. Taylor expanded in u0 around 0 91.2%

    \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)\right)}\right) \]

14. Final simplification 91.2%

    \[\leadsto \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 - u0 \cdot -0.3333333333333333\right)\right)\right)\right) \]
15. Add Preprocessing

Alternative 8: 87.2% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.4%

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

   1. *-commutative 56.4%

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

   2. sub-neg 56.4%

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

   3. log1p-define 99.0%

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

3. Simplified 99.0%

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

   1. log1p-undefine 56.4%

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

   2. flip-+ 53.2%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 5.0%

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

   5. sqr-neg 5.0%

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

   6. sqrt-unprod 5.0%

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

   7. add-sqr-sqrt 5.0%

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

   8. unsub-neg 5.0%

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

   9. rem-exp-log 5.0%

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

   10. log1p-undefine 5.0%

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

   11. log-div 5.0%

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

   12. metadata-eval 5.0%

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

   13. sqr-neg 5.0%

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

   14. pow2 5.0%

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

   15. add-log-exp 4.4%

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

   16. add-sqr-sqrt -0.0%

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

   17. sqrt-unprod 89.3%

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

   18. sqr-neg 89.3%

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

   19. sqrt-unprod 89.1%

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

6. Applied egg-rr 89.3%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \color{blue}{\left(\log \left(1 - {u0}^{2}\right) - \mathsf{log1p}\left(u0\right)\right)} \]
7. Step-by-step derivation

   1. sub-neg 89.3%

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

   2. log1p-define 98.8%

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

8. Simplified 98.8%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(-{u0}^{2}\right) - \mathsf{log1p}\left(u0\right)\right)} \]
9. Step-by-step derivation

   1. add-cube-cbrt 97.4%

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

   2. pow3 97.5%

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

10. Applied egg-rr 97.5%

    \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(\mathsf{log1p}\left(-{u0}^{2}\right) - \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(u0\right)}\right)}^{3}}\right) \]
11. Step-by-step derivation

    1. associate-*l* 97.4%

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

    2. rem-cube-cbrt 98.8%

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

12. Applied egg-rr 98.8%

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

13. Taylor expanded in u0 around 0 86.9%

    \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + 0.5 \cdot \left(\alpha \cdot u0\right)\right)\right)} \]
14. Add Preprocessing

Alternative 9: 74.3% 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 56.4%

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

   1. *-commutative 56.4%

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

   2. sub-neg 56.4%

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

   3. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u0 around 0 74.2%

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

   1. mul-1-neg 74.2%

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

7. Simplified 74.2%

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

   1. associate-*l* 74.3%

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

   2. add-sqr-sqrt -0.0%

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

   3. sqrt-unprod 4.4%

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

   4. sqr-neg 4.4%

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

   5. sqrt-prod 4.4%

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

   6. add-sqr-sqrt 4.4%

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

   7. add-sqr-sqrt -0.0%

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

   8. sqrt-unprod 74.3%

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

   9. sqr-neg 74.3%

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

   10. sqrt-prod 74.0%

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

   11. add-sqr-sqrt 74.3%

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

9. Applied egg-rr 74.3%

   \[\leadsto \color{blue}{\alpha \cdot \left(\alpha \cdot u0\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))