Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.5% → 96.6%

Time: 7.8s

Alternatives: 11

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: 55.5% 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: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;u0 \leq 0.002739999908953905:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(\frac{-1}{\left(\left({\alpha}^{-2} \cdot \alpha\right) \cdot {\alpha}^{1.5}\right) \cdot {\alpha}^{1.5}} \cdot {\alpha}^{3}\right) \cdot \alpha\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= u0 0.002739999908953905)
     (+ (* (* 0.5 u0) t_0) t_0)
     (*
      (log (- 1.0 u0))
      (*
       (*
        (/
         -1.0
         (* (* (* (pow alpha -2.0) alpha) (pow alpha 1.5)) (pow alpha 1.5)))
        (pow alpha 3.0))
       alpha)))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if (u0 <= 0.002739999908953905f) {
		tmp = ((0.5f * u0) * t_0) + t_0;
	} else {
		tmp = logf((1.0f - u0)) * (((-1.0f / (((powf(alpha, -2.0f) * alpha) * powf(alpha, 1.5f)) * powf(alpha, 1.5f))) * powf(alpha, 3.0f)) * alpha);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if (u0 <= 0.002739999908953905e0) then
        tmp = ((0.5e0 * u0) * t_0) + t_0
    else
        tmp = log((1.0e0 - u0)) * ((((-1.0e0) / ((((alpha ** (-2.0e0)) * alpha) * (alpha ** 1.5e0)) * (alpha ** 1.5e0))) * (alpha ** 3.0e0)) * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.002739999908953905))
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	else
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(Float32(Float32(-1.0) / Float32(Float32(Float32((alpha ^ Float32(-2.0)) * alpha) * (alpha ^ Float32(1.5))) * (alpha ^ Float32(1.5)))) * (alpha ^ Float32(3.0))) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if (u0 <= single(0.002739999908953905))
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	else
		tmp = log((single(1.0) - u0)) * (((single(-1.0) / ((((alpha ^ single(-2.0)) * alpha) * (alpha ^ single(1.5))) * (alpha ^ single(1.5)))) * (alpha ^ single(3.0))) * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00273999991

   1. Initial program 43.5%

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{\left(0 + \left(-\alpha\right) \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{0 - \left(-\alpha\right) \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      3. neg-sub0 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\mathsf{neg}\left(\left(-\alpha\right) \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \log \left(1 - u0\right) \]

      5. lift-neg.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)}\right)} \cdot \log \left(1 - u0\right) \]

      7. remove-double-neg N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      10. lower--.f32 N/A

          \[\leadsto \frac{\color{blue}{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \frac{0 - \color{blue}{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      12. lower-*.f32 43.5

          \[\leadsto \frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   5. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 28.4

          \[\leadsto \mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \]

   7. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.1%

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

   if 0.00273999991 < u0

   1. Initial program 91.9%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      3. flip3-- N/A

         \[\leadsto \left(\color{blue}{\frac{{0}^{3} - {\alpha}^{3}}{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      7. mul0-lft N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\alpha \cdot \alpha + \color{blue}{0}\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\alpha \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      10. lift-neg.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right)} \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      12. div-inv N/A

          \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\left(\color{blue}{0} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      14. sub0-neg N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({\alpha}^{3}\right)\right)}\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(\color{blue}{{\alpha}^{3}} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      16. lower-*.f32 N/A

          \[\leadsto \left(\color{blue}{\left({\alpha}^{3} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      17. lower-pow.f32 N/A

          \[\leadsto \left(\left(\color{blue}{{\alpha}^{3}} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      18. lower-/.f32 92.0

          \[\leadsto \left(\left({\alpha}^{3} \cdot \color{blue}{\frac{1}{\left(-\alpha\right) \cdot \alpha}}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 92.0%

      \[\leadsto \left(\color{blue}{\left({\alpha}^{3} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   6. Applied rewrites 92.3%

      \[\leadsto \left(\left({\alpha}^{3} \cdot \frac{1}{\color{blue}{{\alpha}^{1.5} \cdot \left({\alpha}^{1.5} \cdot \left(\left(-{\alpha}^{-2}\right) \cdot \alpha\right)\right)}}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.002739999908953905:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(\frac{-1}{\left(\left({\alpha}^{-2} \cdot \alpha\right) \cdot {\alpha}^{1.5}\right) \cdot {\alpha}^{1.5}} \cdot {\alpha}^{3}\right) \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;u0 \leq 0.002739999908953905:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{-1}{\left(\frac{1}{\alpha} \cdot {\alpha}^{1.5}\right) \cdot {\alpha}^{1.5}} \cdot {\alpha}^{3}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= u0 0.002739999908953905)
     (+ (* (* 0.5 u0) t_0) t_0)
     (*
      (*
       (*
        (/ -1.0 (* (* (/ 1.0 alpha) (pow alpha 1.5)) (pow alpha 1.5)))
        (pow alpha 3.0))
       alpha)
      (log (- 1.0 u0))))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if (u0 <= 0.002739999908953905f) {
		tmp = ((0.5f * u0) * t_0) + t_0;
	} else {
		tmp = (((-1.0f / (((1.0f / alpha) * powf(alpha, 1.5f)) * powf(alpha, 1.5f))) * powf(alpha, 3.0f)) * alpha) * logf((1.0f - u0));
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if (u0 <= 0.002739999908953905e0) then
        tmp = ((0.5e0 * u0) * t_0) + t_0
    else
        tmp = ((((-1.0e0) / (((1.0e0 / alpha) * (alpha ** 1.5e0)) * (alpha ** 1.5e0))) * (alpha ** 3.0e0)) * alpha) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.002739999908953905))
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(1.0) / alpha) * (alpha ^ Float32(1.5))) * (alpha ^ Float32(1.5)))) * (alpha ^ Float32(3.0))) * alpha) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if (u0 <= single(0.002739999908953905))
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	else
		tmp = (((single(-1.0) / (((single(1.0) / alpha) * (alpha ^ single(1.5))) * (alpha ^ single(1.5)))) * (alpha ^ single(3.0))) * alpha) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00273999991

   1. Initial program 43.5%

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{\left(0 + \left(-\alpha\right) \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{0 - \left(-\alpha\right) \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      3. neg-sub0 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\mathsf{neg}\left(\left(-\alpha\right) \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \log \left(1 - u0\right) \]

      5. lift-neg.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)}\right)} \cdot \log \left(1 - u0\right) \]

      7. remove-double-neg N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      10. lower--.f32 N/A

          \[\leadsto \frac{\color{blue}{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \frac{0 - \color{blue}{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      12. lower-*.f32 43.5

          \[\leadsto \frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   5. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 28.4

          \[\leadsto \mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \]

   7. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.1%

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

   if 0.00273999991 < u0

   1. Initial program 91.9%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      3. flip3-- N/A

         \[\leadsto \left(\color{blue}{\frac{{0}^{3} - {\alpha}^{3}}{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      7. mul0-lft N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\alpha \cdot \alpha + \color{blue}{0}\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\alpha \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      10. lift-neg.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right)} \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      12. div-inv N/A

          \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\left(\color{blue}{0} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      14. sub0-neg N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({\alpha}^{3}\right)\right)}\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(\color{blue}{{\alpha}^{3}} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      16. lower-*.f32 N/A

          \[\leadsto \left(\color{blue}{\left({\alpha}^{3} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      17. lower-pow.f32 N/A

          \[\leadsto \left(\left(\color{blue}{{\alpha}^{3}} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      18. lower-/.f32 92.0

          \[\leadsto \left(\left({\alpha}^{3} \cdot \color{blue}{\frac{1}{\left(-\alpha\right) \cdot \alpha}}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 92.0%

      \[\leadsto \left(\color{blue}{\left({\alpha}^{3} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   6. Applied rewrites 91.9%

      \[\leadsto \left(\left({\alpha}^{3} \cdot \frac{1}{\color{blue}{\frac{0 \cdot \left(\alpha \cdot \alpha\right) - \left(\alpha \cdot \alpha\right) \cdot \left({\alpha}^{3} \cdot \alpha\right)}{{\alpha}^{3} \cdot \alpha}}}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   8. Applied rewrites 92.2%

      \[\leadsto \left(\left({\alpha}^{3} \cdot \frac{1}{\color{blue}{{\alpha}^{1.5} \cdot \left({\alpha}^{1.5} \cdot \left(-\frac{1}{\alpha}\right)\right)}}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.002739999908953905:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{-1}{\left(\frac{1}{\alpha} \cdot {\alpha}^{1.5}\right) \cdot {\alpha}^{1.5}} \cdot {\alpha}^{3}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;u0 \leq 0.002739999908953905:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\alpha}{{\alpha}^{3} \cdot \alpha} \cdot \alpha} \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= u0 0.002739999908953905)
     (+ (* (* 0.5 u0) t_0) t_0)
     (*
      (/ -1.0 (* (/ alpha (* (pow alpha 3.0) alpha)) alpha))
      (log (- 1.0 u0))))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if (u0 <= 0.002739999908953905f) {
		tmp = ((0.5f * u0) * t_0) + t_0;
	} else {
		tmp = (-1.0f / ((alpha / (powf(alpha, 3.0f) * alpha)) * alpha)) * logf((1.0f - u0));
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if (u0 <= 0.002739999908953905e0) then
        tmp = ((0.5e0 * u0) * t_0) + t_0
    else
        tmp = ((-1.0e0) / ((alpha / ((alpha ** 3.0e0) * alpha)) * alpha)) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.002739999908953905))
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	else
		tmp = Float32(Float32(Float32(-1.0) / Float32(Float32(alpha / Float32((alpha ^ Float32(3.0)) * alpha)) * alpha)) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if (u0 <= single(0.002739999908953905))
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	else
		tmp = (single(-1.0) / ((alpha / ((alpha ^ single(3.0)) * alpha)) * alpha)) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00273999991

   1. Initial program 43.5%

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{\left(0 + \left(-\alpha\right) \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{0 - \left(-\alpha\right) \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      3. neg-sub0 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\mathsf{neg}\left(\left(-\alpha\right) \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \log \left(1 - u0\right) \]

      5. lift-neg.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)}\right)} \cdot \log \left(1 - u0\right) \]

      7. remove-double-neg N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      10. lower--.f32 N/A

          \[\leadsto \frac{\color{blue}{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \frac{0 - \color{blue}{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      12. lower-*.f32 43.5

          \[\leadsto \frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   5. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 28.4

          \[\leadsto \mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \]

   7. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.1%

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

   if 0.00273999991 < u0

   1. Initial program 91.9%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. flip-- N/A

         \[\leadsto \left(\color{blue}{\frac{0 \cdot 0 - \alpha \cdot \alpha}{0 + \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{0} - \alpha \cdot \alpha}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      6. neg-sub0 N/A

         \[\leadsto \left(\frac{\color{blue}{\mathsf{neg}\left(\alpha \cdot \alpha\right)}}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \left(\frac{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      8. lift-neg.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(-\alpha\right)} \cdot \alpha}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      9. lift-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(-\alpha\right) \cdot \alpha}}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      10. +-lft-identity N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 91.9

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

   4. Applied rewrites 91.9%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. *-inverses N/A

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

      7. lift-/.f32 N/A

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

      8. lower-/.f32 91.8

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

      9. lift-/.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-neg.f32 N/A

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

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

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

      13. lift-*.f32 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-neg.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. pow2 N/A

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

      18. pow-flip N/A

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

      19. lower-pow.f32 N/A

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

      20. metadata-eval 91.9

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

   6. Applied rewrites 91.9%

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

      1. lift-neg.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lift-*.f32 N/A

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

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

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

      10. lift-neg.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-inverses N/A

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

      13. associate-/r* N/A

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

      14. remove-double-neg N/A

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

      15. lift-*.f32 N/A

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

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

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

      17. lift-neg.f32 N/A

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

      18. lift-*.f32 N/A

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

      19. +-lft-identity N/A

          \[\leadsto \frac{1}{\frac{\alpha \cdot \alpha}{\color{blue}{0 + \left(\mathsf{neg}\left(\left(-\alpha\right) \cdot \alpha\right)\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}} \cdot \log \left(1 - u0\right) \]

      20. cancel-sign-sub-inv N/A

          \[\leadsto \frac{1}{\frac{\alpha \cdot \alpha}{\color{blue}{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}} \cdot \log \left(1 - u0\right) \]

      21. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\alpha \cdot \alpha}{0 - \color{blue}{\left(\left(-\alpha\right) \cdot \alpha\right)} \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

      22. lift-neg.f32 N/A

          \[\leadsto \frac{1}{\frac{\alpha \cdot \alpha}{0 - \left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

      23. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\alpha \cdot \alpha}{0 - \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(-\alpha\right) \cdot \alpha\right)}}} \cdot \log \left(1 - u0\right) \]

      24. lift-neg.f32 N/A

          \[\leadsto \frac{1}{\frac{\alpha \cdot \alpha}{0 - \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

   8. Applied rewrites 92.1%

      \[\leadsto \frac{1}{\color{blue}{\alpha \cdot \frac{\alpha}{{\alpha}^{3} \cdot \left(-\alpha\right)}}} \cdot \log \left(1 - u0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.002739999908953905:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\alpha}{{\alpha}^{3} \cdot \alpha} \cdot \alpha} \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;u0 \leq 0.002739999908953905:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{-1}{\alpha \cdot \alpha} \cdot {\alpha}^{3}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= u0 0.002739999908953905)
     (+ (* (* 0.5 u0) t_0) t_0)
     (*
      (* (* (/ -1.0 (* alpha alpha)) (pow alpha 3.0)) alpha)
      (log (- 1.0 u0))))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if (u0 <= 0.002739999908953905f) {
		tmp = ((0.5f * u0) * t_0) + t_0;
	} else {
		tmp = (((-1.0f / (alpha * alpha)) * powf(alpha, 3.0f)) * alpha) * logf((1.0f - u0));
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if (u0 <= 0.002739999908953905e0) then
        tmp = ((0.5e0 * u0) * t_0) + t_0
    else
        tmp = ((((-1.0e0) / (alpha * alpha)) * (alpha ** 3.0e0)) * alpha) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.002739999908953905))
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-1.0) / Float32(alpha * alpha)) * (alpha ^ Float32(3.0))) * alpha) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if (u0 <= single(0.002739999908953905))
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	else
		tmp = (((single(-1.0) / (alpha * alpha)) * (alpha ^ single(3.0))) * alpha) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00273999991

   1. Initial program 43.5%

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{\left(0 + \left(-\alpha\right) \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{0 - \left(-\alpha\right) \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      3. neg-sub0 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\mathsf{neg}\left(\left(-\alpha\right) \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \log \left(1 - u0\right) \]

      5. lift-neg.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)}\right)} \cdot \log \left(1 - u0\right) \]

      7. remove-double-neg N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      10. lower--.f32 N/A

          \[\leadsto \frac{\color{blue}{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \frac{0 - \color{blue}{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      12. lower-*.f32 43.5

          \[\leadsto \frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   5. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 28.4

          \[\leadsto \mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \]

   7. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.1%

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

   if 0.00273999991 < u0

   1. Initial program 91.9%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      3. flip3-- N/A

         \[\leadsto \left(\color{blue}{\frac{{0}^{3} - {\alpha}^{3}}{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      7. mul0-lft N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\alpha \cdot \alpha + \color{blue}{0}\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\alpha \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      10. lift-neg.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right)} \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      12. div-inv N/A

          \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\left(\color{blue}{0} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      14. sub0-neg N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({\alpha}^{3}\right)\right)}\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(\color{blue}{{\alpha}^{3}} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      16. lower-*.f32 N/A

          \[\leadsto \left(\color{blue}{\left({\alpha}^{3} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      17. lower-pow.f32 N/A

          \[\leadsto \left(\left(\color{blue}{{\alpha}^{3}} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      18. lower-/.f32 92.0

          \[\leadsto \left(\left({\alpha}^{3} \cdot \color{blue}{\frac{1}{\left(-\alpha\right) \cdot \alpha}}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 92.0%

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

      1. lift-/.f32 N/A

         \[\leadsto \left(\left({\alpha}^{3} \cdot \color{blue}{\frac{1}{\left(-\alpha\right) \cdot \alpha}}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      2. frac-2neg N/A

         \[\leadsto \left(\left({\alpha}^{3} \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(-\alpha\right) \cdot \alpha\right)}}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\left({\alpha}^{3} \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(-\alpha\right) \cdot \alpha\right)}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \left(\left({\alpha}^{3} \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(-\alpha\right) \cdot \alpha}\right)}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      5. lift-neg.f32 N/A

         \[\leadsto \left(\left({\alpha}^{3} \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right)}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \left(\left({\alpha}^{3} \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)}\right)}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      7. lift-*.f32 N/A

         \[\leadsto \left(\left({\alpha}^{3} \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\alpha \cdot \alpha}\right)\right)\right)}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      8. remove-double-neg N/A

         \[\leadsto \left(\left({\alpha}^{3} \cdot \frac{-1}{\color{blue}{\alpha \cdot \alpha}}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      9. lower-/.f32 92.0

         \[\leadsto \left(\left({\alpha}^{3} \cdot \color{blue}{\frac{-1}{\alpha \cdot \alpha}}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   6. Applied rewrites 92.0%

      \[\leadsto \left(\left({\alpha}^{3} \cdot \color{blue}{\frac{-1}{\alpha \cdot \alpha}}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.002739999908953905:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{-1}{\alpha \cdot \alpha} \cdot {\alpha}^{3}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;u0 \leq 0.002739999908953905:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{\left(-\alpha\right)}^{3}}{\alpha \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= u0 0.002739999908953905)
     (+ (* (* 0.5 u0) t_0) t_0)
     (* (* (/ (pow (- alpha) 3.0) (* alpha alpha)) alpha) (log (- 1.0 u0))))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if (u0 <= 0.002739999908953905f) {
		tmp = ((0.5f * u0) * t_0) + t_0;
	} else {
		tmp = ((powf(-alpha, 3.0f) / (alpha * alpha)) * alpha) * logf((1.0f - u0));
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if (u0 <= 0.002739999908953905e0) then
        tmp = ((0.5e0 * u0) * t_0) + t_0
    else
        tmp = (((-alpha ** 3.0e0) / (alpha * alpha)) * alpha) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.002739999908953905))
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	else
		tmp = Float32(Float32(Float32((Float32(-alpha) ^ Float32(3.0)) / Float32(alpha * alpha)) * alpha) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if (u0 <= single(0.002739999908953905))
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	else
		tmp = (((-alpha ^ single(3.0)) / (alpha * alpha)) * alpha) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00273999991

   1. Initial program 43.5%

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{\left(0 + \left(-\alpha\right) \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{0 - \left(-\alpha\right) \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      3. neg-sub0 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\mathsf{neg}\left(\left(-\alpha\right) \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \log \left(1 - u0\right) \]

      5. lift-neg.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)}\right)} \cdot \log \left(1 - u0\right) \]

      7. remove-double-neg N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      10. lower--.f32 N/A

          \[\leadsto \frac{\color{blue}{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \frac{0 - \color{blue}{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      12. lower-*.f32 43.5

          \[\leadsto \frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   5. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 28.4

          \[\leadsto \mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \]

   7. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.1%

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

   if 0.00273999991 < u0

   1. Initial program 91.9%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      3. flip3-- N/A

         \[\leadsto \left(\color{blue}{\frac{{0}^{3} - {\alpha}^{3}}{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{{0}^{3} - {\alpha}^{3}}{\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      5. +-lft-identity N/A

         \[\leadsto \left(\frac{{0}^{3} - {\alpha}^{3}}{\color{blue}{\alpha \cdot \alpha + 0 \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      6. mul0-lft N/A

         \[\leadsto \left(\frac{{0}^{3} - {\alpha}^{3}}{\alpha \cdot \alpha + \color{blue}{0}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      7. +-rgt-identity N/A

         \[\leadsto \left(\frac{{0}^{3} - {\alpha}^{3}}{\color{blue}{\alpha \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \left(\color{blue}{\frac{{0}^{3} - {\alpha}^{3}}{\alpha \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{0} - {\alpha}^{3}}{\alpha \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      10. sub0-neg N/A

          \[\leadsto \left(\frac{\color{blue}{\mathsf{neg}\left({\alpha}^{3}\right)}}{\alpha \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      11. cube-neg N/A

          \[\leadsto \left(\frac{\color{blue}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}}{\alpha \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      12. lift-neg.f32 N/A

          \[\leadsto \left(\frac{{\color{blue}{\left(-\alpha\right)}}^{3}}{\alpha \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      13. lower-pow.f32 N/A

          \[\leadsto \left(\frac{\color{blue}{{\left(-\alpha\right)}^{3}}}{\alpha \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      14. lower-*.f32 91.9

          \[\leadsto \left(\frac{{\left(-\alpha\right)}^{3}}{\color{blue}{\alpha \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 91.9%

      \[\leadsto \left(\color{blue}{\frac{{\left(-\alpha\right)}^{3}}{\alpha \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.002739999908953905:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{\left(-\alpha\right)}^{3}}{\alpha \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;u0 \leq 0.002739999908953905:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{-1}{\alpha} \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= u0 0.002739999908953905)
     (+ (* (* 0.5 u0) t_0) t_0)
     (* (* (* (/ -1.0 alpha) (* alpha alpha)) alpha) (log (- 1.0 u0))))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if (u0 <= 0.002739999908953905f) {
		tmp = ((0.5f * u0) * t_0) + t_0;
	} else {
		tmp = (((-1.0f / alpha) * (alpha * alpha)) * alpha) * logf((1.0f - u0));
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if (u0 <= 0.002739999908953905e0) then
        tmp = ((0.5e0 * u0) * t_0) + t_0
    else
        tmp = ((((-1.0e0) / alpha) * (alpha * alpha)) * alpha) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.002739999908953905))
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-1.0) / alpha) * Float32(alpha * alpha)) * alpha) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if (u0 <= single(0.002739999908953905))
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	else
		tmp = (((single(-1.0) / alpha) * (alpha * alpha)) * alpha) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00273999991

   1. Initial program 43.5%

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{\left(0 + \left(-\alpha\right) \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{0 - \left(-\alpha\right) \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      3. neg-sub0 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\mathsf{neg}\left(\left(-\alpha\right) \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \log \left(1 - u0\right) \]

      5. lift-neg.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)}\right)} \cdot \log \left(1 - u0\right) \]

      7. remove-double-neg N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      10. lower--.f32 N/A

          \[\leadsto \frac{\color{blue}{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \frac{0 - \color{blue}{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      12. lower-*.f32 43.5

          \[\leadsto \frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   5. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 28.4

          \[\leadsto \mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \]

   7. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.1%

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

   if 0.00273999991 < u0

   1. Initial program 91.9%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      3. flip3-- N/A

         \[\leadsto \left(\color{blue}{\frac{{0}^{3} - {\alpha}^{3}}{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      7. mul0-lft N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\alpha \cdot \alpha + \color{blue}{0}\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\alpha \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      10. lift-neg.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right)} \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      12. div-inv N/A

          \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\left(\color{blue}{0} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      14. sub0-neg N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({\alpha}^{3}\right)\right)}\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(\color{blue}{{\alpha}^{3}} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      16. lower-*.f32 N/A

          \[\leadsto \left(\color{blue}{\left({\alpha}^{3} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      17. lower-pow.f32 N/A

          \[\leadsto \left(\left(\color{blue}{{\alpha}^{3}} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      18. lower-/.f32 92.0

          \[\leadsto \left(\left({\alpha}^{3} \cdot \color{blue}{\frac{1}{\left(-\alpha\right) \cdot \alpha}}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 92.0%

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

      1. lift-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left({\alpha}^{3} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\left(-\alpha\right) \cdot \alpha} \cdot {\alpha}^{3}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      3. lift-pow.f32 N/A

         \[\leadsto \left(\left(\frac{1}{\left(-\alpha\right) \cdot \alpha} \cdot \color{blue}{{\alpha}^{3}}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. cube-mult N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 91.9

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

      9. lift-/.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-neg.f32 N/A

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

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

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

      13. lift-*.f32 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-neg.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. pow2 N/A

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

      18. pow-flip N/A

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

      19. lower-pow.f32 N/A

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

      20. metadata-eval 91.9

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-/.f32 91.9

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

   9. Applied rewrites 91.9%

      \[\leadsto \left(\left(\color{blue}{\frac{-1}{\alpha}} \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.002739999908953905:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{-1}{\alpha} \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;1 - u0 \leq 0.9972599744796753:\\ \;\;\;\;\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \alpha}{\alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= (- 1.0 u0) 0.9972599744796753)
     (* (/ (* (* (- alpha) alpha) alpha) alpha) (log (- 1.0 u0)))
     (+ (* (* 0.5 u0) t_0) t_0))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if ((1.0f - u0) <= 0.9972599744796753f) {
		tmp = (((-alpha * alpha) * alpha) / alpha) * logf((1.0f - u0));
	} else {
		tmp = ((0.5f * u0) * t_0) + t_0;
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if ((1.0e0 - u0) <= 0.9972599744796753e0) then
        tmp = (((-alpha * alpha) * alpha) / alpha) * log((1.0e0 - u0))
    else
        tmp = ((0.5e0 * u0) * t_0) + t_0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9972599744796753))
		tmp = Float32(Float32(Float32(Float32(Float32(-alpha) * alpha) * alpha) / alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9972599744796753))
		tmp = (((-alpha * alpha) * alpha) / alpha) * log((single(1.0) - u0));
	else
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.997259974

   1. Initial program 91.9%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. flip-- N/A

         \[\leadsto \left(\color{blue}{\frac{0 \cdot 0 - \alpha \cdot \alpha}{0 + \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{0} - \alpha \cdot \alpha}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      6. neg-sub0 N/A

         \[\leadsto \left(\frac{\color{blue}{\mathsf{neg}\left(\alpha \cdot \alpha\right)}}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \left(\frac{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      8. lift-neg.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(-\alpha\right)} \cdot \alpha}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      9. lift-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(-\alpha\right) \cdot \alpha}}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      10. +-lft-identity N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 91.9

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

   4. Applied rewrites 91.9%

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

   if 0.997259974 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 43.5%

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{\left(0 + \left(-\alpha\right) \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{0 - \left(-\alpha\right) \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      3. neg-sub0 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\mathsf{neg}\left(\left(-\alpha\right) \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \log \left(1 - u0\right) \]

      5. lift-neg.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)}\right)} \cdot \log \left(1 - u0\right) \]

      7. remove-double-neg N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      10. lower--.f32 N/A

          \[\leadsto \frac{\color{blue}{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \frac{0 - \color{blue}{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      12. lower-*.f32 43.5

          \[\leadsto \frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   5. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 28.4

          \[\leadsto \mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \]

   7. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.1%

      \[\leadsto \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(0.5 \cdot u0\right) + \color{blue}{\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972599744796753:\\ \;\;\;\;\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \alpha}{\alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;1 - u0 \leq 0.9972599744796753:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= (- 1.0 u0) 0.9972599744796753)
     (* (* (- alpha) alpha) (log (- 1.0 u0)))
     (+ (* (* 0.5 u0) t_0) t_0))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if ((1.0f - u0) <= 0.9972599744796753f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = ((0.5f * u0) * t_0) + t_0;
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if ((1.0e0 - u0) <= 0.9972599744796753e0) then
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    else
        tmp = ((0.5e0 * u0) * t_0) + t_0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9972599744796753))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9972599744796753))
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	else
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.997259974

   1. Initial program 91.9%

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

   if 0.997259974 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 43.5%

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{\left(0 + \left(-\alpha\right) \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{0 - \left(-\alpha\right) \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      3. neg-sub0 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\mathsf{neg}\left(\left(-\alpha\right) \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \log \left(1 - u0\right) \]

      5. lift-neg.f32 N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)}\right)} \cdot \log \left(1 - u0\right) \]

      7. remove-double-neg N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      10. lower--.f32 N/A

          \[\leadsto \frac{\color{blue}{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \frac{0 - \color{blue}{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

      12. lower-*.f32 43.5

          \[\leadsto \frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   5. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 28.4

          \[\leadsto \mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \]

   7. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.1%

      \[\leadsto \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(0.5 \cdot u0\right) + \color{blue}{\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972599744796753:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \left(0.5 \cdot u0\right) \cdot t\_0 + t\_0 \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0))) (+ (* (* 0.5 u0) t_0) t_0)))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	return ((0.5f * u0) * t_0) + t_0;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    t_0 = (alpha * alpha) * u0
    code = ((0.5e0 * u0) * t_0) + t_0
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	return Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0)
end

MATLAB

function tmp = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = ((single(0.5) * u0) * t_0) + t_0;
end

Derivation

1. Initial program 55.1%

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

   1. +-lft-identity N/A

      \[\leadsto \color{blue}{\left(0 + \left(-\alpha\right) \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

   2. flip-+ N/A

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{0 - \left(-\alpha\right) \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   3. neg-sub0 N/A

      \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\mathsf{neg}\left(\left(-\alpha\right) \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \log \left(1 - u0\right) \]

   5. lift-neg.f32 N/A

      \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

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

      \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)}\right)} \cdot \log \left(1 - u0\right) \]

   7. remove-double-neg N/A

      \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   8. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   9. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

   10. lower--.f32 N/A

       \[\leadsto \frac{\color{blue}{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{0 - \color{blue}{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

   12. lower-*.f32 55.1

       \[\leadsto \frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

4. Applied rewrites 55.1%

   \[\leadsto \color{blue}{\frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

5. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft1-in N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 28.0

       \[\leadsto \mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \]

7. Applied rewrites 27.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 89.1%

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

10. Final simplification 89.1%

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

Alternative 10: 87.3% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.1%

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

   1. +-lft-identity N/A

      \[\leadsto \color{blue}{\left(0 + \left(-\alpha\right) \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

   2. flip-+ N/A

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{0 - \left(-\alpha\right) \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   3. neg-sub0 N/A

      \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\mathsf{neg}\left(\left(-\alpha\right) \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \log \left(1 - u0\right) \]

   5. lift-neg.f32 N/A

      \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

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

      \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)}\right)} \cdot \log \left(1 - u0\right) \]

   7. remove-double-neg N/A

      \[\leadsto \frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   8. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

   9. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

   10. lower--.f32 N/A

       \[\leadsto \frac{\color{blue}{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{0 - \color{blue}{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right) \]

   12. lower-*.f32 55.1

       \[\leadsto \frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

4. Applied rewrites 55.1%

   \[\leadsto \color{blue}{\frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

5. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft1-in N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 28.0

       \[\leadsto \mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \]

7. Applied rewrites 27.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 88.9%

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

10. Final simplification 88.9%

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

Alternative 11: 74.9% accurate, 10.5× speedup

Math

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

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(Float32(alpha * alpha) * u0)
end

MATLAB

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 75.9

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

5. Applied rewrites 75.9%

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

6. Final simplification 75.9%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot u0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024241 
(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))))