Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.0% → 96.7%

Time: 6.6s

Alternatives: 9

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: 56.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.0035000001080334187:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(0.5 \cdot u0\right) \cdot u0 + u0\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(\frac{-1}{\alpha \cdot \alpha} \cdot {\alpha}^{3}\right) \cdot \alpha\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.0035000001080334187)
   (* (* alpha alpha) (+ (* (* 0.5 u0) u0) u0))
   (*
    (log (- 1.0 u0))
    (* (* (/ -1.0 (* alpha alpha)) (pow alpha 3.0)) alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.0035000001080334187f) {
		tmp = (alpha * alpha) * (((0.5f * u0) * u0) + u0);
	} else {
		tmp = logf((1.0f - u0)) * (((-1.0f / (alpha * alpha)) * 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) :: tmp
    if (u0 <= 0.0035000001080334187e0) then
        tmp = (alpha * alpha) * (((0.5e0 * u0) * u0) + u0)
    else
        tmp = log((1.0e0 - u0)) * ((((-1.0e0) / (alpha * alpha)) * (alpha ** 3.0e0)) * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0035000001080334187))
		tmp = Float32(Float32(alpha * alpha) * Float32(Float32(Float32(Float32(0.5) * u0) * u0) + u0));
	else
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(Float32(Float32(-1.0) / Float32(alpha * alpha)) * (alpha ^ Float32(3.0))) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.0035000001080334187))
		tmp = (alpha * alpha) * (((single(0.5) * u0) * u0) + u0);
	else
		tmp = log((single(1.0) - u0)) * (((single(-1.0) / (alpha * alpha)) * (alpha ^ single(3.0))) * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00350000011

   1. Initial program 41.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right)} \]
   4. 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. associate-*r* N/A

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

      3. *-commutative N/A

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

      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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 86.3

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

   5. Applied rewrites 85.8%

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

   7. Applied rewrites 97.6%

      \[\leadsto \left(\left(1 - -0.5 \cdot u0\right) \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 97.7%

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

   if 0.00350000011 < u0

   1. Initial program 93.3%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 93.4

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

   4. Applied rewrites 93.4%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.0035000001080334187:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(0.5 \cdot u0\right) \cdot u0 + u0\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(\frac{-1}{\alpha \cdot \alpha} \cdot {\alpha}^{3}\right) \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.0035000001080334187:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(0.5 \cdot u0\right) \cdot u0 + u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{\left(-\alpha\right)}^{3}}{\alpha \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.0035000001080334187)
   (* (* alpha alpha) (+ (* (* 0.5 u0) u0) u0))
   (* (* (/ (pow (- alpha) 3.0) (* alpha alpha)) alpha) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.0035000001080334187f) {
		tmp = (alpha * alpha) * (((0.5f * u0) * u0) + u0);
	} 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) :: tmp
    if (u0 <= 0.0035000001080334187e0) then
        tmp = (alpha * alpha) * (((0.5e0 * u0) * u0) + u0)
    else
        tmp = (((-alpha ** 3.0e0) / (alpha * alpha)) * alpha) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0035000001080334187))
		tmp = Float32(Float32(alpha * alpha) * Float32(Float32(Float32(Float32(0.5) * u0) * u0) + u0));
	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)
	tmp = single(0.0);
	if (u0 <= single(0.0035000001080334187))
		tmp = (alpha * alpha) * (((single(0.5) * u0) * u0) + u0);
	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 < 0.00350000011

   1. Initial program 41.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right)} \]
   4. 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. associate-*r* N/A

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

      3. *-commutative N/A

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

      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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 86.3

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

   5. Applied rewrites 85.8%

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

   7. Applied rewrites 97.6%

      \[\leadsto \left(\left(1 - -0.5 \cdot u0\right) \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 97.7%

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

   if 0.00350000011 < u0

   1. Initial program 93.3%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 93.4

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

   4. Applied rewrites 93.4%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.0035000001080334187:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(0.5 \cdot u0\right) \cdot u0 + u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{\left(-\alpha\right)}^{3}}{\alpha \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.0035000001080334187:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(0.5 \cdot u0\right) \cdot u0 + u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\alpha}^{3}}{-\alpha} \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.0035000001080334187)
   (* (* alpha alpha) (+ (* (* 0.5 u0) u0) u0))
   (* (/ (pow alpha 3.0) (- alpha)) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.0035000001080334187f) {
		tmp = (alpha * alpha) * (((0.5f * u0) * u0) + u0);
	} else {
		tmp = (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) :: tmp
    if (u0 <= 0.0035000001080334187e0) then
        tmp = (alpha * alpha) * (((0.5e0 * u0) * u0) + u0)
    else
        tmp = ((alpha ** 3.0e0) / -alpha) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0035000001080334187))
		tmp = Float32(Float32(alpha * alpha) * Float32(Float32(Float32(Float32(0.5) * u0) * u0) + u0));
	else
		tmp = Float32(Float32((alpha ^ Float32(3.0)) / Float32(-alpha)) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.0035000001080334187))
		tmp = (alpha * alpha) * (((single(0.5) * u0) * u0) + u0);
	else
		tmp = ((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 < 0.00350000011

   1. Initial program 41.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right)} \]
   4. 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. associate-*r* N/A

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

      3. *-commutative N/A

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

      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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 86.3

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

   5. Applied rewrites 86.3%

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

   7. Applied rewrites 97.6%

      \[\leadsto \left(\left(1 - -0.5 \cdot u0\right) \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 97.7%

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

   if 0.00350000011 < u0

   1. Initial program 93.3%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. div-inv N/A

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

      13. +-lft-identity N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. +-lft-identity N/A

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

      17. lower-/.f32 93.3

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

   4. Applied rewrites 93.3%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. un-div-inv N/A

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

      4. frac-2neg N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. lift-neg.f32 N/A

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

      9. lift-*.f32 N/A

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

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

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

      11. lift-*.f32 N/A

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

      12. cube-mult N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left({\alpha}^{\color{blue}{\left(\frac{6}{2}\right)}}\right)\right)\right)}{\mathsf{neg}\left(\alpha\right)} \cdot \log \left(1 - u0\right) \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{{\alpha}^{\left(\frac{6}{2}\right)}}}{\mathsf{neg}\left(\alpha\right)} \cdot \log \left(1 - u0\right) \]

      15. lift-neg.f32 N/A

          \[\leadsto \frac{{\alpha}^{\left(\frac{6}{2}\right)}}{\color{blue}{-\alpha}} \cdot \log \left(1 - u0\right) \]

      16. lower-/.f32 N/A

          \[\leadsto \color{blue}{\frac{{\alpha}^{\left(\frac{6}{2}\right)}}{-\alpha}} \cdot \log \left(1 - u0\right) \]

      17. metadata-eval N/A

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

      18. lower-pow.f32 93.3

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

   6. Applied rewrites 93.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.0035000001080334187:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(0.5 \cdot u0\right) \cdot u0 + u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\alpha}^{3}}{-\alpha} \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.0035000001080334187:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(0.5 \cdot u0\right) \cdot u0 + u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\alpha}} \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.0035000001080334187)
   (* (* alpha alpha) (+ (* (* 0.5 u0) u0) u0))
   (* (/ alpha (/ -1.0 alpha)) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.0035000001080334187f) {
		tmp = (alpha * alpha) * (((0.5f * u0) * u0) + u0);
	} else {
		tmp = (alpha / (-1.0f / 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) :: tmp
    if (u0 <= 0.0035000001080334187e0) then
        tmp = (alpha * alpha) * (((0.5e0 * u0) * u0) + u0)
    else
        tmp = (alpha / ((-1.0e0) / alpha)) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0035000001080334187))
		tmp = Float32(Float32(alpha * alpha) * Float32(Float32(Float32(Float32(0.5) * u0) * u0) + u0));
	else
		tmp = Float32(Float32(alpha / Float32(Float32(-1.0) / alpha)) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.0035000001080334187))
		tmp = (alpha * alpha) * (((single(0.5) * u0) * u0) + u0);
	else
		tmp = (alpha / (single(-1.0) / alpha)) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00350000011

   1. Initial program 41.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right)} \]
   4. 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. associate-*r* N/A

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

      3. *-commutative N/A

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

      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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 86.3

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

   5. Applied rewrites 85.8%

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

   7. Applied rewrites 97.6%

      \[\leadsto \left(\left(1 - -0.5 \cdot u0\right) \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 97.7%

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

   if 0.00350000011 < u0

   1. Initial program 93.3%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. *-commutative N/A

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

      3. lift-neg.f32 N/A

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

      4. neg-sub0 N/A

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

      5. flip3-- N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f32 N/A

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

      9. metadata-eval N/A

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

      10. +-lft-identity N/A

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

      11. mul0-lft N/A

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

      12. +-rgt-identity N/A

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

      13. clear-num N/A

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

      14. +-rgt-identity N/A

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

      15. mul0-lft N/A

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

      16. +-lft-identity N/A

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

      17. metadata-eval N/A

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

      18. flip3-- N/A

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

      19. neg-sub0 N/A

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

      20. lift-neg.f32 N/A

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

      21. lower-/.f32 93.3

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

   4. Applied rewrites 93.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.0035000001080334187:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(0.5 \cdot u0\right) \cdot u0 + u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\alpha}} \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.0035000001080334187:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(0.5 \cdot u0\right) \cdot u0 + u0\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.0035000001080334187)
   (* (* alpha alpha) (+ (* (* 0.5 u0) u0) u0))
   (* (log (- 1.0 u0)) (* (- alpha) alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.0035000001080334187f) {
		tmp = (alpha * alpha) * (((0.5f * u0) * u0) + u0);
	} else {
		tmp = logf((1.0f - u0)) * (-alpha * alpha);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if (u0 <= 0.0035000001080334187e0) then
        tmp = (alpha * alpha) * (((0.5e0 * u0) * u0) + u0)
    else
        tmp = log((1.0e0 - u0)) * (-alpha * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0035000001080334187))
		tmp = Float32(Float32(alpha * alpha) * Float32(Float32(Float32(Float32(0.5) * u0) * u0) + u0));
	else
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(-alpha) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.0035000001080334187))
		tmp = (alpha * alpha) * (((single(0.5) * u0) * u0) + u0);
	else
		tmp = log((single(1.0) - u0)) * (-alpha * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00350000011

   1. Initial program 41.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right)} \]
   4. 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. associate-*r* N/A

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

      3. *-commutative N/A

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

      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. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 86.3

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

   5. Applied rewrites 86.3%

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

   7. Applied rewrites 97.6%

      \[\leadsto \left(\left(1 - -0.5 \cdot u0\right) \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 97.7%

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

   if 0.00350000011 < u0

   1. Initial program 93.3%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.0035000001080334187:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(0.5 \cdot u0\right) \cdot u0 + u0\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right)} \]
4. 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. associate-*r* N/A

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

   3. *-commutative N/A

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

   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. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 73.0

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

5. Applied rewrites 73.0%

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

7. Applied rewrites 85.2%

   \[\leadsto \left(\left(1 - -0.5 \cdot u0\right) \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right) \]
8. Step-by-step derivation

9. Applied rewrites 85.2%

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

10. Final simplification 85.2%

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

Alternative 7: 87.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right)} \]
4. 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. associate-*r* N/A

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

   3. *-commutative N/A

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

   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. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 73.0

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

5. Applied rewrites 73.0%

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

7. Applied rewrites 85.2%

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

Alternative 8: 74.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 73.0

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

5. Applied rewrites 73.0%

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

7. Applied rewrites 73.1%

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

8. Final simplification 73.1%

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

Alternative 9: 74.4% 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 56.7%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 73.0

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

5. Applied rewrites 73.0%

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

Reproduce

herbie shell --seed 2024331 
(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))))