Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.7% → 96.7%

Time: 5.8s

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: 56.7% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{1}{\frac{-1}{{\alpha}^{3}} \cdot \alpha}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot u0\right) \cdot u0 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9972000122070313)
   (* (log (- 1.0 u0)) (/ 1.0 (* (/ -1.0 (pow alpha 3.0)) alpha)))
   (* (- (* (* -0.5 u0) u0) u0) (* (- alpha) alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9972000122070313f) {
		tmp = logf((1.0f - u0)) * (1.0f / ((-1.0f / powf(alpha, 3.0f)) * alpha));
	} else {
		tmp = (((-0.5f * u0) * u0) - 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 ((1.0e0 - u0) <= 0.9972000122070313e0) then
        tmp = log((1.0e0 - u0)) * (1.0e0 / (((-1.0e0) / (alpha ** 3.0e0)) * alpha))
    else
        tmp = ((((-0.5e0) * u0) * u0) - u0) * (-alpha * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9972000122070313))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(1.0) / Float32(Float32(Float32(-1.0) / (alpha ^ Float32(3.0))) * alpha)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-0.5) * u0) * u0) - u0) * Float32(Float32(-alpha) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9972000122070313))
		tmp = log((single(1.0) - u0)) * (single(1.0) / ((single(-1.0) / (alpha ^ single(3.0))) * alpha));
	else
		tmp = (((single(-0.5) * u0) * u0) - u0) * (-alpha * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99720001

   1. Initial program 92.0%

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

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

      13. lower-/.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 92.0

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

   4. Applied rewrites 92.0%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f32 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lift-*.f32 N/A

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

      10. lift-neg.f32 N/A

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

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

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

      12. lift-*.f32 N/A

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

      13. remove-double-neg N/A

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

      14. lift-*.f32 N/A

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

      15. unpow3 N/A

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

      16. lift-pow.f32 N/A

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

      17. lower-/.f32 92.1

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

   6. Applied rewrites 92.1%

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

   if 0.99720001 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 44.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.2%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{1}{\frac{-1}{{\alpha}^{3}} \cdot \alpha}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot u0\right) \cdot u0 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\left(\frac{1}{\frac{\alpha}{\frac{\alpha}{\frac{-1}{\alpha}}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot u0\right) \cdot u0 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9972000122070313)
   (* (* (/ 1.0 (/ alpha (/ alpha (/ -1.0 alpha)))) alpha) (log (- 1.0 u0)))
   (* (- (* (* -0.5 u0) u0) u0) (* (- alpha) alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9972000122070313f) {
		tmp = ((1.0f / (alpha / (alpha / (-1.0f / alpha)))) * alpha) * logf((1.0f - u0));
	} else {
		tmp = (((-0.5f * u0) * u0) - 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 ((1.0e0 - u0) <= 0.9972000122070313e0) then
        tmp = ((1.0e0 / (alpha / (alpha / ((-1.0e0) / alpha)))) * alpha) * log((1.0e0 - u0))
    else
        tmp = ((((-0.5e0) * u0) * u0) - u0) * (-alpha * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9972000122070313))
		tmp = Float32(Float32(Float32(Float32(1.0) / Float32(alpha / Float32(alpha / Float32(Float32(-1.0) / alpha)))) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-0.5) * u0) * u0) - u0) * Float32(Float32(-alpha) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9972000122070313))
		tmp = ((single(1.0) / (alpha / (alpha / (single(-1.0) / alpha)))) * alpha) * log((single(1.0) - u0));
	else
		tmp = (((single(-0.5) * u0) * u0) - u0) * (-alpha * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99720001

   1. Initial program 92.0%

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

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

      13. lower-/.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 92.0

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

   4. Applied rewrites 92.0%

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

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      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. associate-/r* N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-/.f32 91.9

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

   6. Applied rewrites 91.9%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

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

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

      4. lift-*.f32 N/A

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

      5. neg-mul-1 N/A

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

      6. metadata-eval N/A

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

      7. *-inverses N/A

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

      8. distribute-frac-neg2 N/A

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

      9. lift-neg.f32 N/A

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

      10. associate-/r/ N/A

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

      11. lift-neg.f32 N/A

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

      12. distribute-frac-neg N/A

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

      13. distribute-frac-neg2 N/A

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

      14. lift-*.f32 N/A

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

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

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

      16. lift-neg.f32 N/A

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

      17. lift-*.f32 N/A

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

      18. lift-/.f32 N/A

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

      19. lower-/.f32 91.9

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

      20. lift-/.f32 N/A

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

      21. lift-*.f32 N/A

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

      22. associate-/r* N/A

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

      23. lift-neg.f32 N/A

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

      24. distribute-frac-neg2 N/A

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

      25. *-inverses N/A

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

      26. metadata-eval N/A

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

      27. lower-/.f32 92.1

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

   8. Applied rewrites 92.1%

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

   if 0.99720001 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 44.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.2%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\left(\frac{1}{\frac{\alpha}{\frac{\alpha}{\frac{-1}{\alpha}}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot u0\right) \cdot u0 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\frac{-1}{\frac{\frac{\alpha}{\alpha \cdot \alpha}}{\alpha}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot u0\right) \cdot u0 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9972000122070313)
   (* (/ -1.0 (/ (/ alpha (* alpha alpha)) alpha)) (log (- 1.0 u0)))
   (* (- (* (* -0.5 u0) u0) u0) (* (- alpha) alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9972000122070313f) {
		tmp = (-1.0f / ((alpha / (alpha * alpha)) / alpha)) * logf((1.0f - u0));
	} else {
		tmp = (((-0.5f * u0) * u0) - 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 ((1.0e0 - u0) <= 0.9972000122070313e0) then
        tmp = ((-1.0e0) / ((alpha / (alpha * alpha)) / alpha)) * log((1.0e0 - u0))
    else
        tmp = ((((-0.5e0) * u0) * u0) - u0) * (-alpha * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9972000122070313))
		tmp = Float32(Float32(Float32(-1.0) / Float32(Float32(alpha / Float32(alpha * alpha)) / alpha)) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-0.5) * u0) * u0) - u0) * Float32(Float32(-alpha) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9972000122070313))
		tmp = (single(-1.0) / ((alpha / (alpha * alpha)) / alpha)) * log((single(1.0) - u0));
	else
		tmp = (((single(-0.5) * u0) * u0) - u0) * (-alpha * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99720001

   1. Initial program 92.0%

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

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

      13. lower-/.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 92.0

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

   4. Applied rewrites 92.0%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 92.1

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

   6. Applied rewrites 92.1%

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

   if 0.99720001 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 44.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.2%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\frac{-1}{\frac{\frac{\alpha}{\alpha \cdot \alpha}}{\alpha}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot u0\right) \cdot u0 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.00279999990016222:\\ \;\;\;\;\left(\left(-0.5 \cdot u0\right) \cdot u0 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\alpha}{\left(\alpha \cdot \alpha\right) \cdot \alpha}} \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.00279999990016222)
   (* (- (* (* -0.5 u0) u0) u0) (* (- alpha) alpha))
   (* (/ -1.0 (/ alpha (* (* alpha alpha) alpha))) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.00279999990016222f) {
		tmp = (((-0.5f * u0) * u0) - u0) * (-alpha * alpha);
	} 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) :: tmp
    if (u0 <= 0.00279999990016222e0) then
        tmp = ((((-0.5e0) * u0) * u0) - u0) * (-alpha * alpha)
    else
        tmp = ((-1.0e0) / (alpha / ((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(0.00279999990016222))
		tmp = Float32(Float32(Float32(Float32(Float32(-0.5) * u0) * u0) - u0) * Float32(Float32(-alpha) * alpha));
	else
		tmp = Float32(Float32(Float32(-1.0) / Float32(alpha / Float32(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(0.00279999990016222))
		tmp = (((single(-0.5) * u0) * u0) - u0) * (-alpha * alpha);
	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 < 0.0027999999

   1. Initial program 44.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.2%

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

   if 0.0027999999 < u0

   1. Initial program 92.0%

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

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

      13. lower-/.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 92.0

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

   4. Applied rewrites 92.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00279999990016222:\\ \;\;\;\;\left(\left(-0.5 \cdot u0\right) \cdot u0 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\alpha}{\left(\alpha \cdot \alpha\right) \cdot \alpha}} \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.00279999990016222:\\ \;\;\;\;\left(\left(-0.5 \cdot u0\right) \cdot u0 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\alpha}} \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.00279999990016222)
   (* (- (* (* -0.5 u0) u0) u0) (* (- alpha) alpha))
   (* (/ alpha (/ -1.0 alpha)) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.00279999990016222f) {
		tmp = (((-0.5f * u0) * u0) - u0) * (-alpha * alpha);
	} 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 <= 0.00279999990016222e0) then
        tmp = ((((-0.5e0) * u0) * u0) - u0) * (-alpha * alpha)
    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(0.00279999990016222))
		tmp = Float32(Float32(Float32(Float32(Float32(-0.5) * u0) * u0) - u0) * Float32(Float32(-alpha) * alpha));
	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(0.00279999990016222))
		tmp = (((single(-0.5) * u0) * u0) - u0) * (-alpha * alpha);
	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 < 0.0027999999

   1. Initial program 44.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.2%

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

   if 0.0027999999 < u0

   1. Initial program 92.0%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. distribute-lft-neg-out N/A

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

      4. neg-sub0 N/A

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

      5. flip3-- N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. +-lft-identity N/A

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

      9. mul0-lft N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. mul0-rgt N/A

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

      13. mul0-lft N/A

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

      14. mul0-lft N/A

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

      15. mul0-lft N/A

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

      16. lower-*.f32 N/A

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

   4. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\left(\left(-{\alpha}^{6}\right) \cdot \frac{1}{{\alpha}^{4}}\right)} \cdot \log \left(1 - u0\right) \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(-{\alpha}^{6}\right) \cdot \frac{1}{{\alpha}^{4}}\right)} \cdot \log \left(1 - u0\right) \]

      2. lift-neg.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left({\alpha}^{6}\right)\right)} \cdot \frac{1}{{\alpha}^{4}}\right) \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({\alpha}^{6} \cdot \frac{1}{{\alpha}^{4}}\right)\right)} \cdot \log \left(1 - u0\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left({\alpha}^{6} \cdot \left(\mathsf{neg}\left(\frac{1}{{\alpha}^{4}}\right)\right)\right)} \cdot \log \left(1 - u0\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left({\alpha}^{6} \cdot \left(\mathsf{neg}\left(\frac{1}{{\alpha}^{4}}\right)\right)\right)} \cdot \log \left(1 - u0\right) \]

      6. lower-neg.f32 91.9

         \[\leadsto \left({\alpha}^{6} \cdot \color{blue}{\left(-\frac{1}{{\alpha}^{4}}\right)}\right) \cdot \log \left(1 - u0\right) \]

      7. lift-/.f32 N/A

         \[\leadsto \left({\alpha}^{6} \cdot \left(-\color{blue}{\frac{1}{{\alpha}^{4}}}\right)\right) \cdot \log \left(1 - u0\right) \]

      8. lift-pow.f32 N/A

         \[\leadsto \left({\alpha}^{6} \cdot \left(-\frac{1}{\color{blue}{{\alpha}^{4}}}\right)\right) \cdot \log \left(1 - u0\right) \]

      9. pow-flip N/A

         \[\leadsto \left({\alpha}^{6} \cdot \left(-\color{blue}{{\alpha}^{\left(\mathsf{neg}\left(4\right)\right)}}\right)\right) \cdot \log \left(1 - u0\right) \]

      10. lower-pow.f32 N/A

          \[\leadsto \left({\alpha}^{6} \cdot \left(-\color{blue}{{\alpha}^{\left(\mathsf{neg}\left(4\right)\right)}}\right)\right) \cdot \log \left(1 - u0\right) \]

      11. metadata-eval 92.1

          \[\leadsto \left({\alpha}^{6} \cdot \left(-{\alpha}^{\color{blue}{-4}}\right)\right) \cdot \log \left(1 - u0\right) \]

   6. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\left({\alpha}^{6} \cdot \left(-{\alpha}^{-4}\right)\right)} \cdot \log \left(1 - u0\right) \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\left({\alpha}^{6} \cdot \left(-{\alpha}^{-4}\right)\right)} \cdot \log \left(1 - u0\right) \]

      2. lift-neg.f32 N/A

         \[\leadsto \left({\alpha}^{6} \cdot \color{blue}{\left(\mathsf{neg}\left({\alpha}^{-4}\right)\right)}\right) \cdot \log \left(1 - u0\right) \]

      3. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({\alpha}^{6} \cdot {\alpha}^{-4}\right)\right)} \cdot \log \left(1 - u0\right) \]

      4. lift-pow.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\alpha}^{6}} \cdot {\alpha}^{-4}\right)\right) \cdot \log \left(1 - u0\right) \]

      5. lift-pow.f32 N/A

         \[\leadsto \left(\mathsf{neg}\left({\alpha}^{6} \cdot \color{blue}{{\alpha}^{-4}}\right)\right) \cdot \log \left(1 - u0\right) \]

      6. pow-prod-up N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\alpha}^{\left(6 + -4\right)}}\right)\right) \cdot \log \left(1 - u0\right) \]

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 N/A

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

      10. neg-mul-1 N/A

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

      11. metadata-eval N/A

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

      12. *-inverses N/A

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

      13. distribute-frac-neg2 N/A

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

      14. lift-neg.f32 N/A

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

      15. associate-/r/ N/A

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

      16. lift-neg.f32 N/A

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

      17. distribute-frac-neg N/A

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

      18. distribute-frac-neg2 N/A

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

      19. lift-*.f32 N/A

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

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

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

      21. lift-neg.f32 N/A

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

      22. lift-*.f32 N/A

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

      23. lift-/.f32 N/A

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

      24. lower-/.f32 92.0

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

      25. lift-/.f32 N/A

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

      26. lift-*.f32 N/A

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

   8. Applied rewrites 92.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00279999990016222:\\ \;\;\;\;\left(\left(-0.5 \cdot u0\right) \cdot u0 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\alpha}} \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\alpha\right) \cdot \alpha\\ \mathbf{if}\;u0 \leq 0.00279999990016222:\\ \;\;\;\;\left(\left(-0.5 \cdot u0\right) \cdot u0 - u0\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (- alpha) alpha)))
   (if (<= u0 0.00279999990016222)
     (* (- (* (* -0.5 u0) u0) u0) t_0)
     (* t_0 (log (- 1.0 u0))))))

C

float code(float alpha, float u0) {
	float t_0 = -alpha * alpha;
	float tmp;
	if (u0 <= 0.00279999990016222f) {
		tmp = (((-0.5f * u0) * u0) - u0) * t_0;
	} else {
		tmp = t_0 * 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
    if (u0 <= 0.00279999990016222e0) then
        tmp = ((((-0.5e0) * u0) * u0) - u0) * t_0
    else
        tmp = t_0 * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(-alpha) * alpha)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.00279999990016222))
		tmp = Float32(Float32(Float32(Float32(Float32(-0.5) * u0) * u0) - u0) * t_0);
	else
		tmp = Float32(t_0 * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = -alpha * alpha;
	tmp = single(0.0);
	if (u0 <= single(0.00279999990016222))
		tmp = (((single(-0.5) * u0) * u0) - u0) * t_0;
	else
		tmp = t_0 * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.0027999999

   1. Initial program 44.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.2%

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

   if 0.0027999999 < u0

   1. Initial program 92.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00279999990016222:\\ \;\;\;\;\left(\left(-0.5 \cdot u0\right) \cdot u0 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.0%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 75.5

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

5. Applied rewrites 75.5%

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

7. Applied rewrites 88.7%

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

9. Applied rewrites 88.7%

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

10. Final simplification 88.7%

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

Alternative 8: 86.4% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.0%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 75.5

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

5. Applied rewrites 75.5%

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

7. Applied rewrites 88.4%

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

8. Final simplification 88.4%

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

Alternative 9: 73.9% 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 56.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 75.5

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

5. Applied rewrites 75.5%

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

6. Final simplification 75.5%

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

Reproduce

herbie shell --seed 2024242 
(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))))