Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.9% → 89.8%

Time: 6.4s

Alternatives: 7

Speedup: 10.5×

Specification

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 55.9% 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: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998624920845032:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{-1}{\frac{\alpha \cdot \alpha}{{\alpha}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{{\alpha}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998624920845032)
   (* (log (- 1.0 u0)) (/ -1.0 (/ (* alpha alpha) (pow alpha 4.0))))
   (/ u0 (pow alpha -2.0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998624920845032f) {
		tmp = logf((1.0f - u0)) * (-1.0f / ((alpha * alpha) / powf(alpha, 4.0f)));
	} else {
		tmp = u0 / powf(alpha, -2.0f);
	}
	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) <= 0.9998624920845032e0) then
        tmp = log((1.0e0 - u0)) * ((-1.0e0) / ((alpha * alpha) / (alpha ** 4.0e0)))
    else
        tmp = u0 / (alpha ** (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998624920845032))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(-1.0) / Float32(Float32(alpha * alpha) / (alpha ^ Float32(4.0)))));
	else
		tmp = Float32(u0 / (alpha ^ Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998624920845032))
		tmp = log((single(1.0) - u0)) * (single(-1.0) / ((alpha * alpha) / (alpha ^ single(4.0))));
	else
		tmp = u0 / (alpha ^ single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999862492

   1. Initial program 86.8%

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

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

      6. +-lft-identity N/A

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

      7. clear-num N/A

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

      8. lower-/.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. metadata-eval N/A

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

      12. sub0-neg N/A

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

      13. lower-neg.f32 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. pow-prod-up N/A

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

      17. lower-pow.f32 N/A

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

      18. metadata-eval 87.0

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

   4. Applied rewrites 87.0%

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

   if 0.999862492 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 36.3%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   7. Applied rewrites 90.7%

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

   9. Applied rewrites 90.9%

      \[\leadsto \frac{u0}{\color{blue}{{\alpha}^{-2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998624920845032:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{-1}{\frac{\alpha \cdot \alpha}{{\alpha}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{{\alpha}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998624920845032:\\ \;\;\;\;\frac{-{\alpha}^{4}}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{{\alpha}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998624920845032)
   (* (/ (- (pow alpha 4.0)) (* alpha alpha)) (log (- 1.0 u0)))
   (/ u0 (pow alpha -2.0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998624920845032f) {
		tmp = (-powf(alpha, 4.0f) / (alpha * alpha)) * logf((1.0f - u0));
	} else {
		tmp = u0 / powf(alpha, -2.0f);
	}
	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) <= 0.9998624920845032e0) then
        tmp = (-(alpha ** 4.0e0) / (alpha * alpha)) * log((1.0e0 - u0))
    else
        tmp = u0 / (alpha ** (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998624920845032))
		tmp = Float32(Float32(Float32(-(alpha ^ Float32(4.0))) / Float32(alpha * alpha)) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(u0 / (alpha ^ Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998624920845032))
		tmp = (-(alpha ^ single(4.0)) / (alpha * alpha)) * log((single(1.0) - u0));
	else
		tmp = u0 / (alpha ^ single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999862492

   1. Initial program 86.8%

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

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

      6. +-lft-identity N/A

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

      7. lower-/.f32 N/A

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

      8. metadata-eval N/A

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

      9. sub0-neg N/A

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

      10. lower-neg.f32 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. pow-prod-up N/A

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

      14. lower-pow.f32 N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 86.9

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

   4. Applied rewrites 86.9%

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

   if 0.999862492 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 36.3%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   7. Applied rewrites 90.7%

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

   9. Applied rewrites 90.9%

      \[\leadsto \frac{u0}{\color{blue}{{\alpha}^{-2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998624920845032:\\ \;\;\;\;\frac{-1}{{\alpha}^{-2}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{{\alpha}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998624920845032)
   (* (/ -1.0 (pow alpha -2.0)) (log (- 1.0 u0)))
   (/ u0 (pow alpha -2.0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998624920845032f) {
		tmp = (-1.0f / powf(alpha, -2.0f)) * logf((1.0f - u0));
	} else {
		tmp = u0 / powf(alpha, -2.0f);
	}
	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) <= 0.9998624920845032e0) then
        tmp = ((-1.0e0) / (alpha ** (-2.0e0))) * log((1.0e0 - u0))
    else
        tmp = u0 / (alpha ** (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998624920845032))
		tmp = Float32(Float32(Float32(-1.0) / (alpha ^ Float32(-2.0))) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(u0 / (alpha ^ Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998624920845032))
		tmp = (single(-1.0) / (alpha ^ single(-2.0))) * log((single(1.0) - u0));
	else
		tmp = u0 / (alpha ^ single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999862492

   1. Initial program 86.8%

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

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

      6. +-lft-identity N/A

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

      7. clear-num N/A

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

      8. lower-/.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. metadata-eval N/A

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

      12. sub0-neg N/A

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

      13. lower-neg.f32 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. pow-prod-up N/A

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

      17. lower-pow.f32 N/A

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

      18. metadata-eval 87.0

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

   4. Applied rewrites 87.0%

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

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. clear-num N/A

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

      4. lift-neg.f32 N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. pow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{{\alpha}^{4}}{\color{blue}{{\alpha}^{2}}}\right)\right) \cdot \log \left(1 - u0\right) \]

      9. pow-div N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\alpha}^{\left(4 - 2\right)}}\right)\right) \cdot \log \left(1 - u0\right) \]

      10. metadata-eval N/A

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

      11. pow2 N/A

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

      12. lift-*.f32 N/A

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

      13. remove-double-div N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f32 N/A

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

      17. lift-*.f32 N/A

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

      18. pow2 N/A

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

      19. pow-flip N/A

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

      20. lower-pow.f32 N/A

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

      21. metadata-eval 86.9

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

   6. Applied rewrites 86.9%

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

   if 0.999862492 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 36.3%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   7. Applied rewrites 90.7%

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

   9. Applied rewrites 90.9%

      \[\leadsto \frac{u0}{\color{blue}{{\alpha}^{-2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998624920845032:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\alpha}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{{\alpha}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998624920845032)
   (* (/ alpha (/ -1.0 alpha)) (log (- 1.0 u0)))
   (/ u0 (pow alpha -2.0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998624920845032f) {
		tmp = (alpha / (-1.0f / alpha)) * logf((1.0f - u0));
	} else {
		tmp = u0 / powf(alpha, -2.0f);
	}
	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) <= 0.9998624920845032e0) then
        tmp = (alpha / ((-1.0e0) / alpha)) * log((1.0e0 - u0))
    else
        tmp = u0 / (alpha ** (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998624920845032))
		tmp = Float32(Float32(alpha / Float32(Float32(-1.0) / alpha)) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(u0 / (alpha ^ Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998624920845032))
		tmp = (alpha / (single(-1.0) / alpha)) * log((single(1.0) - u0));
	else
		tmp = u0 / (alpha ^ single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999862492

   1. Initial program 86.8%

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

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

      7. +-lft-identity N/A

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

      8. mul0-lft N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. --rgt-identity N/A

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

      12. lower-/.f32 N/A

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

      13. metadata-eval N/A

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

      14. sub0-neg N/A

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

      15. lower-neg.f32 N/A

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

      16. pow2 N/A

          \[\leadsto \frac{-{\color{blue}{\left({\alpha}^{2}\right)}}^{3}}{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

      17. pow-pow N/A

          \[\leadsto \frac{-\color{blue}{{\alpha}^{\left(2 \cdot 3\right)}}}{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

      18. lower-pow.f32 N/A

          \[\leadsto \frac{-\color{blue}{{\alpha}^{\left(2 \cdot 3\right)}}}{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

      19. metadata-eval N/A

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

      20. pow2 N/A

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

      21. pow2 N/A

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

      22. pow-prod-up N/A

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

      23. lower-pow.f32 N/A

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

      24. metadata-eval 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift-/.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. distribute-frac-neg N/A

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

      4. lift-pow.f32 N/A

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

      5. lift-pow.f32 N/A

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

      6. pow-div 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. distribute-rgt-neg-in N/A

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

      10. lift-neg.f32 N/A

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

      11. remove-double-div N/A

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

      12. lift-/.f32 N/A

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

      13. un-div-inv N/A

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

      14. lower-/.f32 86.9

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

      15. lift-/.f32 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. lift-neg.f32 N/A

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

      19. remove-double-neg N/A

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

      20. lower-/.f32 86.9

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

   6. Applied rewrites 86.9%

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

   if 0.999862492 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 36.3%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   7. Applied rewrites 90.7%

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

   9. Applied rewrites 90.9%

      \[\leadsto \frac{u0}{\color{blue}{{\alpha}^{-2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998624920845032:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{{\alpha}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998624920845032)
   (* (* (- alpha) alpha) (log (- 1.0 u0)))
   (/ u0 (pow alpha -2.0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998624920845032f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = u0 / powf(alpha, -2.0f);
	}
	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) <= 0.9998624920845032e0) then
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    else
        tmp = u0 / (alpha ** (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998624920845032))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(u0 / (alpha ^ Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998624920845032))
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	else
		tmp = u0 / (alpha ^ single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999862492

   1. Initial program 86.8%

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

   if 0.999862492 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 36.3%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   7. Applied rewrites 90.7%

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

   9. Applied rewrites 90.9%

      \[\leadsto \frac{u0}{\color{blue}{{\alpha}^{-2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{{\alpha}^{-2}} \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (/ u0 (pow alpha -2.0)))

C

float code(float alpha, float u0) {
	return u0 / powf(alpha, -2.0f);
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = u0 / (alpha ** (-2.0e0))
end function

Julia

function code(alpha, u0)
	return Float32(u0 / (alpha ^ Float32(-2.0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = u0 / (alpha ^ single(-2.0));
end

Derivation

1. Initial program 58.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 72.2

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

5. Applied rewrites 72.2%

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

7. Applied rewrites 72.1%

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

9. Applied rewrites 72.2%

   \[\leadsto \frac{u0}{\color{blue}{{\alpha}^{-2}}} \]
10. Add Preprocessing

Alternative 7: 74.5% 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 58.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 72.2

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

5. Applied rewrites 72.2%

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

Reproduce

herbie shell --seed 2024332 
(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))))