Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.2% → 75.2%

Time: 6.8s

Alternatives: 6

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: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 75.2% 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 54.3%

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

3. Taylor expanded in u0 around 0

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

   1. *-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 76.1

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

5. Applied rewrites 76.1%

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

6. Final simplification 76.1%

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

Alternative 2: 49.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.00015999999595806003:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, u0, 1\right) \cdot u0 - \left(-u0\right) \cdot u0\right) \cdot \alpha\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \alpha}{\alpha} \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.00015999999595806003)
   (* (* (- (* (fma -0.5 u0 1.0) u0) (* (- u0) u0)) alpha) alpha)
   (* (/ (* (* (- alpha) alpha) alpha) alpha) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.00015999999595806003f) {
		tmp = (((fmaf(-0.5f, u0, 1.0f) * u0) - (-u0 * u0)) * alpha) * alpha;
	} else {
		tmp = (((-alpha * alpha) * alpha) / alpha) * logf((1.0f - u0));
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.00015999999595806003))
		tmp = Float32(Float32(Float32(Float32(fma(Float32(-0.5), u0, Float32(1.0)) * u0) - Float32(Float32(-u0) * u0)) * alpha) * alpha);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-alpha) * alpha) * alpha) / alpha) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.59999996e-4

   1. Initial program 34.7%

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

   3. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. neg-mul-1 N/A

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

      6. lower-*.f32 N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f32 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f32 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 92.5

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

   5. Applied rewrites 92.5%

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in u0 around 0

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

   10. Applied rewrites 92.6%

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

   11. Taylor expanded in u0 around 0

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

   13. Applied rewrites 92.1%

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

   if 1.59999996e-4 < u0

   1. Initial program 87.5%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

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

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

      8. lift-neg.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. +-lft-identity N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 87.6

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

   4. Applied rewrites 87.6%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00015999999595806003:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, u0, 1\right) \cdot u0 - \left(-u0\right) \cdot u0\right) \cdot \alpha\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \alpha}{\alpha} \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.00015999999595806003:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, u0, 1\right) \cdot u0 - \left(-u0\right) \cdot u0\right) \cdot \alpha\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{\alpha}{\frac{-1}{\alpha}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.00015999999595806003)
   (* (* (- (* (fma -0.5 u0 1.0) u0) (* (- u0) u0)) alpha) alpha)
   (* (log (- 1.0 u0)) (/ alpha (/ -1.0 alpha)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.00015999999595806003f) {
		tmp = (((fmaf(-0.5f, u0, 1.0f) * u0) - (-u0 * u0)) * alpha) * alpha;
	} else {
		tmp = logf((1.0f - u0)) * (alpha / (-1.0f / alpha));
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.00015999999595806003))
		tmp = Float32(Float32(Float32(Float32(fma(Float32(-0.5), u0, Float32(1.0)) * u0) - Float32(Float32(-u0) * u0)) * alpha) * alpha);
	else
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(alpha / Float32(Float32(-1.0) / alpha)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.59999996e-4

   1. Initial program 34.7%

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

   3. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. neg-mul-1 N/A

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

      6. lower-*.f32 N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f32 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f32 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 92.5

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

   5. Applied rewrites 92.5%

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in u0 around 0

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

   10. Applied rewrites 92.6%

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

   11. Taylor expanded in u0 around 0

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

   13. Applied rewrites 92.1%

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

   if 1.59999996e-4 < u0

   1. Initial program 87.5%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-neg.f32 N/A

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

      4. neg-sub0 N/A

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

      5. flip3-- N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f32 N/A

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

      9. metadata-eval N/A

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

      10. +-lft-identity N/A

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

      11. mul0-lft N/A

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

      12. +-rgt-identity N/A

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

      13. clear-num N/A

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

      14. +-rgt-identity N/A

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

      15. mul0-lft N/A

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

      16. +-lft-identity N/A

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

      17. metadata-eval N/A

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

      18. flip3-- N/A

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

      19. neg-sub0 N/A

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

      20. lift-neg.f32 N/A

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

      21. lower-/.f32 87.6

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

   4. Applied rewrites 87.6%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00015999999595806003:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, u0, 1\right) \cdot u0 - \left(-u0\right) \cdot u0\right) \cdot \alpha\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{\alpha}{\frac{-1}{\alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.00015999999595806003:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, u0, 1\right) \cdot u0 - \left(-u0\right) \cdot u0\right) \cdot \alpha\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.00015999999595806003)
   (* (* (- (* (fma -0.5 u0 1.0) u0) (* (- u0) u0)) alpha) alpha)
   (* (* (- alpha) alpha) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.00015999999595806003f) {
		tmp = (((fmaf(-0.5f, u0, 1.0f) * u0) - (-u0 * u0)) * alpha) * alpha;
	} else {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.00015999999595806003))
		tmp = Float32(Float32(Float32(Float32(fma(Float32(-0.5), u0, Float32(1.0)) * u0) - Float32(Float32(-u0) * u0)) * alpha) * alpha);
	else
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.59999996e-4

   1. Initial program 34.7%

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

   3. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. neg-mul-1 N/A

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

      6. lower-*.f32 N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f32 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f32 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 92.5

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

   5. Applied rewrites 92.5%

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in u0 around 0

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

   10. Applied rewrites 92.6%

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

   11. Taylor expanded in u0 around 0

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

   13. Applied rewrites 92.1%

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

   if 1.59999996e-4 < u0

   1. Initial program 87.5%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00015999999595806003:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, u0, 1\right) \cdot u0 - \left(-u0\right) \cdot u0\right) \cdot \alpha\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.00015999999595806003:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, u0, 1\right) \cdot u0 - \left(-u0\right) \cdot u0\right) \cdot \alpha\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.00015999999595806003)
   (* (* (- (* (fma -0.5 u0 1.0) u0) (* (- u0) u0)) alpha) alpha)
   (* (* (log (- 1.0 u0)) (- alpha)) alpha)))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.00015999999595806003f) {
		tmp = (((fmaf(-0.5f, u0, 1.0f) * u0) - (-u0 * u0)) * alpha) * alpha;
	} else {
		tmp = (logf((1.0f - u0)) * -alpha) * alpha;
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.00015999999595806003))
		tmp = Float32(Float32(Float32(Float32(fma(Float32(-0.5), u0, Float32(1.0)) * u0) - Float32(Float32(-u0) * u0)) * alpha) * alpha);
	else
		tmp = Float32(Float32(log(Float32(Float32(1.0) - u0)) * Float32(-alpha)) * alpha);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.59999996e-4

   1. Initial program 34.7%

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

   3. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. neg-mul-1 N/A

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

      6. lower-*.f32 N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f32 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f32 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 92.5

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

   5. Applied rewrites 92.5%

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in u0 around 0

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

   10. Applied rewrites 92.6%

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

   11. Taylor expanded in u0 around 0

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

   13. Applied rewrites 92.1%

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

   if 1.59999996e-4 < u0

   1. Initial program 87.5%

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

   3. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. neg-mul-1 N/A

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

      6. lower-*.f32 N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f32 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f32 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 48.2

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

   5. Applied rewrites 48.2%

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

   7. Applied rewrites 87.5%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00015999999595806003:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, u0, 1\right) \cdot u0 - \left(-u0\right) \cdot u0\right) \cdot \alpha\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 24.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in alpha around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. neg-mul-1 N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. mul-1-neg N/A

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

   9. lower-*.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 76.1

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

5. Applied rewrites 76.1%

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

7. Applied rewrites 76.7%

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

8. Taylor expanded in u0 around 0

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

10. Applied rewrites 76.7%

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

11. Taylor expanded in u0 around 0

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

13. Applied rewrites 76.4%

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

14. Final simplification 76.7%

    \[\leadsto \left(\left(\mathsf{fma}\left(-0.5, u0, 1\right) \cdot u0 - \left(-u0\right) \cdot u0\right) \cdot \alpha\right) \cdot \alpha \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024271 
(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))))