Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.1% → 74.5%

Time: 6.2s

Alternatives: 5

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: 56.1% 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: 74.5% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = (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 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 75.4

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

5. Applied rewrites 75.4%

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

7. Applied rewrites 75.4%

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

Alternative 2: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.003000000026077032:\\ \;\;\;\;\frac{\left(\alpha \cdot \alpha\right) \cdot \left(\left(\left(\alpha \cdot \mathsf{fma}\left(-0.3333333333333333, u0, -0.5\right)\right) \cdot u0 - \alpha\right) \cdot u0\right)}{-\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\alpha}} \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.003000000026077032)
   (/
    (*
     (* alpha alpha)
     (* (- (* (* alpha (fma -0.3333333333333333 u0 -0.5)) u0) alpha) u0))
    (- alpha))
   (* (/ alpha (/ -1.0 alpha)) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.003000000026077032f) {
		tmp = ((alpha * alpha) * ((((alpha * fmaf(-0.3333333333333333f, u0, -0.5f)) * u0) - alpha) * u0)) / -alpha;
	} else {
		tmp = (alpha / (-1.0f / alpha)) * logf((1.0f - u0));
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.003000000026077032))
		tmp = Float32(Float32(Float32(alpha * alpha) * Float32(Float32(Float32(Float32(alpha * fma(Float32(-0.3333333333333333), u0, Float32(-0.5))) * u0) - alpha) * u0)) / Float32(-alpha));
	else
		tmp = Float32(Float32(alpha / Float32(Float32(-1.0) / alpha)) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00300000003

   1. Initial program 42.2%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lift-neg.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. +-lft-identity N/A

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

      12. lower-/.f32 42.2

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

   4. Applied rewrites 42.2%

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

   5. Taylor expanded in u0 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 85.7

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

   7. Applied rewrites 85.7%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. un-div-inv N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 85.7

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

   9. Applied rewrites 85.7%

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

   10. Taylor expanded in u0 around 0

       \[\leadsto \frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(-1 \cdot \alpha + u0 \cdot \left(\frac{-1}{2} \cdot \alpha + \frac{-1}{3} \cdot \left(\alpha \cdot u0\right)\right)\right)\right)}}{\alpha} \]
   11. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

   12. Applied rewrites 97.2%

       \[\leadsto \frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(\left(\alpha \cdot \mathsf{fma}\left(-0.3333333333333333, u0, -0.5\right)\right) \cdot u0 - \alpha\right) \cdot u0\right)}}{\alpha} \]

   if 0.00300000003 < u0

   1. Initial program 94.8%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. *-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 94.9

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

   4. Applied rewrites 94.9%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.003000000026077032:\\ \;\;\;\;\frac{\left(\alpha \cdot \alpha\right) \cdot \left(\left(\left(\alpha \cdot \mathsf{fma}\left(-0.3333333333333333, u0, -0.5\right)\right) \cdot u0 - \alpha\right) \cdot u0\right)}{-\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\alpha}} \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.996999979019165:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha \cdot \alpha\right) \cdot \left(\left(\left(\alpha \cdot \mathsf{fma}\left(-0.3333333333333333, u0, -0.5\right)\right) \cdot u0 - \alpha\right) \cdot u0\right)}{-\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.996999979019165)
   (* (* (- alpha) alpha) (log (- 1.0 u0)))
   (/
    (*
     (* alpha alpha)
     (* (- (* (* alpha (fma -0.3333333333333333 u0 -0.5)) u0) alpha) u0))
    (- alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.996999979019165f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = ((alpha * alpha) * ((((alpha * fmaf(-0.3333333333333333f, u0, -0.5f)) * u0) - alpha) * u0)) / -alpha;
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.996999979019165))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(alpha * alpha) * Float32(Float32(Float32(Float32(alpha * fma(Float32(-0.3333333333333333), u0, Float32(-0.5))) * u0) - alpha) * u0)) / Float32(-alpha));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996999979

   1. Initial program 94.8%

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

   if 0.996999979 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 42.2%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lift-neg.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. +-lft-identity N/A

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

      12. lower-/.f32 42.2

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

   4. Applied rewrites 42.2%

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

   5. Taylor expanded in u0 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 85.7

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

   7. Applied rewrites 85.7%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. un-div-inv N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 85.7

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

   9. Applied rewrites 85.7%

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

   10. Taylor expanded in u0 around 0

       \[\leadsto \frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(-1 \cdot \alpha + u0 \cdot \left(\frac{-1}{2} \cdot \alpha + \frac{-1}{3} \cdot \left(\alpha \cdot u0\right)\right)\right)\right)}}{\alpha} \]
   11. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

   12. Applied rewrites 97.4%

       \[\leadsto \frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(\left(\alpha \cdot \mathsf{fma}\left(-0.3333333333333333, u0, -0.5\right)\right) \cdot u0 - \alpha\right) \cdot u0\right)}}{\alpha} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.996999979019165:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha \cdot \alpha\right) \cdot \left(\left(\left(\alpha \cdot \mathsf{fma}\left(-0.3333333333333333, u0, -0.5\right)\right) \cdot u0 - \alpha\right) \cdot u0\right)}{-\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\alpha \cdot \alpha\right) \cdot \left(\left(\left(\alpha \cdot \mathsf{fma}\left(-0.3333333333333333, u0, -0.5\right)\right) \cdot u0 - \alpha\right) \cdot u0\right)}{-\alpha} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (/
  (*
   (* alpha alpha)
   (* (- (* (* alpha (fma -0.3333333333333333 u0 -0.5)) u0) alpha) u0))
  (- alpha)))

C

float code(float alpha, float u0) {
	return ((alpha * alpha) * ((((alpha * fmaf(-0.3333333333333333f, u0, -0.5f)) * u0) - alpha) * u0)) / -alpha;
}

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(alpha * alpha) * Float32(Float32(Float32(Float32(alpha * fma(Float32(-0.3333333333333333), u0, Float32(-0.5))) * u0) - alpha) * u0)) / Float32(-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. Step-by-step derivation

   1. lift-neg.f32 N/A

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

   2. neg-sub0 N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. neg-sub0 N/A

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

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

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

   7. lift-neg.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. div-inv N/A

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

   10. lower-*.f32 N/A

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

   11. +-lft-identity N/A

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

   12. lower-/.f32 54.3

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

4. Applied rewrites 54.3%

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

5. Taylor expanded in u0 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 75.3

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

7. Applied rewrites 75.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. un-div-inv N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 75.3

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

9. Applied rewrites 75.3%

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

10. Taylor expanded in u0 around 0

    \[\leadsto \frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(-1 \cdot \alpha + u0 \cdot \left(\frac{-1}{2} \cdot \alpha + \frac{-1}{3} \cdot \left(\alpha \cdot u0\right)\right)\right)\right)}}{\alpha} \]
11. Step-by-step derivation

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

12. Applied rewrites 87.3%

    \[\leadsto \frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(\left(\alpha \cdot \mathsf{fma}\left(-0.3333333333333333, u0, -0.5\right)\right) \cdot u0 - \alpha\right) \cdot u0\right)}}{\alpha} \]

13. Final simplification 87.8%

    \[\leadsto \frac{\left(\alpha \cdot \alpha\right) \cdot \left(\left(\left(\alpha \cdot \mathsf{fma}\left(-0.3333333333333333, u0, -0.5\right)\right) \cdot u0 - \alpha\right) \cdot u0\right)}{-\alpha} \]
14. Add Preprocessing

Alternative 5: 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 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. 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 75.4

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

5. Applied rewrites 75.4%

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

Reproduce

herbie shell --seed 2024315 
(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))))