Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.4% → 96.8%

Time: 6.9s

Alternatives: 4

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: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9959999918937683:\\ \;\;\;\;\left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(0.5 \cdot u0\right) \cdot u0 + u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9959999918937683)
   (* (* (log (- 1.0 u0)) (- alpha)) alpha)
   (* (* alpha alpha) (+ (* (* 0.5 u0) u0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9959999918937683f) {
		tmp = (logf((1.0f - u0)) * -alpha) * alpha;
	} else {
		tmp = (alpha * alpha) * (((0.5f * u0) * u0) + u0);
	}
	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) <= 0.9959999918937683e0) then
        tmp = (log((1.0e0 - u0)) * -alpha) * alpha
    else
        tmp = (alpha * alpha) * (((0.5e0 * u0) * u0) + u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9959999918937683))
		tmp = Float32(Float32(log(Float32(Float32(1.0) - u0)) * Float32(-alpha)) * alpha);
	else
		tmp = Float32(Float32(alpha * alpha) * Float32(Float32(Float32(Float32(0.5) * u0) * u0) + u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9959999918937683))
		tmp = (log((single(1.0) - u0)) * -alpha) * alpha;
	else
		tmp = (alpha * alpha) * (((single(0.5) * u0) * u0) + u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.995999992

   1. Initial program 93.5%

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative 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 42.1

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

   5. Applied rewrites 42.1%

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

   7. Applied rewrites 93.6%

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

   if 0.995999992 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 45.2%

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

      1. +-lft-identity N/A

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

      2. flip-+ N/A

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

      3. neg-sub0 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-neg.f32 N/A

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

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

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

      7. remove-double-neg N/A

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

      8. lower-/.f32 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-*.f32 45.2

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 84.7

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

   7. Applied rewrites 84.2%

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

   9. Applied rewrites 98.2%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9959999918937683:\\ \;\;\;\;\left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(0.5 \cdot u0\right) \cdot u0 + u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.8%

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

   1. +-lft-identity N/A

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

   2. flip-+ N/A

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

   3. neg-sub0 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-neg.f32 N/A

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

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

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

   7. remove-double-neg N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 58.8

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

4. Applied rewrites 58.8%

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

5. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 72.7

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

7. Applied rewrites 72.4%

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

9. Applied rewrites 86.6%

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

10. Final simplification 86.6%

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

Alternative 3: 87.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.8%

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

   1. +-lft-identity N/A

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

   2. flip-+ N/A

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

   3. neg-sub0 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-neg.f32 N/A

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

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

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

   7. remove-double-neg N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 58.8

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

4. Applied rewrites 58.8%

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

5. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 72.7

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

7. Applied rewrites 72.4%

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

9. Applied rewrites 86.4%

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

Alternative 4: 74.7% 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. *-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 72.7

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

5. Applied rewrites 72.7%

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

6. Final simplification 72.7%

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

Reproduce

herbie shell --seed 2024277 
(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))))