Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.7% → 97.9%

Time: 6.5s

Alternatives: 8

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: 55.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9860000014305115:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \alpha}{\left(-\alpha\right) \cdot \alpha} \cdot \frac{{\alpha}^{3}}{\alpha \cdot \alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.5 + 0.3333333333333333 \cdot u0\right) \cdot u0 - -1\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9860000014305115)
   (*
    (log (- 1.0 u0))
    (*
     (/ (* (* alpha alpha) alpha) (* (- alpha) alpha))
     (/ (pow alpha 3.0) (* alpha alpha))))
   (*
    (* (- (* (+ 0.5 (* 0.3333333333333333 u0)) u0) -1.0) (* alpha alpha))
    u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9860000014305115f) {
		tmp = logf((1.0f - u0)) * ((((alpha * alpha) * alpha) / (-alpha * alpha)) * (powf(alpha, 3.0f) / (alpha * alpha)));
	} else {
		tmp = ((((0.5f + (0.3333333333333333f * u0)) * u0) - -1.0f) * (alpha * alpha)) * 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) <= 0.9860000014305115e0) then
        tmp = log((1.0e0 - u0)) * ((((alpha * alpha) * alpha) / (-alpha * alpha)) * ((alpha ** 3.0e0) / (alpha * alpha)))
    else
        tmp = ((((0.5e0 + (0.3333333333333333e0 * u0)) * u0) - (-1.0e0)) * (alpha * alpha)) * u0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9860000014305115))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(Float32(Float32(alpha * alpha) * alpha) / Float32(Float32(-alpha) * alpha)) * Float32((alpha ^ Float32(3.0)) / Float32(alpha * alpha))));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u0)) * u0) - Float32(-1.0)) * Float32(alpha * alpha)) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9860000014305115))
		tmp = log((single(1.0) - u0)) * ((((alpha * alpha) * alpha) / (-alpha * alpha)) * ((alpha ^ single(3.0)) / (alpha * alpha)));
	else
		tmp = ((((single(0.5) + (single(0.3333333333333333) * u0)) * u0) - single(-1.0)) * (alpha * alpha)) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.986000001

   1. Initial program 94.5%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 94.4

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

   4. Applied rewrites 94.4%

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

   6. Applied rewrites 94.5%

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

      1. lift-pow.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. cube-neg N/A

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

      4. unpow3 N/A

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

      5. lift-*.f32 N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. metadata-eval 94.6

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

   8. Applied rewrites 94.6%

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

   if 0.986000001 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 44.7%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(0.3333333333333333 \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right)\right)\right) \cdot u0} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.3%

      \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(0.3333333333333333, u0, 0.5\right) + 1\right)\right) \cdot u0 \]
   8. Step-by-step derivation

   9. Applied rewrites 98.5%

      \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot 0.3333333333333333 + 0.5\right) + 1\right)\right) \cdot u0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9860000014305115:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \alpha}{\left(-\alpha\right) \cdot \alpha} \cdot \frac{{\alpha}^{3}}{\alpha \cdot \alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.5 + 0.3333333333333333 \cdot u0\right) \cdot u0 - -1\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9829999804496765:\\ \;\;\;\;\frac{{\left(-\alpha\right)}^{3} \cdot \alpha}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.5 + 0.3333333333333333 \cdot u0\right) \cdot u0 - -1\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9829999804496765)
   (* (/ (* (pow (- alpha) 3.0) alpha) (* alpha alpha)) (log (- 1.0 u0)))
   (*
    (* (- (* (+ 0.5 (* 0.3333333333333333 u0)) u0) -1.0) (* alpha alpha))
    u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9829999804496765f) {
		tmp = ((powf(-alpha, 3.0f) * alpha) / (alpha * alpha)) * logf((1.0f - u0));
	} else {
		tmp = ((((0.5f + (0.3333333333333333f * u0)) * u0) - -1.0f) * (alpha * alpha)) * 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) <= 0.9829999804496765e0) then
        tmp = (((-alpha ** 3.0e0) * alpha) / (alpha * alpha)) * log((1.0e0 - u0))
    else
        tmp = ((((0.5e0 + (0.3333333333333333e0 * u0)) * u0) - (-1.0e0)) * (alpha * alpha)) * u0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9829999804496765))
		tmp = Float32(Float32(Float32((Float32(-alpha) ^ Float32(3.0)) * alpha) / Float32(alpha * alpha)) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u0)) * u0) - Float32(-1.0)) * Float32(alpha * alpha)) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9829999804496765))
		tmp = (((-alpha ^ single(3.0)) * alpha) / (alpha * alpha)) * log((single(1.0) - u0));
	else
		tmp = ((((single(0.5) + (single(0.3333333333333333) * u0)) * u0) - single(-1.0)) * (alpha * alpha)) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.98299998

   1. Initial program 94.9%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. flip3-- N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. +-lft-identity N/A

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

      8. mul0-lft N/A

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

      9. +-rgt-identity N/A

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

      10. lower-/.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. metadata-eval N/A

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

      13. sub0-neg N/A

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

      14. cube-neg N/A

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

      15. lift-neg.f32 N/A

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

      16. lower-pow.f32 N/A

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

      17. lower-*.f32 95.0

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

   4. Applied rewrites 95.0%

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

   if 0.98299998 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 45.3%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 82.2%

      \[\leadsto \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(0.3333333333333333 \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right)\right)\right) \cdot u0} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.9%

      \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(0.3333333333333333, u0, 0.5\right) + 1\right)\right) \cdot u0 \]
   8. Step-by-step derivation

   9. Applied rewrites 98.4%

      \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot 0.3333333333333333 + 0.5\right) + 1\right)\right) \cdot u0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9829999804496765:\\ \;\;\;\;\frac{{\left(-\alpha\right)}^{3} \cdot \alpha}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.5 + 0.3333333333333333 \cdot u0\right) \cdot u0 - -1\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9829999804496765:\\ \;\;\;\;\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.5 + 0.3333333333333333 \cdot u0\right) \cdot u0 - -1\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9829999804496765)
   (*
    (/ (* (* (- alpha) alpha) (* alpha alpha)) (* alpha alpha))
    (log (- 1.0 u0)))
   (*
    (* (- (* (+ 0.5 (* 0.3333333333333333 u0)) u0) -1.0) (* alpha alpha))
    u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9829999804496765f) {
		tmp = (((-alpha * alpha) * (alpha * alpha)) / (alpha * alpha)) * logf((1.0f - u0));
	} else {
		tmp = ((((0.5f + (0.3333333333333333f * u0)) * u0) - -1.0f) * (alpha * alpha)) * 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) <= 0.9829999804496765e0) then
        tmp = (((-alpha * alpha) * (alpha * alpha)) / (alpha * alpha)) * log((1.0e0 - u0))
    else
        tmp = ((((0.5e0 + (0.3333333333333333e0 * u0)) * u0) - (-1.0e0)) * (alpha * alpha)) * u0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9829999804496765))
		tmp = Float32(Float32(Float32(Float32(Float32(-alpha) * alpha) * Float32(alpha * alpha)) / Float32(alpha * alpha)) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u0)) * u0) - Float32(-1.0)) * Float32(alpha * alpha)) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9829999804496765))
		tmp = (((-alpha * alpha) * (alpha * alpha)) / (alpha * alpha)) * log((single(1.0) - u0));
	else
		tmp = ((((single(0.5) + (single(0.3333333333333333) * u0)) * u0) - single(-1.0)) * (alpha * alpha)) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.98299998

   1. Initial program 94.9%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 95.0

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

   4. Applied rewrites 95.0%

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

   if 0.98299998 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 45.3%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 82.2%

      \[\leadsto \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(0.3333333333333333 \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right)\right)\right) \cdot u0} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.9%

      \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(0.3333333333333333, u0, 0.5\right) + 1\right)\right) \cdot u0 \]
   8. Step-by-step derivation

   9. Applied rewrites 98.4%

      \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot 0.3333333333333333 + 0.5\right) + 1\right)\right) \cdot u0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9829999804496765:\\ \;\;\;\;\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.5 + 0.3333333333333333 \cdot u0\right) \cdot u0 - -1\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9860000014305115:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.5 + 0.3333333333333333 \cdot u0\right) \cdot u0 - -1\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9860000014305115)
   (* (log (- 1.0 u0)) (* (- alpha) alpha))
   (*
    (* (- (* (+ 0.5 (* 0.3333333333333333 u0)) u0) -1.0) (* alpha alpha))
    u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9860000014305115f) {
		tmp = logf((1.0f - u0)) * (-alpha * alpha);
	} else {
		tmp = ((((0.5f + (0.3333333333333333f * u0)) * u0) - -1.0f) * (alpha * alpha)) * 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) <= 0.9860000014305115e0) then
        tmp = log((1.0e0 - u0)) * (-alpha * alpha)
    else
        tmp = ((((0.5e0 + (0.3333333333333333e0 * u0)) * u0) - (-1.0e0)) * (alpha * alpha)) * u0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9860000014305115))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(-alpha) * alpha));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u0)) * u0) - Float32(-1.0)) * Float32(alpha * alpha)) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9860000014305115))
		tmp = log((single(1.0) - u0)) * (-alpha * alpha);
	else
		tmp = ((((single(0.5) + (single(0.3333333333333333) * u0)) * u0) - single(-1.0)) * (alpha * alpha)) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.986000001

   1. Initial program 94.5%

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

   if 0.986000001 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 44.7%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(0.3333333333333333 \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right)\right)\right) \cdot u0} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.3%

      \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(0.3333333333333333, u0, 0.5\right) + 1\right)\right) \cdot u0 \]
   8. Step-by-step derivation

   9. Applied rewrites 98.5%

      \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot 0.3333333333333333 + 0.5\right) + 1\right)\right) \cdot u0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9860000014305115:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.5 + 0.3333333333333333 \cdot u0\right) \cdot u0 - -1\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{-1}{\left(0.3333333333333333 \cdot u0\right) \cdot u0 - 1} + 0.5 \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0 \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (*
   (+ (/ -1.0 (- (* (* 0.3333333333333333 u0) u0) 1.0)) (* 0.5 u0))
   (* alpha alpha))
  u0))

C

float code(float alpha, float u0) {
	return (((-1.0f / (((0.3333333333333333f * u0) * u0) - 1.0f)) + (0.5f * u0)) * (alpha * alpha)) * u0;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = ((((-1.0e0) / (((0.3333333333333333e0 * u0) * u0) - 1.0e0)) + (0.5e0 * u0)) * (alpha * alpha)) * u0
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(0.3333333333333333) * u0) * u0) - Float32(1.0))) + Float32(Float32(0.5) * u0)) * Float32(alpha * alpha)) * u0)
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (((single(-1.0) / (((single(0.3333333333333333) * u0) * u0) - single(1.0))) + (single(0.5) * u0)) * (alpha * alpha)) * u0;
end

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 74.8%

   \[\leadsto \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(0.3333333333333333 \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right)\right)\right) \cdot u0} \]
6. Step-by-step derivation

7. Applied rewrites 92.1%

   \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(0.5 \cdot u0 + \left(1 + \left(0.3333333333333333 \cdot u0\right) \cdot u0\right)\right)\right) \cdot u0 \]
8. Step-by-step derivation

9. Applied rewrites 92.1%

   \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(0.5 \cdot u0 + \frac{{\left(\left(0.3333333333333333 \cdot u0\right) \cdot u0\right)}^{2} - 1}{\left(0.3333333333333333 \cdot u0\right) \cdot u0 - 1}\right)\right) \cdot u0 \]

10. Taylor expanded in u0 around 0

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

12. Applied rewrites 92.2%

    \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(0.5 \cdot u0 + \frac{-1}{\left(0.3333333333333333 \cdot u0\right) \cdot u0 - 1}\right)\right) \cdot u0 \]

13. Final simplification 92.2%

    \[\leadsto \left(\left(\frac{-1}{\left(0.3333333333333333 \cdot u0\right) \cdot u0 - 1} + 0.5 \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0 \]
14. Add Preprocessing

Alternative 6: 91.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(0.5 + 0.3333333333333333 \cdot u0\right) \cdot u0 - -1\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0 \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- (* (+ 0.5 (* 0.3333333333333333 u0)) u0) -1.0) (* alpha alpha)) u0))

C

float code(float alpha, float u0) {
	return ((((0.5f + (0.3333333333333333f * u0)) * u0) - -1.0f) * (alpha * alpha)) * u0;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = ((((0.5e0 + (0.3333333333333333e0 * u0)) * u0) - (-1.0e0)) * (alpha * alpha)) * u0
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(Float32(Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u0)) * u0) - Float32(-1.0)) * Float32(alpha * alpha)) * u0)
end

MATLAB

function tmp = code(alpha, u0)
	tmp = ((((single(0.5) + (single(0.3333333333333333) * u0)) * u0) - single(-1.0)) * (alpha * alpha)) * u0;
end

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 74.8%

   \[\leadsto \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(0.3333333333333333 \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right)\right)\right) \cdot u0} \]
6. Step-by-step derivation

7. Applied rewrites 87.4%

   \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(0.3333333333333333, u0, 0.5\right) + 1\right)\right) \cdot u0 \]
8. Step-by-step derivation

9. Applied rewrites 92.2%

   \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot 0.3333333333333333 + 0.5\right) + 1\right)\right) \cdot u0 \]

10. Final simplification 92.2%

    \[\leadsto \left(\left(\left(0.5 + 0.3333333333333333 \cdot u0\right) \cdot u0 - -1\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0 \]
11. Add Preprocessing

Alternative 7: 87.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 74.8%

   \[\leadsto \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(0.3333333333333333 \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right)\right)\right) \cdot u0} \]
6. Step-by-step derivation

7. Applied rewrites 87.4%

   \[\leadsto \left(\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(0.3333333333333333, u0, 0.5\right) + 1\right)\right) \cdot u0 \]

8. Taylor expanded in u0 around 0

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

10. Applied rewrites 87.7%

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

11. Final simplification 87.7%

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

Alternative 8: 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 53.8%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]

   2. unpow2 N/A

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

   3. lower-*.f32 75.5

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

5. Applied rewrites 75.5%

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

Reproduce

herbie shell --seed 2024296 
(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))))