Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.3% → 97.9%

Time: 6.8s

Alternatives: 9

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: 56.3% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9880499839782715:\\ \;\;\;\;\frac{{\alpha}^{4}}{\frac{-1}{\frac{\alpha \cdot \alpha}{{\alpha}^{4}}}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(\left(-0.3333333333333333 \cdot u0 + -0.5\right) \cdot u0 - 1\right) \cdot u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9880499839782715)
   (*
    (/ (pow alpha 4.0) (/ -1.0 (/ (* alpha alpha) (pow alpha 4.0))))
    (log (- 1.0 u0)))
   (*
    (* (- alpha) alpha)
    (* (- (* (+ (* -0.3333333333333333 u0) -0.5) u0) 1.0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9880499839782715f) {
		tmp = (powf(alpha, 4.0f) / (-1.0f / ((alpha * alpha) / powf(alpha, 4.0f)))) * logf((1.0f - u0));
	} else {
		tmp = (-alpha * alpha) * (((((-0.3333333333333333f * u0) + -0.5f) * u0) - 1.0f) * 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) <= 0.9880499839782715e0) then
        tmp = ((alpha ** 4.0e0) / ((-1.0e0) / ((alpha * alpha) / (alpha ** 4.0e0)))) * log((1.0e0 - u0))
    else
        tmp = (-alpha * alpha) * ((((((-0.3333333333333333e0) * u0) + (-0.5e0)) * u0) - 1.0e0) * u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9880499839782715))
		tmp = Float32(Float32((alpha ^ Float32(4.0)) / Float32(Float32(-1.0) / Float32(Float32(alpha * alpha) / (alpha ^ Float32(4.0))))) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(-alpha) * alpha) * Float32(Float32(Float32(Float32(Float32(Float32(-0.3333333333333333) * u0) + Float32(-0.5)) * u0) - Float32(1.0)) * u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9880499839782715))
		tmp = ((alpha ^ single(4.0)) / (single(-1.0) / ((alpha * alpha) / (alpha ^ single(4.0))))) * log((single(1.0) - u0));
	else
		tmp = (-alpha * alpha) * (((((single(-0.3333333333333333) * u0) + single(-0.5)) * u0) - single(1.0)) * u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.988049984

   1. Initial program 94.7%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

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

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

      4. neg-sub0 N/A

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

      5. flip-- N/A

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

      6. +-lft-identity N/A

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

      7. lower-/.f32 N/A

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

      8. metadata-eval N/A

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

      9. sub0-neg N/A

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

      10. lower-neg.f32 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. pow-prod-up N/A

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

      14. lower-pow.f32 N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 94.8

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

   4. Applied rewrites 94.8%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-div N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. lift-*.f32 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. sqr-neg N/A

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

      14. lift-neg.f32 N/A

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

      15. lift-neg.f32 N/A

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

      16. swap-sqr N/A

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

      17. lift-*.f32 N/A

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. lift-*.f32 N/A

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

      21. clear-num N/A

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

      22. lower-/.f32 N/A

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

      23. lower-/.f32 94.9

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

      24. lift-*.f32 N/A

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

   6. Applied rewrites 95.0%

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

   if 0.988049984 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 45.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 83.2

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 95.0%

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

   9. Applied rewrites 98.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9880499839782715:\\ \;\;\;\;\frac{{\alpha}^{4}}{\frac{-1}{\frac{\alpha \cdot \alpha}{{\alpha}^{4}}}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(\left(-0.3333333333333333 \cdot u0 + -0.5\right) \cdot u0 - 1\right) \cdot u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9880499839782715:\\ \;\;\;\;\left({\alpha}^{4} \cdot \frac{-1}{\alpha \cdot \alpha}\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(\left(-0.3333333333333333 \cdot u0 + -0.5\right) \cdot u0 - 1\right) \cdot u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9880499839782715)
   (* (* (pow alpha 4.0) (/ -1.0 (* alpha alpha))) (log (- 1.0 u0)))
   (*
    (* (- alpha) alpha)
    (* (- (* (+ (* -0.3333333333333333 u0) -0.5) u0) 1.0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9880499839782715f) {
		tmp = (powf(alpha, 4.0f) * (-1.0f / (alpha * alpha))) * logf((1.0f - u0));
	} else {
		tmp = (-alpha * alpha) * (((((-0.3333333333333333f * u0) + -0.5f) * u0) - 1.0f) * 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) <= 0.9880499839782715e0) then
        tmp = ((alpha ** 4.0e0) * ((-1.0e0) / (alpha * alpha))) * log((1.0e0 - u0))
    else
        tmp = (-alpha * alpha) * ((((((-0.3333333333333333e0) * u0) + (-0.5e0)) * u0) - 1.0e0) * u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9880499839782715))
		tmp = Float32(Float32((alpha ^ Float32(4.0)) * Float32(Float32(-1.0) / Float32(alpha * alpha))) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(-alpha) * alpha) * Float32(Float32(Float32(Float32(Float32(Float32(-0.3333333333333333) * u0) + Float32(-0.5)) * u0) - Float32(1.0)) * u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9880499839782715))
		tmp = ((alpha ^ single(4.0)) * (single(-1.0) / (alpha * alpha))) * log((single(1.0) - u0));
	else
		tmp = (-alpha * alpha) * (((((single(-0.3333333333333333) * u0) + single(-0.5)) * u0) - single(1.0)) * u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.988049984

   1. Initial program 94.7%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 94.7

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

   4. Applied rewrites 94.7%

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

   6. Applied rewrites 94.9%

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

   if 0.988049984 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 45.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 83.2

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 95.0%

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

   9. Applied rewrites 98.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9880499839782715:\\ \;\;\;\;\left({\alpha}^{4} \cdot \frac{-1}{\alpha \cdot \alpha}\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(\left(-0.3333333333333333 \cdot u0 + -0.5\right) \cdot u0 - 1\right) \cdot u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9860000014305115f) {
		tmp = (-1.0f / (alpha / ((alpha * alpha) * alpha))) * logf((1.0f - u0));
	} else {
		tmp = (-alpha * alpha) * (((((-0.3333333333333333f * u0) + -0.5f) * u0) - 1.0f) * 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 = ((-1.0e0) / (alpha / ((alpha * alpha) * alpha))) * log((1.0e0 - u0))
    else
        tmp = (-alpha * alpha) * ((((((-0.3333333333333333e0) * u0) + (-0.5e0)) * u0) - 1.0e0) * 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(Float32(Float32(-1.0) / Float32(alpha / Float32(Float32(alpha * alpha) * alpha))) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(-alpha) * alpha) * Float32(Float32(Float32(Float32(Float32(Float32(-0.3333333333333333) * u0) + Float32(-0.5)) * u0) - Float32(1.0)) * 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 = (single(-1.0) / (alpha / ((alpha * alpha) * alpha))) * log((single(1.0) - u0));
	else
		tmp = (-alpha * alpha) * (((((single(-0.3333333333333333) * u0) + single(-0.5)) * u0) - single(1.0)) * 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 95.1%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

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

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

      8. lift-neg.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. +-lft-identity N/A

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

      11. associate-*l/ N/A

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

      12. clear-num N/A

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

      13. lower-/.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 95.2

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

   4. Applied rewrites 95.2%

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

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

   1. Initial program 46.0%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 82.6

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

   5. Applied rewrites 82.6%

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

   7. Applied rewrites 95.4%

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

   9. Applied rewrites 98.4%

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

4. Final simplification 97.9%

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

Alternative 4: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\alpha\right) \cdot \alpha\\ \mathbf{if}\;1 - u0 \leq 0.9860000014305115:\\ \;\;\;\;\frac{t\_0 \cdot \alpha}{\alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(-0.3333333333333333 \cdot u0 + -0.5\right) \cdot u0 - 1\right) \cdot u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (- alpha) alpha)))
   (if (<= (- 1.0 u0) 0.9860000014305115)
     (* (/ (* t_0 alpha) alpha) (log (- 1.0 u0)))
     (* t_0 (* (- (* (+ (* -0.3333333333333333 u0) -0.5) u0) 1.0) u0)))))

C

float code(float alpha, float u0) {
	float t_0 = -alpha * alpha;
	float tmp;
	if ((1.0f - u0) <= 0.9860000014305115f) {
		tmp = ((t_0 * alpha) / alpha) * logf((1.0f - u0));
	} else {
		tmp = t_0 * (((((-0.3333333333333333f * u0) + -0.5f) * u0) - 1.0f) * u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -alpha * alpha
    if ((1.0e0 - u0) <= 0.9860000014305115e0) then
        tmp = ((t_0 * alpha) / alpha) * log((1.0e0 - u0))
    else
        tmp = t_0 * ((((((-0.3333333333333333e0) * u0) + (-0.5e0)) * u0) - 1.0e0) * u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(-alpha) * alpha)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9860000014305115))
		tmp = Float32(Float32(Float32(t_0 * alpha) / alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(t_0 * Float32(Float32(Float32(Float32(Float32(Float32(-0.3333333333333333) * u0) + Float32(-0.5)) * u0) - Float32(1.0)) * u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = -alpha * alpha;
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9860000014305115))
		tmp = ((t_0 * alpha) / alpha) * log((single(1.0) - u0));
	else
		tmp = t_0 * (((((single(-0.3333333333333333) * u0) + single(-0.5)) * u0) - single(1.0)) * 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 95.1%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

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

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

      8. lift-neg.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. +-lft-identity N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 95.1

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

   4. Applied rewrites 95.1%

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

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

   1. Initial program 46.0%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 82.6

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

   5. Applied rewrites 82.6%

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

   7. Applied rewrites 94.6%

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

   9. Applied rewrites 98.4%

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

Alternative 5: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\alpha\right) \cdot \alpha\\ \mathbf{if}\;1 - u0 \leq 0.9860000014305115:\\ \;\;\;\;t\_0 \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(-0.3333333333333333 \cdot u0 + -0.5\right) \cdot u0 - 1\right) \cdot u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (- alpha) alpha)))
   (if (<= (- 1.0 u0) 0.9860000014305115)
     (* t_0 (log (- 1.0 u0)))
     (* t_0 (* (- (* (+ (* -0.3333333333333333 u0) -0.5) u0) 1.0) u0)))))

C

float code(float alpha, float u0) {
	float t_0 = -alpha * alpha;
	float tmp;
	if ((1.0f - u0) <= 0.9860000014305115f) {
		tmp = t_0 * logf((1.0f - u0));
	} else {
		tmp = t_0 * (((((-0.3333333333333333f * u0) + -0.5f) * u0) - 1.0f) * u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -alpha * alpha
    if ((1.0e0 - u0) <= 0.9860000014305115e0) then
        tmp = t_0 * log((1.0e0 - u0))
    else
        tmp = t_0 * ((((((-0.3333333333333333e0) * u0) + (-0.5e0)) * u0) - 1.0e0) * u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(-alpha) * alpha)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9860000014305115))
		tmp = Float32(t_0 * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(t_0 * Float32(Float32(Float32(Float32(Float32(Float32(-0.3333333333333333) * u0) + Float32(-0.5)) * u0) - Float32(1.0)) * u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = -alpha * alpha;
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9860000014305115))
		tmp = t_0 * log((single(1.0) - u0));
	else
		tmp = t_0 * (((((single(-0.3333333333333333) * u0) + single(-0.5)) * u0) - single(1.0)) * 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 95.1%

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

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

   1. Initial program 46.0%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 82.6

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 94.6%

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

   9. Applied rewrites 98.4%

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

Alternative 6: 91.4% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 74.8

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

5. Applied rewrites 74.5%

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

7. Applied rewrites 87.6%

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

9. Applied rewrites 92.2%

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

Alternative 7: 87.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 74.8

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

5. Applied rewrites 74.5%

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

7. Applied rewrites 87.2%

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

8. Taylor expanded in u0 around 0

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

10. Applied rewrites 87.6%

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

Alternative 8: 74.4% 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 54.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 74.8

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

5. Applied rewrites 74.8%

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

7. Applied rewrites 74.8%

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

Alternative 9: 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 54.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 74.8

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

5. Applied rewrites 74.8%

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

Reproduce

herbie shell --seed 2024313 
(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))))