Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.2% → 99.0%

Time: 6.5s

Alternatives: 7

Speedup: 21.6×

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: 56.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{log1p}\left(-u0\right) \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* (* alpha (- alpha)) (log1p (- u0))))

C

float code(float alpha, float u0) {
	return (alpha * -alpha) * log1pf(-u0);
}

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * Float32(-alpha)) * log1p(Float32(-u0)))
end

Derivation

1. Initial program 52.9%

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

   1. *-commutative 52.9%

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

   2. sub-neg 52.9%

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

   3. log1p-def 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(-\alpha\right) \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* (- alpha) (* alpha (log1p (- u0)))))

C

float code(float alpha, float u0) {
	return -alpha * (alpha * log1pf(-u0));
}

Julia

function code(alpha, u0)
	return Float32(Float32(-alpha) * Float32(alpha * log1p(Float32(-u0))))
end

Derivation

1. Initial program 52.9%

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

   1. associate-*l* 52.9%

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

   2. sub-neg 52.9%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 3: 91.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(u0 \cdot 0.3333333333333333\right) \cdot \left(u0 \cdot u0\right)\\ t_1 := \left(u0 \cdot u0\right) \cdot 0.5\\ \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \frac{t_0 \cdot t_0 - t_1 \cdot t_1}{t_0 - t_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* u0 0.3333333333333333) (* u0 u0))) (t_1 (* (* u0 u0) 0.5)))
   (* (* alpha alpha) (+ u0 (/ (- (* t_0 t_0) (* t_1 t_1)) (- t_0 t_1))))))

C

float code(float alpha, float u0) {
	float t_0 = (u0 * 0.3333333333333333f) * (u0 * u0);
	float t_1 = (u0 * u0) * 0.5f;
	return (alpha * alpha) * (u0 + (((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: t_1
    t_0 = (u0 * 0.3333333333333333e0) * (u0 * u0)
    t_1 = (u0 * u0) * 0.5e0
    code = (alpha * alpha) * (u0 + (((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)))
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(u0 * Float32(0.3333333333333333)) * Float32(u0 * u0))
	t_1 = Float32(Float32(u0 * u0) * Float32(0.5))
	return Float32(Float32(alpha * alpha) * Float32(u0 + Float32(Float32(Float32(t_0 * t_0) - Float32(t_1 * t_1)) / Float32(t_0 - t_1))))
end

MATLAB

function tmp = code(alpha, u0)
	t_0 = (u0 * single(0.3333333333333333)) * (u0 * u0);
	t_1 = (u0 * u0) * single(0.5);
	tmp = (alpha * alpha) * (u0 + (((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)));
end

Derivation

1. Initial program 52.9%

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

   1. associate-*l* 52.9%

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

   2. sub-neg 52.9%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u0 around 0 92.4%

   \[\leadsto \color{blue}{u0 \cdot {\alpha}^{2} + \left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. *-commutative 92.4%

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} + \left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right) \]

   2. +-commutative 92.4%

      \[\leadsto {\alpha}^{2} \cdot u0 + \color{blue}{\left(0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right) + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right)} \]

   3. associate-*r* 92.4%

      \[\leadsto {\alpha}^{2} \cdot u0 + \left(\color{blue}{\left(0.5 \cdot {u0}^{2}\right) \cdot {\alpha}^{2}} + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right) \]

   4. associate-*r* 92.4%

      \[\leadsto {\alpha}^{2} \cdot u0 + \left(\left(0.5 \cdot {u0}^{2}\right) \cdot {\alpha}^{2} + \color{blue}{\left(0.3333333333333333 \cdot {u0}^{3}\right) \cdot {\alpha}^{2}}\right) \]

   5. distribute-rgt-out 92.4%

      \[\leadsto {\alpha}^{2} \cdot u0 + \color{blue}{{\alpha}^{2} \cdot \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right)} \]

   6. distribute-lft-out 92.5%

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot \left(u0 + \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right)\right)} \]

   7. unpow2 92.5%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \left(u0 + \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right)\right) \]

   8. +-commutative 92.5%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \color{blue}{\left(0.3333333333333333 \cdot {u0}^{3} + 0.5 \cdot {u0}^{2}\right)}\right) \]

   9. cube-mult 92.5%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(0.3333333333333333 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot u0\right)\right)} + 0.5 \cdot {u0}^{2}\right)\right) \]

   10. unpow2 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(0.3333333333333333 \cdot \left(u0 \cdot \color{blue}{{u0}^{2}}\right) + 0.5 \cdot {u0}^{2}\right)\right) \]

   11. associate-*r* 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(\color{blue}{\left(0.3333333333333333 \cdot u0\right) \cdot {u0}^{2}} + 0.5 \cdot {u0}^{2}\right)\right) \]

   12. distribute-rgt-out 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \color{blue}{{u0}^{2} \cdot \left(0.3333333333333333 \cdot u0 + 0.5\right)}\right) \]

   13. unpow2 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \color{blue}{\left(u0 \cdot u0\right)} \cdot \left(0.3333333333333333 \cdot u0 + 0.5\right)\right) \]

   14. *-commutative 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \left(\color{blue}{u0 \cdot 0.3333333333333333} + 0.5\right)\right) \]

6. Simplified 92.5%

   \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \left(u0 \cdot 0.3333333333333333 + 0.5\right)\right)} \]
7. Step-by-step derivation

   1. distribute-rgt-in 92.5%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \color{blue}{\left(\left(u0 \cdot 0.3333333333333333\right) \cdot \left(u0 \cdot u0\right) + 0.5 \cdot \left(u0 \cdot u0\right)\right)}\right) \]

   2. flip-+ 92.5%

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

8. Applied egg-rr 92.5%

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

9. Final simplification 92.5%

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

Alternative 4: 91.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \frac{0.25 - \left(u0 \cdot u0\right) \cdot 0.1111111111111111}{0.5 - u0 \cdot 0.3333333333333333}\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (* alpha alpha)
  (+
   u0
   (*
    (* u0 u0)
    (/
     (- 0.25 (* (* u0 u0) 0.1111111111111111))
     (- 0.5 (* u0 0.3333333333333333)))))))

C

float code(float alpha, float u0) {
	return (alpha * alpha) * (u0 + ((u0 * u0) * ((0.25f - ((u0 * u0) * 0.1111111111111111f)) / (0.5f - (u0 * 0.3333333333333333f)))));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (alpha * alpha) * (u0 + ((u0 * u0) * ((0.25e0 - ((u0 * u0) * 0.1111111111111111e0)) / (0.5e0 - (u0 * 0.3333333333333333e0)))))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * alpha) * Float32(u0 + Float32(Float32(u0 * u0) * Float32(Float32(Float32(0.25) - Float32(Float32(u0 * u0) * Float32(0.1111111111111111))) / Float32(Float32(0.5) - Float32(u0 * Float32(0.3333333333333333)))))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (alpha * alpha) * (u0 + ((u0 * u0) * ((single(0.25) - ((u0 * u0) * single(0.1111111111111111))) / (single(0.5) - (u0 * single(0.3333333333333333))))));
end

Derivation

1. Initial program 52.9%

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

   1. associate-*l* 52.9%

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

   2. sub-neg 52.9%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u0 around 0 92.4%

   \[\leadsto \color{blue}{u0 \cdot {\alpha}^{2} + \left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. *-commutative 92.4%

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} + \left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right) \]

   2. +-commutative 92.4%

      \[\leadsto {\alpha}^{2} \cdot u0 + \color{blue}{\left(0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right) + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right)} \]

   3. associate-*r* 92.4%

      \[\leadsto {\alpha}^{2} \cdot u0 + \left(\color{blue}{\left(0.5 \cdot {u0}^{2}\right) \cdot {\alpha}^{2}} + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right) \]

   4. associate-*r* 92.4%

      \[\leadsto {\alpha}^{2} \cdot u0 + \left(\left(0.5 \cdot {u0}^{2}\right) \cdot {\alpha}^{2} + \color{blue}{\left(0.3333333333333333 \cdot {u0}^{3}\right) \cdot {\alpha}^{2}}\right) \]

   5. distribute-rgt-out 92.4%

      \[\leadsto {\alpha}^{2} \cdot u0 + \color{blue}{{\alpha}^{2} \cdot \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right)} \]

   6. distribute-lft-out 92.5%

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot \left(u0 + \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right)\right)} \]

   7. unpow2 92.5%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \left(u0 + \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right)\right) \]

   8. +-commutative 92.5%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \color{blue}{\left(0.3333333333333333 \cdot {u0}^{3} + 0.5 \cdot {u0}^{2}\right)}\right) \]

   9. cube-mult 92.5%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(0.3333333333333333 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot u0\right)\right)} + 0.5 \cdot {u0}^{2}\right)\right) \]

   10. unpow2 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(0.3333333333333333 \cdot \left(u0 \cdot \color{blue}{{u0}^{2}}\right) + 0.5 \cdot {u0}^{2}\right)\right) \]

   11. associate-*r* 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(\color{blue}{\left(0.3333333333333333 \cdot u0\right) \cdot {u0}^{2}} + 0.5 \cdot {u0}^{2}\right)\right) \]

   12. distribute-rgt-out 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \color{blue}{{u0}^{2} \cdot \left(0.3333333333333333 \cdot u0 + 0.5\right)}\right) \]

   13. unpow2 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \color{blue}{\left(u0 \cdot u0\right)} \cdot \left(0.3333333333333333 \cdot u0 + 0.5\right)\right) \]

   14. *-commutative 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \left(\color{blue}{u0 \cdot 0.3333333333333333} + 0.5\right)\right) \]

6. Simplified 92.5%

   \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \left(u0 \cdot 0.3333333333333333 + 0.5\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative 92.5%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \color{blue}{\left(0.5 + u0 \cdot 0.3333333333333333\right)}\right) \]

   2. flip-+ 92.5%

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

   3. metadata-eval 92.5%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \frac{\color{blue}{0.25} - \left(u0 \cdot 0.3333333333333333\right) \cdot \left(u0 \cdot 0.3333333333333333\right)}{0.5 - u0 \cdot 0.3333333333333333}\right) \]

   4. swap-sqr 92.5%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \frac{0.25 - \color{blue}{\left(u0 \cdot u0\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)}}{0.5 - u0 \cdot 0.3333333333333333}\right) \]

   5. metadata-eval 92.5%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \frac{0.25 - \left(u0 \cdot u0\right) \cdot \color{blue}{0.1111111111111111}}{0.5 - u0 \cdot 0.3333333333333333}\right) \]

8. Applied egg-rr 92.5%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \color{blue}{\frac{0.25 - \left(u0 \cdot u0\right) \cdot 0.1111111111111111}{0.5 - u0 \cdot 0.3333333333333333}}\right) \]

9. Final simplification 92.5%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \frac{0.25 - \left(u0 \cdot u0\right) \cdot 0.1111111111111111}{0.5 - u0 \cdot 0.3333333333333333}\right) \]

Alternative 5: 91.3% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.9%

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

   1. associate-*l* 52.9%

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

   2. sub-neg 52.9%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u0 around 0 92.4%

   \[\leadsto \color{blue}{u0 \cdot {\alpha}^{2} + \left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. *-commutative 92.4%

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} + \left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right) \]

   2. +-commutative 92.4%

      \[\leadsto {\alpha}^{2} \cdot u0 + \color{blue}{\left(0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right) + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right)} \]

   3. associate-*r* 92.4%

      \[\leadsto {\alpha}^{2} \cdot u0 + \left(\color{blue}{\left(0.5 \cdot {u0}^{2}\right) \cdot {\alpha}^{2}} + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right) \]

   4. associate-*r* 92.4%

      \[\leadsto {\alpha}^{2} \cdot u0 + \left(\left(0.5 \cdot {u0}^{2}\right) \cdot {\alpha}^{2} + \color{blue}{\left(0.3333333333333333 \cdot {u0}^{3}\right) \cdot {\alpha}^{2}}\right) \]

   5. distribute-rgt-out 92.4%

      \[\leadsto {\alpha}^{2} \cdot u0 + \color{blue}{{\alpha}^{2} \cdot \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right)} \]

   6. distribute-lft-out 92.5%

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot \left(u0 + \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right)\right)} \]

   7. unpow2 92.5%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \left(u0 + \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right)\right) \]

   8. +-commutative 92.5%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \color{blue}{\left(0.3333333333333333 \cdot {u0}^{3} + 0.5 \cdot {u0}^{2}\right)}\right) \]

   9. cube-mult 92.5%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(0.3333333333333333 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot u0\right)\right)} + 0.5 \cdot {u0}^{2}\right)\right) \]

   10. unpow2 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(0.3333333333333333 \cdot \left(u0 \cdot \color{blue}{{u0}^{2}}\right) + 0.5 \cdot {u0}^{2}\right)\right) \]

   11. associate-*r* 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(\color{blue}{\left(0.3333333333333333 \cdot u0\right) \cdot {u0}^{2}} + 0.5 \cdot {u0}^{2}\right)\right) \]

   12. distribute-rgt-out 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \color{blue}{{u0}^{2} \cdot \left(0.3333333333333333 \cdot u0 + 0.5\right)}\right) \]

   13. unpow2 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \color{blue}{\left(u0 \cdot u0\right)} \cdot \left(0.3333333333333333 \cdot u0 + 0.5\right)\right) \]

   14. *-commutative 92.5%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \left(\color{blue}{u0 \cdot 0.3333333333333333} + 0.5\right)\right) \]

6. Simplified 92.5%

   \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \left(u0 \cdot 0.3333333333333333 + 0.5\right)\right)} \]

7. Final simplification 92.5%

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

Alternative 6: 87.1% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.9%

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

   1. associate-*l* 52.9%

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

   2. sub-neg 52.9%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u0 around 0 88.4%

   \[\leadsto \color{blue}{u0 \cdot {\alpha}^{2} + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)} \]
5. Step-by-step derivation

   1. associate-*r* 88.4%

      \[\leadsto u0 \cdot {\alpha}^{2} + \color{blue}{\left(0.5 \cdot {u0}^{2}\right) \cdot {\alpha}^{2}} \]

   2. distribute-rgt-out 88.4%

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot \left(u0 + 0.5 \cdot {u0}^{2}\right)} \]

   3. unpow2 88.4%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \left(u0 + 0.5 \cdot {u0}^{2}\right) \]

   4. unpow2 88.4%

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

6. Simplified 88.4%

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

7. Final simplification 88.4%

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

Alternative 7: 74.3% accurate, 21.6× speedup

Math

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

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(u0 * Float32(alpha * alpha))
end

MATLAB

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

Derivation

1. Initial program 52.9%

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

   1. associate-*l* 52.9%

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

   2. sub-neg 52.9%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u0 around 0 75.7%

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

   1. *-commutative 75.7%

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

   2. unpow2 75.7%

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

6. Simplified 75.7%

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

7. Final simplification 75.7%

   \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) \]

Reproduce

herbie shell --seed 2023240 
(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))))