Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.2% → 99.0%

Time: 6.7s

Alternatives: 8

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

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

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(alpha * Float32(alpha * Float32(-log1p(Float32(-u0)))))
end

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 2: 91.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

float code(float alpha, float u0) {
	return alpha * (alpha * (u0 - ((u0 * (u0 * -0.5f)) + ((u0 * u0) * (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 * (-0.5e0))) + ((u0 * u0) * (u0 * (-0.3333333333333333e0))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 90.9%

   \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + \left(-1 \cdot \left(u0 \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)\right)} \]
5. Step-by-step derivation

   1. associate-+r+ 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)} \]

   2. +-commutative 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right)\right)} \]

   3. associate-*r* 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \left(\color{blue}{\left(-0.5 \cdot {u0}^{2}\right) \cdot \alpha} + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right)\right) \]

   4. *-commutative 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \left(\color{blue}{\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right)} + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right)\right) \]

   5. associate-*r* 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right) + \left(\color{blue}{\left(-0.3333333333333333 \cdot {u0}^{3}\right) \cdot \alpha} + -1 \cdot \left(u0 \cdot \alpha\right)\right)\right) \]

   6. associate-*r* 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right) + \left(\left(-0.3333333333333333 \cdot {u0}^{3}\right) \cdot \alpha + \color{blue}{\left(-1 \cdot u0\right) \cdot \alpha}\right)\right) \]

   7. distribute-rgt-out 91.0%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right) + \color{blue}{\alpha \cdot \left(-0.3333333333333333 \cdot {u0}^{3} + -1 \cdot u0\right)}\right) \]

   8. distribute-lft-out 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(\alpha \cdot \left(-0.5 \cdot {u0}^{2} + \left(-0.3333333333333333 \cdot {u0}^{3} + -1 \cdot u0\right)\right)\right)} \]

   9. associate-+l+ 91.0%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\left(\left(-0.5 \cdot {u0}^{2} + -0.3333333333333333 \cdot {u0}^{3}\right) + -1 \cdot u0\right)}\right) \]

   10. mul-1-neg 91.0%

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

   11. unsub-neg 91.0%

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

   12. cube-mult 91.0%

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

   13. unpow2 91.0%

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

   14. associate-*r* 91.0%

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

   15. distribute-rgt-out 91.0%

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

   16. unpow2 91.0%

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

6. Simplified 91.0%

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

   1. distribute-lft-in 91.0%

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

   2. associate-*r* 91.0%

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

   3. *-commutative 91.0%

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

8. Applied egg-rr 91.0%

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

9. Final simplification 91.0%

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

Alternative 3: 91.2% accurate, 7.2× speedup

Math

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

FPCore

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

C

float code(float alpha, float u0) {
	return (alpha * alpha) * (u0 + ((u0 * u0) * (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) * (0.5e0 + (u0 * 0.3333333333333333e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 90.8%

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

   1. *-commutative 90.8%

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

   2. +-commutative 90.8%

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

   3. associate-*r* 90.8%

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

   4. associate-*r* 90.8%

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

   5. distribute-rgt-out 90.8%

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

   6. distribute-lft-out 90.8%

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

   7. unpow2 90.8%

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

   8. cube-mult 90.8%

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

   9. unpow2 90.8%

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

   10. associate-*r* 90.8%

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

   11. distribute-rgt-out 90.8%

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

   12. unpow2 90.8%

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

6. Simplified 90.8%

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

7. Final simplification 90.8%

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

Alternative 4: 91.2% accurate, 7.2× speedup

Math

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

FPCore

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

C

float code(float alpha, float u0) {
	return alpha * (alpha * (u0 - ((u0 * u0) * (-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) * ((-0.5e0) + (u0 * (-0.3333333333333333e0))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 90.9%

   \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + \left(-1 \cdot \left(u0 \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)\right)} \]
5. Step-by-step derivation

   1. associate-+r+ 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)} \]

   2. +-commutative 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right)\right)} \]

   3. associate-*r* 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \left(\color{blue}{\left(-0.5 \cdot {u0}^{2}\right) \cdot \alpha} + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right)\right) \]

   4. *-commutative 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \left(\color{blue}{\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right)} + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right)\right) \]

   5. associate-*r* 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right) + \left(\color{blue}{\left(-0.3333333333333333 \cdot {u0}^{3}\right) \cdot \alpha} + -1 \cdot \left(u0 \cdot \alpha\right)\right)\right) \]

   6. associate-*r* 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right) + \left(\left(-0.3333333333333333 \cdot {u0}^{3}\right) \cdot \alpha + \color{blue}{\left(-1 \cdot u0\right) \cdot \alpha}\right)\right) \]

   7. distribute-rgt-out 91.0%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right) + \color{blue}{\alpha \cdot \left(-0.3333333333333333 \cdot {u0}^{3} + -1 \cdot u0\right)}\right) \]

   8. distribute-lft-out 90.9%

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(\alpha \cdot \left(-0.5 \cdot {u0}^{2} + \left(-0.3333333333333333 \cdot {u0}^{3} + -1 \cdot u0\right)\right)\right)} \]

   9. associate-+l+ 91.0%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\left(\left(-0.5 \cdot {u0}^{2} + -0.3333333333333333 \cdot {u0}^{3}\right) + -1 \cdot u0\right)}\right) \]

   10. mul-1-neg 91.0%

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

   11. unsub-neg 91.0%

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

   12. cube-mult 91.0%

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

   13. unpow2 91.0%

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

   14. associate-*r* 91.0%

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

   15. distribute-rgt-out 91.0%

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

   16. unpow2 91.0%

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

6. Simplified 91.0%

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

7. Final simplification 91.0%

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

Alternative 5: 87.0% accurate, 8.3× speedup

Math

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

FPCore

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

C

float code(float alpha, float u0) {
	return alpha * ((alpha * u0) - (alpha * (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) - (alpha * (u0 * (u0 * (-0.5e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 86.6%

   \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(-1 \cdot \left(u0 \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative 86.6%

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

   2. mul-1-neg 86.6%

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

   3. unsub-neg 86.6%

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

   4. associate-*r* 86.6%

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

   5. distribute-rgt-out-- 86.6%

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

   6. *-commutative 86.6%

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

   7. unpow2 86.6%

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

   8. associate-*l* 86.6%

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

6. Simplified 86.6%

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

   1. sub-neg 86.6%

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

   2. distribute-rgt-in 86.6%

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

8. Applied egg-rr 86.6%

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

9. Final simplification 86.6%

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

Alternative 6: 87.0% 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 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 86.6%

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

   1. +-commutative 86.6%

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

   2. associate-*r* 86.6%

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

   3. distribute-rgt-out 86.5%

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

   4. unpow2 86.5%

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

   5. unpow2 86.5%

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

6. Simplified 86.5%

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

7. Final simplification 86.5%

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

Alternative 7: 87.0% accurate, 9.8× speedup

Math

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

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(alpha * Float32(alpha * Float32(u0 - Float32(u0 * Float32(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 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 86.6%

   \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(-1 \cdot \left(u0 \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative 86.6%

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

   2. mul-1-neg 86.6%

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

   3. unsub-neg 86.6%

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

   4. associate-*r* 86.6%

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

   5. distribute-rgt-out-- 86.6%

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

   6. *-commutative 86.6%

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

   7. unpow2 86.6%

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

   8. associate-*l* 86.6%

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

6. Simplified 86.6%

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

7. Final simplification 86.6%

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

Alternative 8: 74.3% accurate, 21.6× speedup

Math

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

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(alpha * Float32(alpha * u0))
end

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 72.9%

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

   1. *-commutative 72.9%

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

   2. unpow2 72.9%

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

   3. associate-*l* 73.0%

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

6. Simplified 73.0%

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

7. Final simplification 73.0%

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

Reproduce

herbie shell --seed 2023242 
(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))))