Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.9% → 99.0%

Time: 9.0s

Alternatives: 9

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: 55.9% 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

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

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

Derivation

1. Initial program 57.3%

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

   1. associate-*l* 57.3%

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

   2. sub-neg 57.3%

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

   3. log1p-def 98.9%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

   1. associate-*l* 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in alpha around 0 57.3%

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

   1. mul-1-neg 57.3%

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

   2. *-commutative 57.3%

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

   3. distribute-rgt-neg-in 57.3%

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

   4. sub-neg 57.3%

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

   5. mul-1-neg 57.3%

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

   6. log1p-def 98.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot u0\right)} \cdot \left(-{\alpha}^{2}\right) \]

   7. mul-1-neg 98.9%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{-u0}\right) \cdot \left(-{\alpha}^{2}\right) \]

   8. unpow2 98.9%

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

   9. distribute-rgt-neg-in 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 3: 91.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

   1. associate-*l* 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in u0 around 0 89.3%

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

   1. +-commutative 89.3%

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

   2. mul-1-neg 89.3%

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

   3. unsub-neg 89.3%

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

   4. *-commutative 89.3%

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

   5. associate-*r* 89.3%

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

   6. *-commutative 89.3%

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

   7. associate-*r* 89.3%

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

   8. distribute-rgt-out 89.3%

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

   9. cube-mult 89.3%

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

   10. unpow2 89.3%

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

   11. associate-*r* 89.3%

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

   12. distribute-rgt-out 89.3%

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

   13. unpow2 89.3%

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

   14. *-commutative 89.3%

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

6. Simplified 89.3%

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

7. Final simplification 89.3%

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

Alternative 4: 91.1% 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 57.3%

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

   1. associate-*l* 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in u0 around 0 89.2%

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

   1. +-commutative 89.2%

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

   2. mul-1-neg 89.2%

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

   3. unsub-neg 89.2%

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

   4. cube-mult 89.2%

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

   5. unpow2 89.2%

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

   6. associate-*r* 89.2%

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

   7. distribute-rgt-out 89.2%

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

   8. unpow2 89.2%

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

6. Simplified 89.2%

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

7. Final simplification 89.2%

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

Alternative 5: 91.1% 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 57.3%

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

2. Taylor expanded in u0 around 0 89.2%

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

   1. +-commutative 89.2%

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

   2. mul-1-neg 89.2%

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

   3. unsub-neg 89.2%

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

   4. cube-mult 89.2%

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

   5. unpow2 89.2%

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

   6. associate-*r* 89.2%

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

   7. distribute-rgt-out 89.2%

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

   8. unpow2 89.2%

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

4. Simplified 89.2%

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

5. Final simplification 89.2%

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

Alternative 6: 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 57.3%

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

   1. associate-*l* 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in u0 around 0 84.1%

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

   1. +-commutative 84.1%

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

   2. *-commutative 84.1%

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

   3. associate-*l* 84.1%

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

   4. distribute-lft-out 84.2%

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

   5. unpow2 84.2%

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

   6. unpow2 84.2%

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

6. Simplified 84.2%

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

7. Taylor expanded in alpha around 0 84.2%

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

   1. unpow2 84.2%

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

   2. *-commutative 84.2%

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

   3. unpow2 84.2%

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

   4. associate-*r* 84.2%

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

   5. associate-*l* 84.3%

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

9. Simplified 84.3%

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

10. Final simplification 84.3%

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

Alternative 7: 87.0% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

   1. associate-*l* 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in u0 around 0 84.3%

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

   1. +-commutative 84.3%

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

   2. mul-1-neg 84.3%

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

   3. unsub-neg 84.3%

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

   4. *-commutative 84.3%

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

   5. unpow2 84.3%

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

   6. *-commutative 84.3%

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

6. Simplified 84.3%

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

7. Taylor expanded in u0 around 0 84.3%

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

   1. mul-1-neg 84.3%

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

   2. distribute-lft-neg-in 84.3%

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

   3. *-commutative 84.3%

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

   4. *-commutative 84.3%

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

   5. unpow2 84.3%

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

   6. associate-*r* 84.3%

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

   7. associate-*r* 84.3%

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

   8. *-commutative 84.3%

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

   9. distribute-rgt-out 84.3%

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

   10. *-commutative 84.3%

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

   11. associate-*r* 84.3%

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

   12. *-commutative 84.3%

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

   13. *-commutative 84.3%

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

9. Simplified 84.3%

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

10. Final simplification 84.3%

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

Alternative 8: 74.4% 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 57.3%

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

   1. associate-*l* 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in alpha around 0 57.3%

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

   1. mul-1-neg 57.3%

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

   2. *-commutative 57.3%

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

   3. distribute-rgt-neg-in 57.3%

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

   4. sub-neg 57.3%

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

   5. mul-1-neg 57.3%

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

   6. log1p-def 98.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot u0\right)} \cdot \left(-{\alpha}^{2}\right) \]

   7. mul-1-neg 98.9%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{-u0}\right) \cdot \left(-{\alpha}^{2}\right) \]

   8. unpow2 98.9%

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

   9. distribute-rgt-neg-in 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in u0 around 0 72.4%

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

   1. *-commutative 72.4%

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

   2. unpow2 72.4%

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

   3. associate-*r* 72.4%

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

   4. *-commutative 72.4%

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

   5. *-commutative 72.4%

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

9. Simplified 72.4%

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

10. Final simplification 72.4%

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

Alternative 9: 74.4% 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 57.3%

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

   1. associate-*l* 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in u0 around 0 72.4%

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

   1. *-commutative 72.4%

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

   2. unpow2 72.4%

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

6. Simplified 72.4%

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

7. Final simplification 72.4%

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

Reproduce

herbie shell --seed 2023271 
(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))))