Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.6% → 99.0%

Time: 6.5s

Alternatives: 6

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: 55.6% 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 53.7%

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

   1. *-commutative 53.7%

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

   2. sub-neg 53.7%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

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 53.7%

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

   1. associate-*l* 53.7%

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

   2. sub-neg 53.7%

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

   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%

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

Alternative 3: 91.6% 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 53.7%

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

   1. associate-*l* 53.7%

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

   2. sub-neg 53.7%

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

   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 92.1%

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

   1. *-commutative 92.1%

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

   2. +-commutative 92.1%

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

   3. associate-*r* 92.1%

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

   4. associate-*r* 92.1%

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

   5. distribute-rgt-out 92.1%

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

   6. distribute-lft-out 92.2%

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

   7. unpow2 92.2%

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

   8. +-commutative 92.2%

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

   9. cube-mult 92.2%

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

   10. unpow2 92.2%

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

   11. associate-*r* 92.2%

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

   12. distribute-rgt-out 92.2%

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

   13. unpow2 92.2%

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

   14. *-commutative 92.2%

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

6. Simplified 92.2%

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

7. Final simplification 92.2%

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

Alternative 4: 87.5% 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 53.7%

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

   1. associate-*l* 53.7%

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

   2. sub-neg 53.7%

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

   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 88.3%

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

   1. +-commutative 88.3%

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

   2. mul-1-neg 88.3%

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

   3. unsub-neg 88.3%

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

   4. associate-*r* 88.3%

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

   5. distribute-rgt-out-- 88.2%

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

   6. *-commutative 88.2%

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

   7. unpow2 88.2%

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

   8. associate-*l* 88.2%

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

6. Simplified 88.2%

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

   1. sub-neg 88.2%

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

   2. distribute-rgt-in 88.3%

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

8. Applied egg-rr 88.3%

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

9. Final simplification 88.3%

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

Alternative 5: 87.5% 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 53.7%

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

   1. associate-*l* 53.7%

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

   2. sub-neg 53.7%

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

   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 88.2%

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

   1. associate-*r* 88.2%

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

   2. distribute-rgt-out 88.3%

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

   3. unpow2 88.3%

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

   4. unpow2 88.3%

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

6. Simplified 88.3%

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

7. Final simplification 88.3%

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

Alternative 6: 74.7% 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 53.7%

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

   1. associate-*l* 53.7%

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

   2. sub-neg 53.7%

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

   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 75.8%

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

   1. *-commutative 75.8%

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

   2. unpow2 75.8%

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

6. Simplified 75.8%

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

7. Final simplification 75.8%

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

Reproduce

herbie shell --seed 2023213 
(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))))