Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.1% → 99.0%

Time: 6.9s

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: 56.1% 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 56.1%

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

   1. *-commutative 56.1%

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

   2. sub-neg 56.1%

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

   3. log1p-def 98.9%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

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 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   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 3: 91.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in u0 around 0 90.3%

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

   1. *-commutative 90.3%

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

   2. associate-*r* 90.3%

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

   3. associate-*r* 90.3%

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

   4. distribute-rgt-out 90.3%

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

   5. distribute-lft-out 90.2%

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

   6. unpow2 90.2%

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

   7. cube-mult 90.2%

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

   8. unpow2 90.2%

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

   9. associate-*r* 90.2%

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

   10. distribute-rgt-out 90.2%

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

   11. unpow2 90.2%

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

6. Simplified 90.2%

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

   1. distribute-lft-in 90.2%

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

   2. associate-*r* 90.2%

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

8. Applied egg-rr 90.2%

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

9. Final simplification 90.2%

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

Alternative 4: 91.2% accurate, 7.2× speedup

Math

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

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 56.1%

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

   1. associate-*l* 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in u0 around 0 90.3%

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

   1. *-commutative 90.3%

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

   2. associate-*r* 90.3%

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

   3. associate-*r* 90.3%

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

   4. distribute-rgt-out 90.3%

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

   5. distribute-lft-out 90.2%

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

   6. unpow2 90.2%

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

   7. cube-mult 90.2%

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

   8. unpow2 90.2%

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

   9. associate-*r* 90.2%

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

   10. distribute-rgt-out 90.2%

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

   11. unpow2 90.2%

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

6. Simplified 90.2%

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

7. Final simplification 90.2%

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

Alternative 5: 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 56.1%

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

   1. associate-*l* 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in u0 around 0 86.3%

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

   1. +-commutative 86.3%

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

   2. associate-*r* 86.3%

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

   3. distribute-rgt-out 86.3%

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

   4. unpow2 86.3%

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

   5. *-commutative 86.3%

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

   6. unpow2 86.3%

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

6. Simplified 86.3%

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

7. Final simplification 86.3%

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

Alternative 6: 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 56.1%

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

   1. associate-*l* 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in u0 around 0 73.0%

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

   1. unpow2 73.0%

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

6. Simplified 73.0%

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

7. Final simplification 73.0%

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

Reproduce

herbie shell --seed 2023195 
(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))))