Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.6% → 99.0%

Time: 12.6s

Alternatives: 4

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(-u0\right) \cdot \left(-{\alpha}^{2}\right) \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* (log1p (- u0)) (- (pow alpha 2.0))))

C

float code(float alpha, float u0) {
	return log1pf(-u0) * -powf(alpha, 2.0f);
}

Julia

function code(alpha, u0)
	return Float32(log1p(Float32(-u0)) * Float32(-(alpha ^ Float32(2.0))))
end

Derivation

1. Initial program 50.3%

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

   1. *-commutative 50.3%

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

   2. associate-*l* 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in alpha around 0 50.3%

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

   1. mul-1-neg 50.3%

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

   2. distribute-rgt-neg-in 50.3%

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

   3. sub-neg 50.3%

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

   4. mul-1-neg 50.3%

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

   5. log1p-define 99.0%

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

   6. mul-1-neg 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

   \[\leadsto \mathsf{log1p}\left(-u0\right) \cdot \left(-{\alpha}^{2}\right) \]
9. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 50.3%

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

   1. associate-*l* 50.4%

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

   2. remove-double-neg 50.4%

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

   3. distribute-lft-neg-out 50.4%

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

   4. distribute-lft-neg-out 50.4%

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

   5. remove-double-neg 50.4%

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

   6. sub-neg 50.4%

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

   7. log1p-define 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. Add Preprocessing

5. Final simplification 98.9%

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

Alternative 3: 74.6% 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 50.3%

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

   1. *-commutative 50.3%

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

   2. associate-*l* 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in u0 around 0 78.2%

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

   1. *-commutative 78.2%

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

7. Simplified 78.2%

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

8. Final simplification 78.2%

   \[\leadsto \alpha \cdot \left(\alpha \cdot u0\right) \]
9. Add Preprocessing

Alternative 4: 74.6% 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 50.3%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0 78.3%

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

   1. mul-1-neg 78.3%

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

5. Simplified 78.3%

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

6. Final simplification 78.3%

   \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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))))