Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.0% → 99.0%

Time: 9.7s

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: 56.0% 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 54.7%

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

   1. distribute-lft-neg-out 54.7%

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

   2. sub-neg 54.7%

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

   3. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

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 54.7%

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

   1. associate-*l* 54.6%

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

   2. distribute-lft-neg-out 54.6%

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

   3. distribute-rgt-neg-in 54.6%

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

   4. distribute-lft-neg-out 54.6%

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

   5. sub-neg 54.6%

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

   6. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 3: 93.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot \left(u0 \cdot \left(\alpha + \alpha \cdot \left(u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  alpha
  (*
   u0
   (+
    alpha
    (* alpha (* u0 (+ 0.5 (* u0 (+ 0.3333333333333333 (* u0 0.25))))))))))

C

float code(float alpha, float u0) {
	return alpha * (u0 * (alpha + (alpha * (u0 * (0.5f + (u0 * (0.3333333333333333f + (u0 * 0.25f))))))));
}

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 + (u0 * (0.3333333333333333e0 + (u0 * 0.25e0))))))))
end function

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(u0 * Float32(alpha + Float32(alpha * Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) + Float32(u0 * Float32(0.25))))))))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (u0 * (alpha + (alpha * (u0 * (single(0.5) + (u0 * (single(0.3333333333333333) + (u0 * single(0.25)))))))));
end

Derivation

1. Initial program 54.7%

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

   1. associate-*l* 54.6%

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

   2. distribute-lft-neg-out 54.6%

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

   3. distribute-rgt-neg-in 54.6%

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

   4. distribute-lft-neg-out 54.6%

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

   5. sub-neg 54.6%

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

   6. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u0 around 0 91.9%

   \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + u0 \cdot \left(0.25 \cdot \left(\alpha \cdot u0\right) + 0.3333333333333333 \cdot \alpha\right)\right)\right)\right)} \]

6. Taylor expanded in alpha around 0 91.9%

   \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + \color{blue}{\alpha \cdot \left(u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}\right)\right) \]
7. Step-by-step derivation

   1. *-commutative 91.9%

      \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + \alpha \cdot \left(u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + \color{blue}{u0 \cdot 0.25}\right)\right)\right)\right)\right) \]

8. Simplified 91.9%

   \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + \color{blue}{\alpha \cdot \left(u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right)\right)}\right)\right) \]
9. Add Preprocessing

Alternative 4: 91.2% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.7%

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

   1. associate-*l* 54.6%

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

   2. distribute-lft-neg-out 54.6%

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

   3. distribute-rgt-neg-in 54.6%

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

   4. distribute-lft-neg-out 54.6%

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

   5. sub-neg 54.6%

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

   6. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u0 around 0 91.9%

   \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + u0 \cdot \left(0.25 \cdot \left(\alpha \cdot u0\right) + 0.3333333333333333 \cdot \alpha\right)\right)\right)\right)} \]

6. Taylor expanded in u0 around 0 89.8%

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

   1. *-commutative 89.8%

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

   2. associate-*l* 89.8%

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

8. Simplified 89.8%

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

9. Taylor expanded in u0 around 0 89.8%

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

    1. +-commutative 89.8%

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

    2. associate-*r* 89.8%

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

    3. *-commutative 89.8%

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

    4. associate-*r* 89.8%

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

    5. distribute-rgt-in 89.8%

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

    6. associate-*r* 89.8%

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

    7. *-commutative 89.8%

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

    8. +-commutative 89.8%

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

    9. *-commutative 89.8%

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

    10. +-commutative 89.8%

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

    11. associate-*r* 89.8%

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

    12. *-commutative 89.8%

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

11. Simplified 89.8%

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

Alternative 5: 87.0% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.7%

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

   1. associate-*l* 54.6%

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

   2. distribute-lft-neg-out 54.6%

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

   3. distribute-rgt-neg-in 54.6%

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

   4. distribute-lft-neg-out 54.6%

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

   5. sub-neg 54.6%

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

   6. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u0 around 0 86.5%

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

Alternative 6: 86.9% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot \left(u0 \cdot \left(\alpha \cdot \left(1 + u0 \cdot 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* alpha (* u0 (* alpha (+ 1.0 (* u0 0.5))))))

C

float code(float alpha, float u0) {
	return alpha * (u0 * (alpha * (1.0f + (u0 * 0.5f))));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (u0 * (alpha * (1.0e0 + (u0 * 0.5e0))))
end function

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(u0 * Float32(alpha * Float32(Float32(1.0) + Float32(u0 * Float32(0.5))))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (u0 * (alpha * (single(1.0) + (u0 * single(0.5)))));
end

Derivation

1. Initial program 54.7%

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

   1. associate-*l* 54.6%

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

   2. distribute-lft-neg-out 54.6%

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

   3. distribute-rgt-neg-in 54.6%

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

   4. distribute-lft-neg-out 54.6%

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

   5. sub-neg 54.6%

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

   6. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u0 around 0 91.9%

   \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + u0 \cdot \left(0.25 \cdot \left(\alpha \cdot u0\right) + 0.3333333333333333 \cdot \alpha\right)\right)\right)\right)} \]

6. Taylor expanded in u0 around 0 89.8%

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

   1. *-commutative 89.8%

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

   2. associate-*l* 89.8%

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

8. Simplified 89.8%

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

9. Taylor expanded in u0 around 0 86.5%

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

    1. associate-*r* 86.5%

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

    2. *-commutative 86.5%

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

    3. *-commutative 86.5%

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

11. Simplified 86.5%

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

12. Taylor expanded in alpha around 0 86.2%

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

    1. *-commutative 86.2%

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

14. Simplified 86.2%

    \[\leadsto \alpha \cdot \left(u0 \cdot \color{blue}{\left(\alpha \cdot \left(1 + u0 \cdot 0.5\right)\right)}\right) \]
15. Add Preprocessing

Alternative 7: 86.9% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(1 + u0 \cdot 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* alpha (* alpha (* u0 (+ 1.0 (* u0 0.5))))))

C

float code(float alpha, float u0) {
	return alpha * (alpha * (u0 * (1.0f + (u0 * 0.5f))));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (alpha * (u0 * (1.0e0 + (u0 * 0.5e0))))
end function

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(alpha * Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(0.5))))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * (u0 * (single(1.0) + (u0 * single(0.5)))));
end

Derivation

1. Initial program 54.7%

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

   1. associate-*l* 54.6%

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

   2. distribute-lft-neg-out 54.6%

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

   3. distribute-rgt-neg-in 54.6%

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

   4. distribute-lft-neg-out 54.6%

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

   5. sub-neg 54.6%

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

   6. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u0 around 0 91.9%

   \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + u0 \cdot \left(0.25 \cdot \left(\alpha \cdot u0\right) + 0.3333333333333333 \cdot \alpha\right)\right)\right)\right)} \]

6. Taylor expanded in u0 around 0 89.8%

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

   1. *-commutative 89.8%

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

   2. associate-*l* 89.8%

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

8. Simplified 89.8%

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

9. Taylor expanded in u0 around 0 86.5%

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

    1. associate-*r* 86.5%

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

    2. *-commutative 86.5%

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

    3. *-commutative 86.5%

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

11. Simplified 86.5%

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

12. Taylor expanded in alpha around 0 86.0%

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

13. Final simplification 86.0%

    \[\leadsto \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(1 + u0 \cdot 0.5\right)\right)\right) \]
14. Add Preprocessing

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 54.7%

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

   1. associate-*l* 54.6%

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

   2. distribute-lft-neg-out 54.6%

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

   3. distribute-rgt-neg-in 54.6%

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

   4. distribute-lft-neg-out 54.6%

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

   5. sub-neg 54.6%

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

   6. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u0 around 0 75.1%

   \[\leadsto \alpha \cdot \color{blue}{\left(\alpha \cdot u0\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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))))