Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.3% → 100.0%

Time: 12.2s

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: 55.3% 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: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \langle \left( \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{log1p}\left(-u0\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (cast (! :precision binary64 (* (* alpha (- alpha)) (log1p (- u0))))))

C

float code(float alpha, float u0) {
	double tmp = (((double) alpha) * -((double) alpha)) * log1p(-((double) u0));
	return (float) tmp;
}

Julia

function code(alpha, u0)
	tmp = Float64(Float64(Float64(alpha) * Float64(-Float64(alpha))) * log1p(Float64(-Float64(u0))))
	return Float32(tmp)
end

Derivation

1. Initial program 100.0%

   \[\langle \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{log1p}\left(-u0\right) \rangle_{\text{binary64}} \]

2. Final simplification 100.0%

   \[\leadsto \langle \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{log1p}\left(-u0\right) \rangle_{\text{binary64}} \]

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 57.2%

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

   1. *-commutative 57.2%

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

   2. associate-*l* 57.2%

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

   3. sub-neg 57.2%

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

   4. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 3: 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 57.2%

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

   1. *-commutative 57.2%

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

   2. sub-neg 57.2%

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

   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 4: 87.4% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

   1. *-commutative 57.2%

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

   2. associate-*l* 57.2%

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

   3. sub-neg 57.2%

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

   4. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 87.1%

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

   1. +-commutative 87.1%

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

   2. fma-def 87.3%

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

   3. *-commutative 87.3%

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

   4. associate-*r* 87.3%

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

   5. unpow2 87.3%

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

6. Simplified 87.3%

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

   1. fma-udef 87.1%

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

   2. *-commutative 87.1%

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

   3. distribute-lft-out 87.1%

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

   4. associate-*r* 87.1%

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

   5. distribute-rgt-out 87.2%

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

   6. *-commutative 87.2%

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

   7. associate-*r* 87.2%

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

   8. associate-*r* 87.2%

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

   9. *-commutative 87.2%

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

   10. unswap-sqr 87.2%

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

   11. pow2 87.2%

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

8. Applied egg-rr 87.2%

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

   1. unpow2 87.2%

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

   2. swap-sqr 87.2%

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

   3. *-commutative 87.2%

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

10. Applied egg-rr 87.2%

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

11. Final simplification 87.2%

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

Alternative 5: 87.4% 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 57.2%

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

   1. *-commutative 57.2%

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

   2. associate-*l* 57.2%

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

   3. sub-neg 57.2%

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

   4. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 87.1%

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

   1. +-commutative 87.1%

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

   2. fma-def 87.3%

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

   3. *-commutative 87.3%

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

   4. associate-*r* 87.3%

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

   5. unpow2 87.3%

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

6. Simplified 87.3%

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

   1. fma-udef 87.1%

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

   2. +-commutative 87.1%

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

   3. *-commutative 87.1%

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

   4. *-commutative 87.1%

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

   5. associate-*l* 87.1%

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

8. Applied egg-rr 87.1%

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

9. Final simplification 87.1%

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

Alternative 6: 87.5% 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 57.2%

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

   1. *-commutative 57.2%

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

   2. associate-*l* 57.2%

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

   3. sub-neg 57.2%

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

   4. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 87.1%

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

   1. +-commutative 87.1%

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

   2. fma-def 87.3%

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

   3. *-commutative 87.3%

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

   4. associate-*r* 87.3%

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

   5. unpow2 87.3%

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

6. Simplified 87.3%

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

   1. fma-udef 87.1%

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

   2. +-commutative 87.1%

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

   3. *-commutative 87.1%

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

   4. associate-*r* 87.1%

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

   5. associate-*r* 87.1%

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

   6. distribute-rgt-out 87.1%

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

   7. *-commutative 87.1%

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

8. Applied egg-rr 87.1%

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

9. Final simplification 87.1%

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

Alternative 7: 87.5% 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 57.2%

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

   1. *-commutative 57.2%

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

   2. associate-*l* 57.2%

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

   3. sub-neg 57.2%

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

   4. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 87.1%

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

   1. +-commutative 87.1%

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

   2. fma-def 87.3%

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

   3. *-commutative 87.3%

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

   4. associate-*r* 87.3%

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

   5. unpow2 87.3%

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

6. Simplified 87.3%

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

   1. fma-udef 87.1%

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

   2. *-commutative 87.1%

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

   3. distribute-lft-out 87.1%

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

   4. *-commutative 87.1%

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

   5. *-commutative 87.1%

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

   6. associate-*l* 87.1%

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

8. Applied egg-rr 87.1%

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

9. Final simplification 87.1%

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

Alternative 8: 75.0% 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 57.2%

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

   1. *-commutative 57.2%

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

   2. associate-*l* 57.2%

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

   3. sub-neg 57.2%

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

   4. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 73.9%

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

   1. unpow2 73.9%

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

   2. associate-*l* 73.8%

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

6. Simplified 73.8%

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

7. Final simplification 73.8%

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

Alternative 9: 74.9% 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 57.2%

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

   1. *-commutative 57.2%

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

   2. associate-*l* 57.2%

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

   3. sub-neg 57.2%

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

   4. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 73.9%

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

   1. unpow2 73.9%

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

6. Simplified 73.9%

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

7. Final simplification 73.9%

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

Reproduce

herbie shell --seed 2023297 
(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))))