Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5

Percentage Accurate: 60.4% → 98.4%

Time: 18.6s

Alternatives: 8

Speedup: 6.1×

Specification

\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]

Math

\[\begin{array}{l} \\ \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log (- 1.0 u0)))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -logf((1.0f - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = -log((single(1.0) - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Initial Program: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log (- 1.0 u0)))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -logf((1.0f - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = -log((single(1.0) - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Alternative 1: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{sin2phi}{alphay}}{-alphay} - {alphax}^{-2} \cdot cos2phi} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (log1p (- u0))
  (- (/ (/ sin2phi alphay) (- alphay)) (* (pow alphax -2.0) cos2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return log1pf(-u0) / (((sin2phi / alphay) / -alphay) - (powf(alphax, -2.0f) * cos2phi));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(sin2phi / alphay) / Float32(-alphay)) - Float32((alphax ^ Float32(-2.0)) * cos2phi)))
end

Derivation

1. Initial program 62.6%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg 62.6%

      \[\leadsto \color{blue}{-\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   2. distribute-neg-frac2 62.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   3. sub-neg 62.6%

      \[\leadsto \frac{\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   4. log1p-define 98.3%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-u0\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   5. neg-sub0 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

   8. associate-/r* 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   9. distribute-neg-frac2 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

   2. div-inv 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]
7. Step-by-step derivation

   1. associate-*r/ 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay} \cdot 1}{alphay}}} \]

   2. *-rgt-identity 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{\color{blue}{\frac{sin2phi}{alphay}}}{alphay}} \]

8. Simplified 98.5%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
9. Step-by-step derivation

   1. distribute-frac-neg2 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{\frac{cos2phi}{alphax}}{alphax}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   2. associate-/r* 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   3. neg-sub0 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   4. div-inv 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{cos2phi \cdot \frac{1}{alphax \cdot alphax}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   5. pow2 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \frac{1}{\color{blue}{{alphax}^{2}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   6. pow-flip 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \color{blue}{{alphax}^{\left(-2\right)}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   7. metadata-eval 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot {alphax}^{\color{blue}{-2}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

10. Applied egg-rr 98.5%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - cos2phi \cdot {alphax}^{-2}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
11. Step-by-step derivation

    1. neg-sub0 98.5%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi \cdot {alphax}^{-2}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    2. distribute-lft-neg-in 98.5%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi\right) \cdot {alphax}^{-2}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    3. *-commutative 98.5%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{{alphax}^{-2} \cdot \left(-cos2phi\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

12. Simplified 98.5%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{{alphax}^{-2} \cdot \left(-cos2phi\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

13. Final simplification 98.5%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{sin2phi}{alphay}}{-alphay} - {alphax}^{-2} \cdot cos2phi} \]
14. Add Preprocessing

Alternative 2: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1:\\ \;\;\;\;\frac{u0 \cdot \left(1 - u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.0)
   (/
    (* u0 (- 1.0 (* u0 -0.5)))
    (+ (/ (/ sin2phi alphay) alphay) (/ cos2phi (* alphax alphax))))
   (/
    (log1p (- u0))
    (- (/ (/ cos2phi alphax) alphax) (/ sin2phi (* alphay alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.0f) {
		tmp = (u0 * (1.0f - (u0 * -0.5f))) / (((sin2phi / alphay) / alphay) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = log1pf(-u0) / (((cos2phi / alphax) / alphax) - (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.0))
		tmp = Float32(Float32(u0 * Float32(Float32(1.0) - Float32(u0 * Float32(-0.5)))) / Float32(Float32(Float32(sin2phi / alphay) / alphay) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(cos2phi / alphax) / alphax) - Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1

   1. Initial program 56.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg 56.7%

         \[\leadsto \color{blue}{-\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. distribute-neg-frac2 56.7%

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      3. sub-neg 56.7%

         \[\leadsto \frac{\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

      4. log1p-define 98.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-u0\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

      5. neg-sub0 98.5%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 98.5%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 98.5%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

      8. associate-/r* 98.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

      9. distribute-neg-frac2 98.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 98.9%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

      2. div-inv 98.8%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]

   6. Applied egg-rr 98.8%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 98.9%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay} \cdot 1}{alphay}}} \]

      2. *-rgt-identity 98.9%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{\color{blue}{\frac{sin2phi}{alphay}}}{alphay}} \]

   8. Simplified 98.9%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
   9. Step-by-step derivation

      1. distribute-frac-neg2 98.9%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{\frac{cos2phi}{alphax}}{alphax}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      2. associate-/r* 98.8%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      3. neg-sub0 98.8%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      4. div-inv 98.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{cos2phi \cdot \frac{1}{alphax \cdot alphax}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      5. pow2 98.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \frac{1}{\color{blue}{{alphax}^{2}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      6. pow-flip 98.9%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \color{blue}{{alphax}^{\left(-2\right)}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      7. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot {alphax}^{\color{blue}{-2}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   10. Applied egg-rr 98.9%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - cos2phi \cdot {alphax}^{-2}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
   11. Step-by-step derivation

       1. neg-sub0 98.9%

          \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi \cdot {alphax}^{-2}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       2. distribute-lft-neg-in 98.9%

          \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi\right) \cdot {alphax}^{-2}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       3. *-commutative 98.9%

          \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{{alphax}^{-2} \cdot \left(-cos2phi\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   12. Simplified 98.9%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{{alphax}^{-2} \cdot \left(-cos2phi\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   13. Taylor expanded in u0 around 0 88.0%

       \[\leadsto \frac{\color{blue}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}}{{alphax}^{-2} \cdot \left(-cos2phi\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
   14. Step-by-step derivation

       1. *-commutative 88.0%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\left(-cos2phi\right) \cdot {alphax}^{-2}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       2. add-sqr-sqrt -0.0%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\left(\sqrt{-cos2phi} \cdot \sqrt{-cos2phi}\right)} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       3. sqrt-unprod 44.2%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\sqrt{\left(-cos2phi\right) \cdot \left(-cos2phi\right)}} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       4. sqr-neg 44.2%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\sqrt{\color{blue}{cos2phi \cdot cos2phi}} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       5. sqrt-unprod 43.0%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\left(\sqrt{cos2phi} \cdot \sqrt{cos2phi}\right)} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       6. add-sqr-sqrt 43.0%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{cos2phi} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       7. metadata-eval 43.0%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{cos2phi \cdot {alphax}^{\color{blue}{\left(-2\right)}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       8. pow-flip 43.0%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{cos2phi \cdot \color{blue}{\frac{1}{{alphax}^{2}}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       9. div-inv 43.0%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       10. unpow2 43.0%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       11. associate-/l/ 43.0%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       12. div-inv 43.0%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       13. frac-2neg 43.0%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{-cos2phi}{-alphax}} \cdot \frac{1}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       14. frac-times 43.0%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{\left(-cos2phi\right) \cdot 1}{\left(-alphax\right) \cdot alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       15. add-sqr-sqrt -0.0%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\color{blue}{\left(\sqrt{-cos2phi} \cdot \sqrt{-cos2phi}\right)} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       16. sqrt-unprod 78.6%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\color{blue}{\sqrt{\left(-cos2phi\right) \cdot \left(-cos2phi\right)}} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       17. sqr-neg 78.6%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\sqrt{\color{blue}{cos2phi \cdot cos2phi}} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       18. sqrt-unprod 87.9%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\color{blue}{\left(\sqrt{cos2phi} \cdot \sqrt{cos2phi}\right)} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       19. add-sqr-sqrt 88.1%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\color{blue}{cos2phi} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   15. Applied egg-rr 88.1%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{cos2phi \cdot 1}{\left(-alphax\right) \cdot alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   if 1 < sin2phi

   1. Initial program 67.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg 67.5%

         \[\leadsto \color{blue}{-\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. distribute-neg-frac2 67.5%

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      3. sub-neg 67.5%

         \[\leadsto \frac{\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

      4. log1p-define 98.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-u0\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

      5. neg-sub0 98.1%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 98.1%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 98.1%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

      8. associate-/r* 98.1%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

      9. distribute-neg-frac2 98.1%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt -0.0%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{\sqrt{-alphax} \cdot \sqrt{-alphax}}} - \frac{sin2phi}{alphay \cdot alphay}} \]

      2. sqrt-unprod 97.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{\sqrt{\left(-alphax\right) \cdot \left(-alphax\right)}}} - \frac{sin2phi}{alphay \cdot alphay}} \]

      3. sqr-neg 97.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\sqrt{\color{blue}{alphax \cdot alphax}}} - \frac{sin2phi}{alphay \cdot alphay}} \]

      4. sqrt-prod 97.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{\sqrt{alphax} \cdot \sqrt{alphax}}} - \frac{sin2phi}{alphay \cdot alphay}} \]

      5. add-sqr-sqrt 97.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

      6. div-inv 97.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied egg-rr 97.7%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]
   7. Step-by-step derivation

      1. associate-*r/ 97.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot 1}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

      2. *-rgt-identity 97.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\frac{cos2phi}{alphax}}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}} \]

   8. Simplified 97.7%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1:\\ \;\;\;\;\frac{u0 \cdot \left(1 - u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{sin2phi}{alphay}}{alphay}\\ \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0 \cdot \left(1 - u0 \cdot -0.5\right)}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ (/ sin2phi alphay) alphay)))
   (if (<= sin2phi 0.009999999776482582)
     (/ (* u0 (- 1.0 (* u0 -0.5))) (+ t_0 (/ cos2phi (* alphax alphax))))
     (/ (log1p (- u0)) (- (/ (/ cos2phi alphax) alphax) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (sin2phi / alphay) / alphay;
	float tmp;
	if (sin2phi <= 0.009999999776482582f) {
		tmp = (u0 * (1.0f - (u0 * -0.5f))) / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = log1pf(-u0) / (((cos2phi / alphax) / alphax) - t_0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(sin2phi / alphay) / alphay)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.009999999776482582))
		tmp = Float32(Float32(u0 * Float32(Float32(1.0) - Float32(u0 * Float32(-0.5)))) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(cos2phi / alphax) / alphax) - t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00999999978

   1. Initial program 56.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg 56.5%

         \[\leadsto \color{blue}{-\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. distribute-neg-frac2 56.5%

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      3. sub-neg 56.5%

         \[\leadsto \frac{\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

      4. log1p-define 98.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-u0\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

      5. neg-sub0 98.6%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 98.6%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 98.6%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

      8. associate-/r* 98.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

      9. distribute-neg-frac2 98.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 98.9%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

      2. div-inv 98.8%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]

   6. Applied egg-rr 98.8%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 98.9%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay} \cdot 1}{alphay}}} \]

      2. *-rgt-identity 98.9%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{\color{blue}{\frac{sin2phi}{alphay}}}{alphay}} \]

   8. Simplified 98.9%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
   9. Step-by-step derivation

      1. distribute-frac-neg2 98.9%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{\frac{cos2phi}{alphax}}{alphax}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      2. associate-/r* 98.8%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      3. neg-sub0 98.8%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      4. div-inv 98.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{cos2phi \cdot \frac{1}{alphax \cdot alphax}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      5. pow2 98.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \frac{1}{\color{blue}{{alphax}^{2}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      6. pow-flip 98.9%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \color{blue}{{alphax}^{\left(-2\right)}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

      7. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot {alphax}^{\color{blue}{-2}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   10. Applied egg-rr 98.9%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - cos2phi \cdot {alphax}^{-2}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
   11. Step-by-step derivation

       1. neg-sub0 98.9%

          \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi \cdot {alphax}^{-2}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       2. distribute-lft-neg-in 98.9%

          \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi\right) \cdot {alphax}^{-2}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       3. *-commutative 98.9%

          \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{{alphax}^{-2} \cdot \left(-cos2phi\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   12. Simplified 98.9%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{{alphax}^{-2} \cdot \left(-cos2phi\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   13. Taylor expanded in u0 around 0 87.6%

       \[\leadsto \frac{\color{blue}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}}{{alphax}^{-2} \cdot \left(-cos2phi\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
   14. Step-by-step derivation

       1. *-commutative 87.6%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\left(-cos2phi\right) \cdot {alphax}^{-2}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       2. add-sqr-sqrt -0.0%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\left(\sqrt{-cos2phi} \cdot \sqrt{-cos2phi}\right)} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       3. sqrt-unprod 41.8%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\sqrt{\left(-cos2phi\right) \cdot \left(-cos2phi\right)}} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       4. sqr-neg 41.8%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\sqrt{\color{blue}{cos2phi \cdot cos2phi}} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       5. sqrt-unprod 40.5%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\left(\sqrt{cos2phi} \cdot \sqrt{cos2phi}\right)} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       6. add-sqr-sqrt 40.5%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{cos2phi} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       7. metadata-eval 40.5%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{cos2phi \cdot {alphax}^{\color{blue}{\left(-2\right)}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       8. pow-flip 40.5%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{cos2phi \cdot \color{blue}{\frac{1}{{alphax}^{2}}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       9. div-inv 40.5%

          \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       10. unpow2 40.5%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       11. associate-/l/ 40.5%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       12. div-inv 40.5%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       13. frac-2neg 40.5%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{-cos2phi}{-alphax}} \cdot \frac{1}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       14. frac-times 40.5%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{\left(-cos2phi\right) \cdot 1}{\left(-alphax\right) \cdot alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       15. add-sqr-sqrt -0.0%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\color{blue}{\left(\sqrt{-cos2phi} \cdot \sqrt{-cos2phi}\right)} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       16. sqrt-unprod 77.8%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\color{blue}{\sqrt{\left(-cos2phi\right) \cdot \left(-cos2phi\right)}} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       17. sqr-neg 77.8%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\sqrt{\color{blue}{cos2phi \cdot cos2phi}} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       18. sqrt-unprod 87.4%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\color{blue}{\left(\sqrt{cos2phi} \cdot \sqrt{cos2phi}\right)} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

       19. add-sqr-sqrt 87.6%

           \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\color{blue}{cos2phi} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   15. Applied egg-rr 87.6%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{cos2phi \cdot 1}{\left(-alphax\right) \cdot alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   if 0.00999999978 < sin2phi

   1. Initial program 67.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg 67.3%

         \[\leadsto \color{blue}{-\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      2. distribute-neg-frac2 67.3%

         \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      3. sub-neg 67.3%

         \[\leadsto \frac{\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

      4. log1p-define 98.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-u0\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

      5. neg-sub0 98.1%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 98.1%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 98.1%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

      8. associate-/r* 98.1%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

      9. distribute-neg-frac2 98.1%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 98.1%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

      2. div-inv 97.9%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]

   6. Applied egg-rr 97.9%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 98.1%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay} \cdot 1}{alphay}}} \]

      2. *-rgt-identity 98.1%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{\color{blue}{\frac{sin2phi}{alphay}}}{alphay}} \]

   8. Simplified 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt -0.0%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{\sqrt{-alphax} \cdot \sqrt{-alphax}}} - \frac{sin2phi}{alphay \cdot alphay}} \]

      2. sqrt-unprod 97.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{\sqrt{\left(-alphax\right) \cdot \left(-alphax\right)}}} - \frac{sin2phi}{alphay \cdot alphay}} \]

      3. sqr-neg 97.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\sqrt{\color{blue}{alphax \cdot alphax}}} - \frac{sin2phi}{alphay \cdot alphay}} \]

      4. sqrt-prod 97.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{\sqrt{alphax} \cdot \sqrt{alphax}}} - \frac{sin2phi}{alphay \cdot alphay}} \]

      5. add-sqr-sqrt 97.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

      6. div-inv 97.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

   10. Applied egg-rr 97.8%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
   11. Step-by-step derivation

       1. associate-*r/ 97.7%

          \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot 1}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

       2. *-rgt-identity 97.7%

          \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\frac{cos2phi}{alphax}}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}} \]

   12. Simplified 97.8%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0 \cdot \left(1 - u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (log1p (- u0))
  (- (/ (/ cos2phi alphax) (- alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return log1pf(-u0) / (((cos2phi / alphax) / -alphax) - (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(cos2phi / alphax) / Float32(-alphax)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 62.6%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg 62.6%

      \[\leadsto \color{blue}{-\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   2. distribute-neg-frac2 62.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   3. sub-neg 62.6%

      \[\leadsto \frac{\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   4. log1p-define 98.3%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-u0\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   5. neg-sub0 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

   8. associate-/r* 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   9. distribute-neg-frac2 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing

5. Final simplification 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}} \]
6. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (log1p (- u0))
  (- (/ (/ cos2phi alphax) (- alphax)) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return log1pf(-u0) / (((cos2phi / alphax) / -alphax) - ((sin2phi / alphay) / alphay));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(cos2phi / alphax) / Float32(-alphax)) - Float32(Float32(sin2phi / alphay) / alphay)))
end

Derivation

1. Initial program 62.6%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg 62.6%

      \[\leadsto \color{blue}{-\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   2. distribute-neg-frac2 62.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   3. sub-neg 62.6%

      \[\leadsto \frac{\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   4. log1p-define 98.3%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-u0\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   5. neg-sub0 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

   8. associate-/r* 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   9. distribute-neg-frac2 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

   2. div-inv 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]
7. Step-by-step derivation

   1. associate-*r/ 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay} \cdot 1}{alphay}}} \]

   2. *-rgt-identity 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{\color{blue}{\frac{sin2phi}{alphay}}}{alphay}} \]

8. Simplified 98.5%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

9. Final simplification 98.5%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
10. Add Preprocessing

Alternative 6: 63.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(u0 \cdot -0.5 + -1\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ (* u0 -0.5) -1.0))
  (- (/ (/ cos2phi alphax) alphax) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * ((u0 * -0.5f) + -1.0f)) / (((cos2phi / alphax) / alphax) - ((sin2phi / alphay) / alphay));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (u0 * ((u0 * (-0.5e0)) + (-1.0e0))) / (((cos2phi / alphax) / alphax) - ((sin2phi / alphay) / alphay))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(u0 * Float32(-0.5)) + Float32(-1.0))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) - Float32(Float32(sin2phi / alphay) / alphay)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * ((u0 * single(-0.5)) + single(-1.0))) / (((cos2phi / alphax) / alphax) - ((sin2phi / alphay) / alphay));
end

Derivation

1. Initial program 62.6%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg 62.6%

      \[\leadsto \color{blue}{-\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   2. distribute-neg-frac2 62.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   3. sub-neg 62.6%

      \[\leadsto \frac{\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   4. log1p-define 98.3%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-u0\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   5. neg-sub0 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

   8. associate-/r* 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   9. distribute-neg-frac2 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

   2. div-inv 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]
7. Step-by-step derivation

   1. associate-*r/ 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay} \cdot 1}{alphay}}} \]

   2. *-rgt-identity 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{\color{blue}{\frac{sin2phi}{alphay}}}{alphay}} \]

8. Simplified 98.5%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
9. Step-by-step derivation

   1. distribute-frac-neg2 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{\frac{cos2phi}{alphax}}{alphax}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   2. associate-/r* 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   3. neg-sub0 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   4. div-inv 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{cos2phi \cdot \frac{1}{alphax \cdot alphax}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   5. pow2 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \frac{1}{\color{blue}{{alphax}^{2}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   6. pow-flip 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \color{blue}{{alphax}^{\left(-2\right)}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   7. metadata-eval 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot {alphax}^{\color{blue}{-2}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

10. Applied egg-rr 98.5%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - cos2phi \cdot {alphax}^{-2}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
11. Step-by-step derivation

    1. neg-sub0 98.5%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi \cdot {alphax}^{-2}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    2. distribute-lft-neg-in 98.5%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi\right) \cdot {alphax}^{-2}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    3. *-commutative 98.5%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{{alphax}^{-2} \cdot \left(-cos2phi\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

12. Simplified 98.5%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{{alphax}^{-2} \cdot \left(-cos2phi\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

13. Taylor expanded in u0 around 0 88.3%

    \[\leadsto \frac{\color{blue}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}}{{alphax}^{-2} \cdot \left(-cos2phi\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
14. Step-by-step derivation

    1. *-commutative 88.3%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\left(-cos2phi\right) \cdot {alphax}^{-2}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    2. add-sqr-sqrt -0.0%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\left(\sqrt{-cos2phi} \cdot \sqrt{-cos2phi}\right)} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    3. sqrt-unprod 68.3%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\sqrt{\left(-cos2phi\right) \cdot \left(-cos2phi\right)}} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    4. sqr-neg 68.3%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\sqrt{\color{blue}{cos2phi \cdot cos2phi}} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    5. sqrt-unprod 67.8%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\left(\sqrt{cos2phi} \cdot \sqrt{cos2phi}\right)} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    6. add-sqr-sqrt 67.8%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{cos2phi} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    7. metadata-eval 67.8%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{cos2phi \cdot {alphax}^{\color{blue}{\left(-2\right)}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    8. pow-flip 67.8%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{cos2phi \cdot \color{blue}{\frac{1}{{alphax}^{2}}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    9. div-inv 67.8%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    10. unpow2 67.8%

        \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    11. associate-/l/ 67.8%

        \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

15. Applied egg-rr 67.8%

    \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

16. Final simplification 67.8%

    \[\leadsto \frac{u0 \cdot \left(u0 \cdot -0.5 + -1\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
17. Add Preprocessing

Alternative 7: 87.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 - u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (- 1.0 (* u0 -0.5)))
  (+ (/ (/ sin2phi alphay) alphay) (/ cos2phi (* alphax alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f - (u0 * -0.5f))) / (((sin2phi / alphay) / alphay) + (cos2phi / (alphax * alphax)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (u0 * (1.0e0 - (u0 * (-0.5e0)))) / (((sin2phi / alphay) / alphay) + (cos2phi / (alphax * alphax)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(1.0) - Float32(u0 * Float32(-0.5)))) / Float32(Float32(Float32(sin2phi / alphay) / alphay) + Float32(cos2phi / Float32(alphax * alphax))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) - (u0 * single(-0.5)))) / (((sin2phi / alphay) / alphay) + (cos2phi / (alphax * alphax)));
end

Derivation

1. Initial program 62.6%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg 62.6%

      \[\leadsto \color{blue}{-\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   2. distribute-neg-frac2 62.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   3. sub-neg 62.6%

      \[\leadsto \frac{\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   4. log1p-define 98.3%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-u0\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   5. neg-sub0 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

   8. associate-/r* 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   9. distribute-neg-frac2 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

   2. div-inv 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]
7. Step-by-step derivation

   1. associate-*r/ 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay} \cdot 1}{alphay}}} \]

   2. *-rgt-identity 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{\color{blue}{\frac{sin2phi}{alphay}}}{alphay}} \]

8. Simplified 98.5%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
9. Step-by-step derivation

   1. distribute-frac-neg2 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{\frac{cos2phi}{alphax}}{alphax}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   2. associate-/r* 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   3. neg-sub0 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   4. div-inv 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{cos2phi \cdot \frac{1}{alphax \cdot alphax}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   5. pow2 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \frac{1}{\color{blue}{{alphax}^{2}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   6. pow-flip 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \color{blue}{{alphax}^{\left(-2\right)}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   7. metadata-eval 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot {alphax}^{\color{blue}{-2}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

10. Applied egg-rr 98.5%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - cos2phi \cdot {alphax}^{-2}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
11. Step-by-step derivation

    1. neg-sub0 98.5%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi \cdot {alphax}^{-2}\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    2. distribute-lft-neg-in 98.5%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi\right) \cdot {alphax}^{-2}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    3. *-commutative 98.5%

       \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{{alphax}^{-2} \cdot \left(-cos2phi\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

12. Simplified 98.5%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{{alphax}^{-2} \cdot \left(-cos2phi\right)} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

13. Taylor expanded in u0 around 0 88.3%

    \[\leadsto \frac{\color{blue}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}}{{alphax}^{-2} \cdot \left(-cos2phi\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
14. Step-by-step derivation

    1. *-commutative 88.3%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\left(-cos2phi\right) \cdot {alphax}^{-2}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    2. add-sqr-sqrt -0.0%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\left(\sqrt{-cos2phi} \cdot \sqrt{-cos2phi}\right)} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    3. sqrt-unprod 68.3%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\sqrt{\left(-cos2phi\right) \cdot \left(-cos2phi\right)}} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    4. sqr-neg 68.3%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\sqrt{\color{blue}{cos2phi \cdot cos2phi}} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    5. sqrt-unprod 67.8%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\left(\sqrt{cos2phi} \cdot \sqrt{cos2phi}\right)} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    6. add-sqr-sqrt 67.8%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{cos2phi} \cdot {alphax}^{-2} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    7. metadata-eval 67.8%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{cos2phi \cdot {alphax}^{\color{blue}{\left(-2\right)}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    8. pow-flip 67.8%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{cos2phi \cdot \color{blue}{\frac{1}{{alphax}^{2}}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    9. div-inv 67.8%

       \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    10. unpow2 67.8%

        \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    11. associate-/l/ 67.8%

        \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    12. div-inv 67.8%

        \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    13. frac-2neg 67.8%

        \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{-cos2phi}{-alphax}} \cdot \frac{1}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    14. frac-times 67.8%

        \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{\left(-cos2phi\right) \cdot 1}{\left(-alphax\right) \cdot alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    15. add-sqr-sqrt -0.0%

        \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\color{blue}{\left(\sqrt{-cos2phi} \cdot \sqrt{-cos2phi}\right)} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    16. sqrt-unprod 84.0%

        \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\color{blue}{\sqrt{\left(-cos2phi\right) \cdot \left(-cos2phi\right)}} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    17. sqr-neg 84.0%

        \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\sqrt{\color{blue}{cos2phi \cdot cos2phi}} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    18. sqrt-unprod 88.2%

        \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\color{blue}{\left(\sqrt{cos2phi} \cdot \sqrt{cos2phi}\right)} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

    19. add-sqr-sqrt 88.3%

        \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\frac{\color{blue}{cos2phi} \cdot 1}{\left(-alphax\right) \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

15. Applied egg-rr 88.3%

    \[\leadsto \frac{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}{\color{blue}{\frac{cos2phi \cdot 1}{\left(-alphax\right) \cdot alphax}} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

16. Final simplification 88.3%

    \[\leadsto \frac{u0 \cdot \left(1 - u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
17. Add Preprocessing

Alternative 8: 56.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} - \frac{\frac{cos2phi}{alphax}}{alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/ u0 (- (/ (/ sin2phi alphay) alphay) (/ (/ cos2phi alphax) alphax))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 / (((sin2phi / alphay) / alphay) - ((cos2phi / alphax) / alphax));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 / (((sin2phi / alphay) / alphay) - ((cos2phi / alphax) / alphax))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(u0 / Float32(Float32(Float32(sin2phi / alphay) / alphay) - Float32(Float32(cos2phi / alphax) / alphax)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = u0 / (((sin2phi / alphay) / alphay) - ((cos2phi / alphax) / alphax));
end

Derivation

1. Initial program 62.6%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg 62.6%

      \[\leadsto \color{blue}{-\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   2. distribute-neg-frac2 62.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 - u0\right)}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   3. sub-neg 62.6%

      \[\leadsto \frac{\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   4. log1p-define 98.3%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-u0\right)}}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   5. neg-sub0 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-\frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

   8. associate-/r* 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   9. distribute-neg-frac2 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt -0.0%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{\sqrt{-alphax} \cdot \sqrt{-alphax}}} - \frac{sin2phi}{alphay \cdot alphay}} \]

   2. sqrt-unprod 74.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{\sqrt{\left(-alphax\right) \cdot \left(-alphax\right)}}} - \frac{sin2phi}{alphay \cdot alphay}} \]

   3. sqr-neg 74.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\sqrt{\color{blue}{alphax \cdot alphax}}} - \frac{sin2phi}{alphay \cdot alphay}} \]

   4. sqrt-prod 74.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{\sqrt{alphax} \cdot \sqrt{alphax}}} - \frac{sin2phi}{alphay \cdot alphay}} \]

   5. add-sqr-sqrt 74.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

   6. div-inv 74.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

6. Applied egg-rr 74.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. associate-*r/ 74.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot 1}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

   2. *-rgt-identity 74.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\color{blue}{\frac{cos2phi}{alphax}}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}} \]

8. Simplified 74.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]
9. Step-by-step derivation

   1. associate-/r* 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

   2. div-inv 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]

10. Applied egg-rr 74.2%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]

11. Taylor expanded in u0 around 0 58.0%

    \[\leadsto \frac{\color{blue}{-1 \cdot u0}}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}} \]
12. Step-by-step derivation

    1. mul-1-neg 58.0%

       \[\leadsto \frac{\color{blue}{-u0}}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}} \]

13. Simplified 58.0%

    \[\leadsto \frac{\color{blue}{-u0}}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}} \]
14. Step-by-step derivation

    1. div-inv 58.1%

       \[\leadsto \frac{-u0}{\frac{\frac{cos2phi}{alphax}}{alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

15. Applied egg-rr 58.1%

    \[\leadsto \frac{-u0}{\frac{\frac{cos2phi}{alphax}}{alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

16. Final simplification 58.1%

    \[\leadsto \frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} - \frac{\frac{cos2phi}{alphax}}{alphax}} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (alphax alphay u0 cos2phi sin2phi)
  :name "Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5"
  :precision binary32
  :pre (and (and (and (and (and (<= 0.0001 alphax) (<= alphax 1.0)) (and (<= 0.0001 alphay) (<= alphay 1.0))) (and (<= 2.328306437e-10 u0) (<= u0 1.0))) (and (<= 0.0 cos2phi) (<= cos2phi 1.0))) (<= 0.0 sin2phi))
  (/ (- (log (- 1.0 u0))) (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))