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

Percentage Accurate: 60.8% → 98.5%

Time: 18.4s

Alternatives: 9

Speedup: 8.9×

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: 60.8% 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: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.8%

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

   1. sub-neg 58.8%

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

   2. log1p-def 97.9%

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

3. Simplified 97.9%

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

   1. clear-num 97.9%

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

   2. associate-/r/ 97.9%

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

   3. pow2 97.9%

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

   4. pow-flip 98.2%

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

   5. metadata-eval 98.2%

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

6. Applied egg-rr 98.2%

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

   1. *-commutative 98.2%

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

   2. add-sqr-sqrt 97.6%

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

   3. pow2 97.6%

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

   4. sqr-pow 97.3%

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

   5. pow2 97.3%

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

   6. unpow-prod-down 97.3%

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

   7. metadata-eval 97.3%

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

   8. inv-pow 97.3%

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

   9. div-inv 97.4%

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

   10. div-inv 97.3%

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

   11. div-inv 97.3%

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

   12. fma-def 97.3%

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

   13. pow2 97.3%

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

   14. pow-flip 97.4%

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

   15. metadata-eval 97.4%

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

8. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot \frac{1}{\mathsf{fma}\left(cos2phi, {alphax}^{-2}, {alphay}^{-2} \cdot sin2phi\right)}} \]
9. Step-by-step derivation

   1. fma-udef 98.0%

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

   2. *-rgt-identity 98.0%

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

   3. metadata-eval 98.0%

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

   4. pow-sqr 97.9%

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

   5. inv-pow 97.9%

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

   6. inv-pow 97.9%

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

   7. frac-times 98.0%

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

   8. metadata-eval 98.0%

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

   9. div-inv 98.0%

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

   10. frac-times 98.0%

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

   11. *-commutative 98.0%

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

   12. metadata-eval 98.0%

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

   13. pow-flip 97.8%

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

   14. div-inv 97.8%

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

   15. pow2 97.8%

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

   16. +-commutative 97.8%

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

   17. associate-/r* 97.8%

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

   18. un-div-inv 97.9%

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

   19. frac-add 97.7%

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

10. Applied egg-rr 97.7%

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

    1. expm1-log1p-u 96.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot \frac{1}{\frac{\frac{sin2phi}{alphay} \cdot alphax + alphay \cdot \frac{cos2phi}{alphax}}{alphay \cdot alphax}}\right)\right)} \]

    2. expm1-udef 50.5%

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

    3. un-div-inv 50.6%

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

    4. +-commutative 50.6%

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

    5. fma-def 50.6%

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

    6. *-commutative 50.6%

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

12. Applied egg-rr 50.6%

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

    1. expm1-def 96.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(alphay, \frac{cos2phi}{alphax}, alphax \cdot \frac{sin2phi}{alphay}\right)}{alphay \cdot alphax}}\right)\right)} \]

    2. expm1-log1p 97.8%

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

    3. associate-/r/ 98.5%

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

    4. *-commutative 98.5%

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

14. Simplified 98.5%

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

15. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.8%

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

   1. sub-neg 58.8%

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

   2. log1p-def 97.9%

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

3. Simplified 97.9%

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

   1. clear-num 97.9%

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

   2. associate-/r/ 97.9%

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

   3. pow2 97.9%

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

   4. pow-flip 98.2%

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

   5. metadata-eval 98.2%

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

6. Applied egg-rr 98.2%

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

7. Final simplification 98.2%

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

Alternative 3: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9959999918937683:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(-\log \left(1 - u0\right)\right)}{sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphay\right)}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9959999918937683)
   (/ (* (* alphay alphay) (- (log (- 1.0 u0)))) sin2phi)
   (/
    (* alphax (* u0 alphay))
    (+ (/ (* alphax sin2phi) alphay) (/ (* alphay cos2phi) alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((1.0f - u0) <= 0.9959999918937683f) {
		tmp = ((alphay * alphay) * -logf((1.0f - u0))) / sin2phi;
	} else {
		tmp = (alphax * (u0 * alphay)) / (((alphax * sin2phi) / alphay) + ((alphay * cos2phi) / alphax));
	}
	return tmp;
}

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
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9959999918937683e0) then
        tmp = ((alphay * alphay) * -log((1.0e0 - u0))) / sin2phi
    else
        tmp = (alphax * (u0 * alphay)) / (((alphax * sin2phi) / alphay) + ((alphay * cos2phi) / alphax))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9959999918937683))
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(-log(Float32(Float32(1.0) - u0)))) / sin2phi);
	else
		tmp = Float32(Float32(alphax * Float32(u0 * alphay)) / Float32(Float32(Float32(alphax * sin2phi) / alphay) + Float32(Float32(alphay * cos2phi) / alphax)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9959999918937683))
		tmp = ((alphay * alphay) * -log((single(1.0) - u0))) / sin2phi;
	else
		tmp = (alphax * (u0 * alphay)) / (((alphax * sin2phi) / alphay) + ((alphay * cos2phi) / alphax));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u0) < 0.995999992

   1. Initial program 92.5%

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

   3. Taylor expanded in cos2phi around 0 67.9%

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

      1. pow2 31.7%

         \[\leadsto \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \cdot u0 \]

   5. Applied egg-rr 67.9%

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

   if 0.995999992 < (-.f32 1 u0)

   1. Initial program 50.9%

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

      1. sub-neg 50.9%

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

      2. log1p-def 98.4%

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

   3. Simplified 98.4%

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

      1. clear-num 98.4%

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

      2. associate-/r/ 98.5%

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

      3. pow2 98.5%

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

      4. pow-flip 98.7%

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

      5. metadata-eval 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. add-sqr-sqrt 98.1%

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

      3. pow2 98.1%

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

      4. sqr-pow 97.8%

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

      5. pow2 97.8%

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

      6. unpow-prod-down 97.8%

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

      7. metadata-eval 97.8%

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

      8. inv-pow 97.8%

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

      9. div-inv 98.0%

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

      10. div-inv 97.8%

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

      11. div-inv 97.8%

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

      12. fma-def 97.8%

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

      13. pow2 97.8%

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

      14. pow-flip 97.9%

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

      15. metadata-eval 97.9%

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

   8. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot \frac{1}{\mathsf{fma}\left(cos2phi, {alphax}^{-2}, {alphay}^{-2} \cdot sin2phi\right)}} \]
   9. Step-by-step derivation

      1. fma-udef 98.5%

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

      2. *-rgt-identity 98.5%

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

      3. metadata-eval 98.5%

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

      4. pow-sqr 98.4%

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

      5. inv-pow 98.4%

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

      6. inv-pow 98.4%

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

      7. frac-times 98.5%

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

      8. metadata-eval 98.5%

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

      9. div-inv 98.5%

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

      10. frac-times 98.5%

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

      11. *-commutative 98.5%

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

      12. metadata-eval 98.5%

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

      13. pow-flip 98.3%

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

      14. div-inv 98.3%

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

      15. pow2 98.3%

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

      16. +-commutative 98.3%

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

      17. associate-/r* 98.3%

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

      18. un-div-inv 98.4%

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

      19. frac-add 98.3%

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

   10. Applied egg-rr 98.3%

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

   11. Taylor expanded in u0 around 0 86.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9959999918937683:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(-\log \left(1 - u0\right)\right)}{sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphay\right)}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

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(Float32(-log1p(Float32(-u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 58.8%

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

   1. sub-neg 58.8%

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

   2. log1p-def 97.9%

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

3. Simplified 97.9%

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

5. Final simplification 97.9%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

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(Float32(-log1p(Float32(-u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)))
end

Derivation

1. Initial program 58.8%

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

   1. sub-neg 58.8%

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

   2. log1p-def 97.9%

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

3. Simplified 97.9%

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

   1. clear-num 97.9%

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

   2. associate-/r/ 97.9%

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

   3. pow2 97.9%

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

   4. pow-flip 98.2%

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

   5. metadata-eval 98.2%

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

6. Applied egg-rr 98.2%

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

   1. *-commutative 77.7%

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

   2. metadata-eval 77.7%

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

   3. pow-flip 77.7%

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

   4. pow2 77.7%

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

   5. div-inv 77.6%

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

   6. associate-/r* 77.7%

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

8. Applied egg-rr 98.0%

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

9. Final simplification 98.0%

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

Alternative 6: 76.2% accurate, 6.8× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (alphax * (u0 * alphay)) / (((alphax * sin2phi) / alphay) + ((alphay * cos2phi) / 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 = (alphax * (u0 * alphay)) / (((alphax * sin2phi) / alphay) + ((alphay * cos2phi) / alphax))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.8%

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

   1. sub-neg 58.8%

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

   2. log1p-def 97.9%

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

3. Simplified 97.9%

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

   1. clear-num 97.9%

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

   2. associate-/r/ 97.9%

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

   3. pow2 97.9%

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

   4. pow-flip 98.2%

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

   5. metadata-eval 98.2%

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

6. Applied egg-rr 98.2%

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

   1. *-commutative 98.2%

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

   2. add-sqr-sqrt 97.6%

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

   3. pow2 97.6%

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

   4. sqr-pow 97.3%

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

   5. pow2 97.3%

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

   6. unpow-prod-down 97.3%

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

   7. metadata-eval 97.3%

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

   8. inv-pow 97.3%

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

   9. div-inv 97.4%

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

   10. div-inv 97.3%

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

   11. div-inv 97.3%

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

   12. fma-def 97.3%

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

   13. pow2 97.3%

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

   14. pow-flip 97.4%

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

   15. metadata-eval 97.4%

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

8. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot \frac{1}{\mathsf{fma}\left(cos2phi, {alphax}^{-2}, {alphay}^{-2} \cdot sin2phi\right)}} \]
9. Step-by-step derivation

   1. fma-udef 98.0%

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

   2. *-rgt-identity 98.0%

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

   3. metadata-eval 98.0%

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

   4. pow-sqr 97.9%

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

   5. inv-pow 97.9%

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

   6. inv-pow 97.9%

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

   7. frac-times 98.0%

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

   8. metadata-eval 98.0%

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

   9. div-inv 98.0%

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

   10. frac-times 98.0%

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

   11. *-commutative 98.0%

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

   12. metadata-eval 98.0%

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

   13. pow-flip 97.8%

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

   14. div-inv 97.8%

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

   15. pow2 97.8%

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

   16. +-commutative 97.8%

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

   17. associate-/r* 97.8%

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

   18. un-div-inv 97.9%

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

   19. frac-add 97.7%

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

10. Applied egg-rr 97.7%

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

11. Taylor expanded in u0 around 0 77.9%

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

12. Final simplification 77.9%

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

Alternative 7: 76.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return 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 = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.8%

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

3. Taylor expanded in u0 around 0 77.6%

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

   1. mul-1-neg 77.6%

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

5. Simplified 77.6%

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

6. Final simplification 77.6%

   \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
7. Add Preprocessing

Alternative 8: 76.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return 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 = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.8%

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

3. Taylor expanded in u0 around 0 77.6%

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

   1. mul-1-neg 77.6%

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

5. Simplified 77.6%

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

   1. clear-num 97.9%

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

   2. associate-/r/ 97.9%

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

   3. pow2 97.9%

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

   4. pow-flip 98.2%

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

   5. metadata-eval 98.2%

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

7. Applied egg-rr 77.7%

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

   1. *-commutative 77.7%

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

   2. metadata-eval 77.7%

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

   3. pow-flip 77.7%

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

   4. pow2 77.7%

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

   5. div-inv 77.6%

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

   6. associate-/r* 77.7%

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

9. Applied egg-rr 77.7%

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

10. Final simplification 77.7%

    \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]
11. Add Preprocessing

Alternative 9: 59.7% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ u0 \cdot \frac{alphay \cdot alphay}{sin2phi} \end{array} \]

FPCore

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

C

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

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 * ((alphay * alphay) / sin2phi)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.8%

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

3. Taylor expanded in u0 around 0 77.6%

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

   1. mul-1-neg 77.6%

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

5. Simplified 77.6%

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

6. Taylor expanded in cos2phi around 0 61.4%

   \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}} \]
7. Step-by-step derivation

   1. associate-/l* 60.5%

      \[\leadsto \color{blue}{\frac{{alphay}^{2}}{\frac{sin2phi}{u0}}} \]

   2. associate-/r/ 61.4%

      \[\leadsto \color{blue}{\frac{{alphay}^{2}}{sin2phi} \cdot u0} \]

8. Simplified 61.4%

   \[\leadsto \color{blue}{\frac{{alphay}^{2}}{sin2phi} \cdot u0} \]
9. Step-by-step derivation

   1. pow2 61.4%

      \[\leadsto \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \cdot u0 \]

10. Applied egg-rr 61.4%

    \[\leadsto \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \cdot u0 \]

11. Final simplification 61.4%

    \[\leadsto u0 \cdot \frac{alphay \cdot alphay}{sin2phi} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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)))))