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

Percentage Accurate: 60.8% → 98.5%

Time: 13.9s

Alternatives: 14

Speedup: 12.7×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

   1. neg-sub0 57.7%

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

   2. div-sub 57.7%

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

   3. --rgt-identity 57.7%

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

   4. div-sub 57.7%

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

   5. --rgt-identity 57.7%

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

   6. neg-sub0 57.7%

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

   7. sub-neg 57.7%

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

   8. log1p-def 97.6%

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

3. Simplified 97.6%

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

   1. +-commutative 97.6%

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

   2. associate-/r* 97.5%

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

   3. associate-/r* 97.6%

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

   4. frac-add 97.3%

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

5. Applied egg-rr 97.3%

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

   1. expm1-log1p-u 95.7%

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

   2. expm1-udef 52.3%

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

   3. associate-/r/ 52.3%

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

   4. *-commutative 52.3%

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

   5. fma-def 52.3%

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

   6. *-commutative 52.3%

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

7. Applied egg-rr 52.3%

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

   1. expm1-def 96.9%

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

   2. expm1-log1p 98.4%

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

9. Simplified 98.4%

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

10. Taylor expanded in alphax around 0 98.6%

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

    1. associate-*l/ 98.4%

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

    2. associate-/l* 98.6%

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

    3. associate-/l* 98.7%

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

12. Applied egg-rr 98.7%

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

13. Final simplification 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

   1. neg-sub0 57.7%

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

   2. div-sub 57.7%

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

   3. --rgt-identity 57.7%

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

   4. div-sub 57.7%

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

   5. --rgt-identity 57.7%

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

   6. neg-sub0 57.7%

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

   7. sub-neg 57.7%

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

   8. log1p-def 97.6%

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

3. Simplified 97.6%

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

   1. +-commutative 97.6%

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

   2. associate-/r* 97.5%

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

   3. associate-/r* 97.6%

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

   4. frac-add 97.3%

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

5. Applied egg-rr 97.3%

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

   1. expm1-log1p-u 95.7%

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

   2. expm1-udef 52.3%

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

   3. associate-/r/ 52.3%

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

   4. *-commutative 52.3%

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

   5. fma-def 52.3%

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

   6. *-commutative 52.3%

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

7. Applied egg-rr 52.3%

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

   1. expm1-def 96.9%

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

   2. expm1-log1p 98.4%

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

9. Simplified 98.4%

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

10. Taylor expanded in alphax around 0 98.6%

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

11. Final simplification 98.6%

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

Alternative 3: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.0012000000569969416:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \frac{u0 - \left(u0 \cdot u0\right) \cdot -0.5}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot \left(-alphay\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.0012000000569969416)
   (*
    (* alphax alphay)
    (/
     (- u0 (* (* u0 u0) -0.5))
     (+ (/ (* alphax sin2phi) alphay) (/ (* alphay cos2phi) alphax))))
   (* (/ (log1p (- u0)) sin2phi) (* alphay (- alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.0012000000569969416f) {
		tmp = (alphax * alphay) * ((u0 - ((u0 * u0) * -0.5f)) / (((alphax * sin2phi) / alphay) + ((alphay * cos2phi) / alphax)));
	} else {
		tmp = (log1pf(-u0) / sin2phi) * (alphay * -alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.0012000000569969416))
		tmp = Float32(Float32(alphax * alphay) * Float32(Float32(u0 - Float32(Float32(u0 * u0) * Float32(-0.5))) / Float32(Float32(Float32(alphax * sin2phi) / alphay) + Float32(Float32(alphay * cos2phi) / alphax))));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) / sin2phi) * Float32(alphay * Float32(-alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.00120000006

   1. Initial program 54.9%

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

      1. neg-sub0 54.9%

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

      2. div-sub 54.9%

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

      3. --rgt-identity 54.9%

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

      4. div-sub 54.9%

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

      5. --rgt-identity 54.9%

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

      6. neg-sub0 54.9%

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

      7. sub-neg 54.9%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

      1. +-commutative 98.6%

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

      2. associate-/r* 98.6%

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

      3. associate-/r* 98.8%

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

      4. frac-add 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. expm1-log1p-u 94.7%

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

      2. expm1-udef 84.4%

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

      3. associate-/r/ 84.4%

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

      4. *-commutative 84.4%

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

      5. fma-def 84.4%

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

      6. *-commutative 84.4%

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

   7. Applied egg-rr 84.4%

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

      1. expm1-def 94.9%

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

      2. expm1-log1p 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in alphax around 0 98.3%

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

   11. Taylor expanded in u0 around 0 86.2%

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

       1. neg-mul-1 37.8%

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

       2. +-commutative 37.8%

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

       3. unsub-neg 37.8%

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

       4. *-commutative 37.8%

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

       5. unpow2 37.8%

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

   13. Simplified 86.2%

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

   if 0.00120000006 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

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

      1. associate-/r* 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in cos2phi around 0 62.1%

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

      1. mul-1-neg 62.1%

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

      2. unpow2 62.1%

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

      3. *-commutative 62.1%

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

   6. Simplified 62.1%

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

   7. Taylor expanded in alphay around 0 62.1%

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

      1. associate-/l* 61.8%

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

      2. sub-neg 61.8%

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

      3. log1p-def 98.1%

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

      4. associate-/l* 98.7%

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

      5. *-commutative 98.7%

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

      6. associate-*l/ 98.6%

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

      7. *-commutative 98.6%

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

      8. unpow2 98.6%

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

   9. Simplified 98.6%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.0012000000569969416:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \frac{u0 - \left(u0 \cdot u0\right) \cdot -0.5}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot \left(-alphay\right)\right)\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

   1. neg-sub0 57.7%

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

   2. div-sub 57.7%

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

   3. --rgt-identity 57.7%

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

   4. div-sub 57.7%

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

   5. --rgt-identity 57.7%

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

   6. neg-sub0 57.7%

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

   7. sub-neg 57.7%

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

   8. log1p-def 97.6%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 5: 98.4% 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(Float32(-log1p(Float32(-u0))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 57.7%

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

   1. neg-sub0 57.7%

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

   2. div-sub 57.7%

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

   3. --rgt-identity 57.7%

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

   4. div-sub 57.7%

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

   5. --rgt-identity 57.7%

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

   6. sub-neg 57.7%

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

   7. +-commutative 57.7%

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

   8. neg-sub0 57.7%

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

   9. associate-+l- 57.7%

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

   10. sub0-neg 57.7%

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

   11. neg-mul-1 57.7%

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

   12. log-prod -0.0%

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

   13. associate--r+ -0.0%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 6: 81.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.00015999999595806003:\\ \;\;\;\;u0 \cdot \frac{1}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi} - \frac{alphay \cdot \left(alphay \cdot \left(u0 \cdot u0\right)\right)}{\frac{sin2phi}{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.00015999999595806003)
     (* u0 (/ 1.0 (+ t_0 (/ cos2phi (* alphax alphax)))))
     (-
      (* (* alphay alphay) (/ u0 sin2phi))
      (/ (* alphay (* alphay (* u0 u0))) (/ sin2phi -0.5))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.00015999999595806003f) {
		tmp = u0 * (1.0f / (t_0 + (cos2phi / (alphax * alphax))));
	} else {
		tmp = ((alphay * alphay) * (u0 / sin2phi)) - ((alphay * (alphay * (u0 * u0))) / (sin2phi / -0.5f));
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.00015999999595806003e0) then
        tmp = u0 * (1.0e0 / (t_0 + (cos2phi / (alphax * alphax))))
    else
        tmp = ((alphay * alphay) * (u0 / sin2phi)) - ((alphay * (alphay * (u0 * u0))) / (sin2phi / (-0.5e0)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00015999999595806003))
		tmp = Float32(u0 * Float32(Float32(1.0) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax)))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi)) - Float32(Float32(alphay * Float32(alphay * Float32(u0 * u0))) / Float32(sin2phi / Float32(-0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.00015999999595806003))
		tmp = u0 * (single(1.0) / (t_0 + (cos2phi / (alphax * alphax))));
	else
		tmp = ((alphay * alphay) * (u0 / sin2phi)) - ((alphay * (alphay * (u0 * u0))) / (sin2phi / single(-0.5)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.59999996e-4

   1. Initial program 54.9%

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

      1. associate-/r* 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in u0 around 0 74.1%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 74.1%

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      2. unpow2 74.1%

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

   6. Simplified 74.1%

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

      1. div-inv 74.2%

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

      2. +-commutative 74.2%

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

   8. Applied egg-rr 74.2%

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

   if 1.59999996e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.6%

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

      1. associate-/r* 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in cos2phi around 0 61.3%

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

      1. mul-1-neg 61.3%

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

      2. unpow2 61.3%

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

      3. *-commutative 61.3%

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

   6. Simplified 61.3%

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

   7. Taylor expanded in u0 around 0 88.0%

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

      1. mul-1-neg 88.0%

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

      2. unsub-neg 88.0%

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

      3. associate-*r/ 88.0%

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

      4. *-commutative 88.0%

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

      5. associate-/l* 88.0%

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

      6. *-commutative 88.0%

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

      7. unpow2 88.0%

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

      8. associate-*l* 88.0%

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

      9. unpow2 88.0%

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

      10. associate-/l* 86.7%

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

      11. associate-/r/ 88.1%

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

      12. unpow2 88.1%

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

   9. Simplified 88.1%

      \[\leadsto -\color{blue}{\left(\frac{alphay \cdot \left(alphay \cdot \left(u0 \cdot u0\right)\right)}{\frac{sin2phi}{-0.5}} - \frac{u0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00015999999595806003:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi} - \frac{alphay \cdot \left(alphay \cdot \left(u0 \cdot u0\right)\right)}{\frac{sin2phi}{-0.5}}\\ \end{array} \]

Alternative 7: 87.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (alphax * alphay) * ((u0 - ((u0 * u0) * -0.5f)) / (((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 * alphay) * ((u0 - ((u0 * u0) * (-0.5e0))) / (((alphax * sin2phi) / alphay) + ((alphay * cos2phi) / alphax)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

   1. neg-sub0 57.7%

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

   2. div-sub 57.7%

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

   3. --rgt-identity 57.7%

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

   4. div-sub 57.7%

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

   5. --rgt-identity 57.7%

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

   6. neg-sub0 57.7%

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

   7. sub-neg 57.7%

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

   8. log1p-def 97.6%

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

3. Simplified 97.6%

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

   1. +-commutative 97.6%

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

   2. associate-/r* 97.5%

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

   3. associate-/r* 97.6%

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

   4. frac-add 97.3%

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

5. Applied egg-rr 97.3%

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

   1. expm1-log1p-u 95.7%

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

   2. expm1-udef 52.3%

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

   3. associate-/r/ 52.3%

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

   4. *-commutative 52.3%

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

   5. fma-def 52.3%

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

   6. *-commutative 52.3%

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

7. Applied egg-rr 52.3%

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

   1. expm1-def 96.9%

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

   2. expm1-log1p 98.4%

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

9. Simplified 98.4%

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

10. Taylor expanded in alphax around 0 98.6%

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

11. Taylor expanded in u0 around 0 87.6%

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

    1. neg-mul-1 66.0%

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

    2. +-commutative 66.0%

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

    3. unsub-neg 66.0%

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

    4. *-commutative 66.0%

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

    5. unpow2 66.0%

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

13. Simplified 87.6%

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

14. Final simplification 87.6%

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

Alternative 8: 81.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.00015999999595806003:\\ \;\;\;\;u0 \cdot \frac{1}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.00015999999595806003)
     (* u0 (/ 1.0 (+ t_0 (/ cos2phi (* alphax alphax)))))
     (/ (* (* alphay alphay) (- u0 (* u0 (* u0 -0.5)))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.00015999999595806003f) {
		tmp = u0 * (1.0f / (t_0 + (cos2phi / (alphax * alphax))));
	} else {
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * -0.5f)))) / sin2phi;
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.00015999999595806003e0) then
        tmp = u0 * (1.0e0 / (t_0 + (cos2phi / (alphax * alphax))))
    else
        tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * (-0.5e0))))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00015999999595806003))
		tmp = Float32(u0 * Float32(Float32(1.0) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax)))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5))))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.00015999999595806003))
		tmp = u0 * (single(1.0) / (t_0 + (cos2phi / (alphax * alphax))));
	else
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * single(-0.5))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.59999996e-4

   1. Initial program 54.9%

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

      1. associate-/r* 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in u0 around 0 74.1%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 74.1%

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      2. unpow2 74.1%

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

   6. Simplified 74.1%

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

      1. div-inv 74.2%

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

      2. +-commutative 74.2%

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

   8. Applied egg-rr 74.2%

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

   if 1.59999996e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.6%

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

      1. associate-/r* 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in cos2phi around 0 61.3%

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

      1. mul-1-neg 61.3%

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

      2. unpow2 61.3%

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

      3. *-commutative 61.3%

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

   6. Simplified 61.3%

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

      1. div-inv 61.3%

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

      2. sub-neg 61.3%

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

      3. log1p-udef 97.6%

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

   8. Applied egg-rr 97.6%

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

   9. Taylor expanded in u0 around 0 87.9%

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

       1. neg-mul-1 87.9%

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

       2. +-commutative 87.9%

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

       3. unsub-neg 87.9%

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

       4. *-commutative 87.9%

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

       5. unpow2 87.9%

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

   11. Simplified 87.9%

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

       1. un-div-inv 88.1%

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

       2. *-commutative 88.1%

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

       3. associate-*l* 88.1%

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

   13. Applied egg-rr 88.1%

       \[\leadsto -\color{blue}{\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot -0.5\right) - u0\right)}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00015999999595806003:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 9: 81.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.00015999999595806003:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.00015999999595806003)
     (/ u0 (+ (/ (/ cos2phi alphax) alphax) t_0))
     (* (/ (* alphay alphay) sin2phi) (- u0 (* u0 (* u0 -0.5)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.00015999999595806003f) {
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	} else {
		tmp = ((alphay * alphay) / sin2phi) * (u0 - (u0 * (u0 * -0.5f)));
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.00015999999595806003e0) then
        tmp = u0 / (((cos2phi / alphax) / alphax) + t_0)
    else
        tmp = ((alphay * alphay) / sin2phi) * (u0 - (u0 * (u0 * (-0.5e0))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00015999999595806003))
		tmp = Float32(u0 / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.00015999999595806003))
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	else
		tmp = ((alphay * alphay) / sin2phi) * (u0 - (u0 * (u0 * single(-0.5))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.59999996e-4

   1. Initial program 54.9%

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

      1. associate-/r* 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in u0 around 0 74.1%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 74.1%

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      2. unpow2 74.1%

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

   6. Simplified 74.1%

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

      1. +-commutative 74.1%

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

      2. clear-num 74.2%

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

      3. frac-add 74.0%

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

      4. associate-/l* 74.0%

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

      5. div-inv 74.0%

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

      6. clear-num 73.9%

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

      7. *-commutative 73.9%

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

      8. *-un-lft-identity 73.9%

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

      9. associate-/l* 73.8%

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

      10. div-inv 73.8%

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

      11. clear-num 74.0%

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

   8. Applied egg-rr 74.0%

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

   9. Taylor expanded in sin2phi around 0 74.1%

      \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   10. Step-by-step derivation

       1. unpow2 74.1%

          \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

       2. associate-/r* 74.1%

          \[\leadsto \frac{u0}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

       3. unpow2 74.1%

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

   11. Simplified 74.1%

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

   if 1.59999996e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.6%

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

      1. associate-/r* 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in cos2phi around 0 61.3%

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

      1. mul-1-neg 61.3%

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

      2. unpow2 61.3%

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

      3. *-commutative 61.3%

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

   6. Simplified 61.3%

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

      1. div-inv 61.3%

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

      2. sub-neg 61.3%

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

      3. log1p-udef 97.6%

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

   8. Applied egg-rr 97.6%

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

   9. Taylor expanded in u0 around 0 87.9%

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

       1. neg-mul-1 87.9%

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

       2. +-commutative 87.9%

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

       3. unsub-neg 87.9%

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

       4. *-commutative 87.9%

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

       5. unpow2 87.9%

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

   11. Simplified 87.9%

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

   12. Taylor expanded in alphay around 0 88.1%

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

       1. associate-/l* 87.6%

          \[\leadsto -\color{blue}{\frac{{alphay}^{2}}{\frac{sin2phi}{-0.5 \cdot {u0}^{2} - u0}}} \]

       2. *-commutative 87.6%

          \[\leadsto -\frac{{alphay}^{2}}{\frac{sin2phi}{\color{blue}{{u0}^{2} \cdot -0.5} - u0}} \]

       3. unpow2 87.6%

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

       4. associate-*r* 87.6%

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

       5. associate-/r/ 88.0%

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

       6. unpow2 88.0%

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

   14. Simplified 88.0%

       \[\leadsto -\color{blue}{\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 \cdot \left(u0 \cdot -0.5\right) - u0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00015999999595806003:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 10: 81.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.00015999999595806003:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.00015999999595806003)
     (/ u0 (+ (/ (/ cos2phi alphax) alphax) t_0))
     (/ (* (* alphay alphay) (- u0 (* u0 (* u0 -0.5)))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.00015999999595806003f) {
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	} else {
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * -0.5f)))) / sin2phi;
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.00015999999595806003e0) then
        tmp = u0 / (((cos2phi / alphax) / alphax) + t_0)
    else
        tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * (-0.5e0))))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00015999999595806003))
		tmp = Float32(u0 / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5))))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.00015999999595806003))
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	else
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * single(-0.5))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.59999996e-4

   1. Initial program 54.9%

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

      1. associate-/r* 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in u0 around 0 74.1%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 74.1%

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      2. unpow2 74.1%

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

   6. Simplified 74.1%

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

      1. +-commutative 74.1%

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

      2. clear-num 74.2%

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

      3. frac-add 74.0%

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

      4. associate-/l* 74.0%

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

      5. div-inv 74.0%

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

      6. clear-num 73.9%

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

      7. *-commutative 73.9%

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

      8. *-un-lft-identity 73.9%

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

      9. associate-/l* 73.8%

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

      10. div-inv 73.8%

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

      11. clear-num 74.0%

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

   8. Applied egg-rr 74.0%

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

   9. Taylor expanded in sin2phi around 0 74.1%

      \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   10. Step-by-step derivation

       1. unpow2 74.1%

          \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

       2. associate-/r* 74.1%

          \[\leadsto \frac{u0}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

       3. unpow2 74.1%

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

   11. Simplified 74.1%

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

   if 1.59999996e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.6%

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

      1. associate-/r* 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in cos2phi around 0 61.3%

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

      1. mul-1-neg 61.3%

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

      2. unpow2 61.3%

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

      3. *-commutative 61.3%

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

   6. Simplified 61.3%

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

      1. div-inv 61.3%

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

      2. sub-neg 61.3%

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

      3. log1p-udef 97.6%

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

   8. Applied egg-rr 97.6%

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

   9. Taylor expanded in u0 around 0 87.9%

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

       1. neg-mul-1 87.9%

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

       2. +-commutative 87.9%

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

       3. unsub-neg 87.9%

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

       4. *-commutative 87.9%

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

       5. unpow2 87.9%

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

   11. Simplified 87.9%

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

       1. un-div-inv 88.1%

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

       2. *-commutative 88.1%

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

       3. associate-*l* 88.1%

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

   13. Applied egg-rr 88.1%

       \[\leadsto -\color{blue}{\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(u0 \cdot -0.5\right) - u0\right)}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00015999999595806003:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 11: 75.7% accurate, 8.9× speedup

Math

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

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(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(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 57.7%

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

   1. associate-/r* 57.7%

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

3. Simplified 57.7%

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

4. Taylor expanded in u0 around 0 75.2%

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
5. Step-by-step derivation

   1. unpow2 75.2%

      \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

   2. unpow2 75.2%

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

6. Simplified 75.2%

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

7. Final simplification 75.2%

   \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

Alternative 12: 75.7% accurate, 8.9× speedup

Math

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

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(Float32(cos2phi / 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 57.7%

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

   1. associate-/r* 57.7%

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

3. Simplified 57.7%

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

4. Taylor expanded in u0 around 0 75.2%

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
5. Step-by-step derivation

   1. unpow2 75.2%

      \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

   2. unpow2 75.2%

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

6. Simplified 75.2%

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

   1. +-commutative 75.2%

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

   2. clear-num 75.2%

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

   3. frac-add 68.1%

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

   4. associate-/l* 68.0%

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

   5. div-inv 68.0%

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

   6. clear-num 67.9%

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

   7. *-commutative 67.9%

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

   8. *-un-lft-identity 67.9%

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

   9. associate-/l* 67.9%

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

   10. div-inv 67.9%

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

   11. clear-num 68.1%

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

8. Applied egg-rr 68.1%

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

9. Taylor expanded in sin2phi around 0 75.2%

   \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
10. Step-by-step derivation

    1. unpow2 75.2%

       \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

    2. associate-/r* 75.2%

       \[\leadsto \frac{u0}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

    3. unpow2 75.2%

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

11. Simplified 75.2%

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

12. Final simplification 75.2%

    \[\leadsto \frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

Alternative 13: 66.5% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.999999998199587e-24)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.999999998199587e-24f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	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 (sin2phi <= 9.999999998199587e-24) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.999999998199587e-24))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(9.999999998199587e-24))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1e-23

   1. Initial program 54.1%

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

      1. associate-/r* 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in u0 around 0 74.8%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 74.8%

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      2. unpow2 74.8%

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

   6. Simplified 74.8%

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

   7. Taylor expanded in cos2phi around inf 62.3%

      \[\leadsto \color{blue}{\frac{u0 \cdot {alphax}^{2}}{cos2phi}} \]
   8. Step-by-step derivation

      1. associate-/l* 62.3%

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}}}} \]

      2. associate-/r/ 62.3%

         \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot {alphax}^{2}} \]

      3. unpow2 62.3%

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

   9. Simplified 62.3%

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

   if 1e-23 < sin2phi

   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. associate-/r* 58.8%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in u0 around 0 75.3%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 75.3%

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      2. unpow2 75.3%

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

   6. Simplified 75.3%

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

   7. Taylor expanded in cos2phi around 0 69.4%

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

      1. associate-/l* 68.7%

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

      2. associate-/r/ 69.5%

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

      3. unpow2 69.5%

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

   9. Simplified 69.5%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 14: 23.9% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

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 * alphax) * (u0 / cos2phi)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

   1. associate-/r* 57.7%

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

3. Simplified 57.7%

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

4. Taylor expanded in u0 around 0 75.2%

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
5. Step-by-step derivation

   1. unpow2 75.2%

      \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

   2. unpow2 75.2%

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

6. Simplified 75.2%

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

7. Taylor expanded in cos2phi around inf 24.6%

   \[\leadsto \color{blue}{\frac{u0 \cdot {alphax}^{2}}{cos2phi}} \]
8. Step-by-step derivation

   1. associate-/l* 24.6%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}}}} \]

   2. associate-/r/ 24.7%

      \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot {alphax}^{2}} \]

   3. unpow2 24.7%

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

9. Simplified 24.7%

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

10. Final simplification 24.7%

    \[\leadsto \left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi} \]

Reproduce

herbie shell --seed 2023258 
(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)))))