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

Percentage Accurate: 60.8% → 98.4%

Time: 18.8s

Alternatives: 27

Speedup: 3.5×

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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.4%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-add N/A

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

   6. associate-/r/ N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. distribute-frac-neg2 N/A

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

   7. distribute-frac-neg N/A

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

   8. lift-log1p.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. sub-neg N/A

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

   11. associate-*l/ N/A

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

   12. lower-/.f32 N/A

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

6. Applied rewrites 98.8%

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

7. Final simplification 98.8%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.4%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-add N/A

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

   6. associate-/r/ N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. distribute-frac-neg2 N/A

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

   7. distribute-frac-neg N/A

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

   8. lift-log1p.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. sub-neg N/A

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

   11. associate-*l/ N/A

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

   12. lower-/.f32 N/A

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

6. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-log1p.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. sub-neg N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. sub-neg N/A

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

   15. lift-neg.f32 N/A

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

   16. lift-log1p.f32 N/A

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

   17. lift-neg.f32 98.7

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

8. Applied rewrites 98.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 98.6

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

10. Applied rewrites 98.6%

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

11. Final simplification 98.6%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.4%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-add N/A

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

   6. associate-/r/ N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

Alternative 4: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.4%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-add N/A

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

   6. associate-/r/ N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.4%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.1

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

   6. lift-+.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

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

Alternative 6: 84.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.0003000000142492354:\\ \;\;\;\;\frac{\left(\left(alphay \cdot alphay\right) \cdot u0\right) \cdot \left(alphax \cdot alphax\right)}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, sin2phi \cdot \left(alphax \cdot alphax\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \left(\left(u0 \cdot u0\right) \cdot u0\right) \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.0003000000142492354)
   (/
    (* (* (* alphay alphay) u0) (* alphax alphax))
    (fma (* alphay alphay) cos2phi (* sin2phi (* alphax alphax))))
   (/
    (*
     (fma
      (fma 0.5 u0 1.0)
      u0
      (* (* (* u0 u0) u0) (fma 0.25 u0 0.3333333333333333)))
     (* alphay alphay))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.0003000000142492354f) {
		tmp = (((alphay * alphay) * u0) * (alphax * alphax)) / fmaf((alphay * alphay), cos2phi, (sin2phi * (alphax * alphax)));
	} else {
		tmp = (fmaf(fmaf(0.5f, u0, 1.0f), u0, (((u0 * u0) * u0) * fmaf(0.25f, u0, 0.3333333333333333f))) * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.0003000000142492354))
		tmp = Float32(Float32(Float32(Float32(alphay * alphay) * u0) * Float32(alphax * alphax)) / fma(Float32(alphay * alphay), cos2phi, Float32(sin2phi * Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(fma(fma(Float32(0.5), u0, Float32(1.0)), u0, Float32(Float32(Float32(u0 * u0) * u0) * fma(Float32(0.25), u0, Float32(0.3333333333333333)))) * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 3.00000014e-4

   1. Initial program 56.8%

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

      1. lift-/.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. frac-add N/A

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

      6. associate-/r/ N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f32 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 74.7

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

   7. Applied rewrites 74.7%

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

   if 3.00000014e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.2%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right) \cdot \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right)} \]

   6. Taylor expanded in sin2phi around inf

      \[\leadsto \frac{{alphay}^{2} \cdot \left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right) + {alphay}^{2} \cdot \left({u0}^{3} \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{\color{blue}{sin2phi}} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.7%

      \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot \left(\left(u0 \cdot u0\right) \cdot u0\right)\right)}{\color{blue}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.0003000000142492354:\\ \;\;\;\;\frac{\left(\left(alphay \cdot alphay\right) \cdot u0\right) \cdot \left(alphax \cdot alphax\right)}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, sin2phi \cdot \left(alphax \cdot alphax\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \left(\left(u0 \cdot u0\right) \cdot u0\right) \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.0003000000142492354:\\ \;\;\;\;\frac{\left(\left(alphay \cdot alphay\right) \cdot u0\right) \cdot \left(alphax \cdot alphax\right)}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, sin2phi \cdot \left(alphax \cdot alphax\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), \left(u0 \cdot u0\right) \cdot u0, \mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.0003000000142492354)
   (/
    (* (* (* alphay alphay) u0) (* alphax alphax))
    (fma (* alphay alphay) cos2phi (* sin2phi (* alphax alphax))))
   (/
    (*
     (fma
      (fma 0.25 u0 0.3333333333333333)
      (* (* u0 u0) u0)
      (* (fma 0.5 u0 1.0) u0))
     (* alphay alphay))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.0003000000142492354f) {
		tmp = (((alphay * alphay) * u0) * (alphax * alphax)) / fmaf((alphay * alphay), cos2phi, (sin2phi * (alphax * alphax)));
	} else {
		tmp = (fmaf(fmaf(0.25f, u0, 0.3333333333333333f), ((u0 * u0) * u0), (fmaf(0.5f, u0, 1.0f) * u0)) * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.0003000000142492354))
		tmp = Float32(Float32(Float32(Float32(alphay * alphay) * u0) * Float32(alphax * alphax)) / fma(Float32(alphay * alphay), cos2phi, Float32(sin2phi * Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), Float32(Float32(u0 * u0) * u0), Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0)) * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 3.00000014e-4

   1. Initial program 56.8%

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

      1. lift-/.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. frac-add N/A

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

      6. associate-/r/ N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f32 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 74.7

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

   7. Applied rewrites 74.7%

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

   if 3.00000014e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.2%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right) \cdot \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right)} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \frac{{alphax}^{2} \cdot \left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right) + {alphax}^{2} \cdot \left({u0}^{3} \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]
   7. Step-by-step derivation

   8. Applied rewrites 7.6%

      \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot \left(\left(u0 \cdot u0\right) \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]

   9. Taylor expanded in sin2phi around inf

      \[\leadsto \frac{{alphay}^{2} \cdot \left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right) + {alphay}^{2} \cdot \left({u0}^{3} \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{\color{blue}{sin2phi}} \]
   10. Step-by-step derivation

   11. Applied rewrites 92.7%

       \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), \left(u0 \cdot u0\right) \cdot u0, \mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right)}{\color{blue}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.0003000000142492354:\\ \;\;\;\;\frac{\left(\left(alphay \cdot alphay\right) \cdot u0\right) \cdot \left(alphax \cdot alphax\right)}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, sin2phi \cdot \left(alphax \cdot alphax\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), \left(u0 \cdot u0\right) \cdot u0, \mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right) \cdot u0, u0, u0\right)}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, sin2phi \cdot \left(alphax \cdot alphax\right)\right)} \cdot \left(\left(alphay \cdot \left(alphax \cdot alphax\right)\right) \cdot alphay\right) \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  (/
   (fma (* (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0) u0 u0)
   (fma (* alphay alphay) cos2phi (* sin2phi (* alphax alphax))))
  (* (* alphay (* alphax alphax)) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (fmaf((fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f) * u0), u0, u0) / fmaf((alphay * alphay), cos2phi, (sin2phi * (alphax * alphax)))) * ((alphay * (alphax * alphax)) * alphay);
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(fma(Float32(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)) * u0), u0, u0) / fma(Float32(alphay * alphay), cos2phi, Float32(sin2phi * Float32(alphax * alphax)))) * Float32(Float32(alphay * Float32(alphax * alphax)) * alphay))
end

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot u0} + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), u0, 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) \cdot u0} + \frac{1}{2}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u0, u0, \frac{1}{2}\right)}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-fma.f32 92.9

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)}, u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 92.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

7. Applied rewrites 93.2%

   \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right) \cdot u0\right) \cdot u0 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   2. lift-+.f32 N/A

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

   5. frac-add N/A

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\color{blue}{\frac{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi}{\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)}}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\frac{cos2phi \cdot \left(alphay \cdot alphay\right) + \color{blue}{\left(alphax \cdot alphax\right) \cdot sin2phi}}{\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)}} \]

   7. lift-fma.f32 N/A

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\frac{\color{blue}{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, \left(alphax \cdot alphax\right) \cdot sin2phi\right)}}{\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)}} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\frac{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, \left(alphax \cdot alphax\right) \cdot sin2phi\right)}{\left(alphax \cdot alphax\right) \cdot \color{blue}{\left(alphay \cdot alphay\right)}}} \]

   9. associate-*l* N/A

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\frac{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, \left(alphax \cdot alphax\right) \cdot sin2phi\right)}{\color{blue}{\left(\left(alphax \cdot alphax\right) \cdot alphay\right) \cdot alphay}}} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\frac{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, \left(alphax \cdot alphax\right) \cdot sin2phi\right)}{\color{blue}{\left(\left(alphax \cdot alphax\right) \cdot alphay\right)} \cdot alphay}} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\frac{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, \left(alphax \cdot alphax\right) \cdot sin2phi\right)}{\color{blue}{\left(\left(alphax \cdot alphax\right) \cdot alphay\right) \cdot alphay}}} \]

   12. associate-/r/ N/A

       \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, \left(alphax \cdot alphax\right) \cdot sin2phi\right)} \cdot \left(\left(\left(alphax \cdot alphax\right) \cdot alphay\right) \cdot alphay\right)} \]

   13. lower-*.f32 N/A

       \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, \left(alphax \cdot alphax\right) \cdot sin2phi\right)} \cdot \left(\left(\left(alphax \cdot alphax\right) \cdot alphay\right) \cdot alphay\right)} \]

9. Applied rewrites 93.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right) \cdot u0, u0, u0\right)}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, sin2phi \cdot \left(alphax \cdot alphax\right)\right)} \cdot \left(\left(alphay \cdot \left(alphax \cdot alphax\right)\right) \cdot alphay\right)} \]
10. Add Preprocessing

Alternative 9: 92.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, sin2phi \cdot \left(alphax \cdot alphax\right)\right)} \cdot \left(\left(alphay \cdot \left(alphax \cdot alphax\right)\right) \cdot alphay\right) \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  (/
   (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
   (fma cos2phi (* alphay alphay) (* sin2phi (* alphax alphax))))
  (* (* alphay (* alphax alphax)) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return ((fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) / fmaf(cos2phi, (alphay * alphay), (sin2phi * (alphax * alphax)))) * ((alphay * (alphax * alphax)) * alphay);
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / fma(cos2phi, Float32(alphay * alphay), Float32(sin2phi * Float32(alphax * alphax)))) * Float32(Float32(alphay * Float32(alphax * alphax)) * alphay))
end

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot u0} + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), u0, 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) \cdot u0} + \frac{1}{2}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u0, u0, \frac{1}{2}\right)}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-fma.f32 92.9

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)}, u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 92.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   2. lift-+.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

   5. frac-add N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\color{blue}{\frac{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi}{\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{\color{blue}{\left(alphay \cdot alphay\right) \cdot cos2phi} + \left(alphax \cdot alphax\right) \cdot sin2phi}{\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{\left(alphay \cdot alphay\right) \cdot cos2phi + \color{blue}{sin2phi \cdot \left(alphax \cdot alphax\right)}}{\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)}} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{\left(alphay \cdot alphay\right) \cdot cos2phi + \color{blue}{sin2phi \cdot \left(alphax \cdot alphax\right)}}{\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)}} \]

   9. lift-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{\color{blue}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, sin2phi \cdot \left(alphax \cdot alphax\right)\right)}}{\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)}} \]

   10. associate-/r/ N/A

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, sin2phi \cdot \left(alphax \cdot alphax\right)\right)} \cdot \left(\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)\right)} \]

   11. lower-*.f32 N/A

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right), u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, sin2phi \cdot \left(alphax \cdot alphax\right)\right)} \cdot \left(\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)\right)} \]

7. Applied rewrites 93.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, \left(alphax \cdot alphax\right) \cdot sin2phi\right)} \cdot \left(\left(\left(alphax \cdot alphax\right) \cdot alphay\right) \cdot alphay\right)} \]

8. Final simplification 93.5%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, sin2phi \cdot \left(alphax \cdot alphax\right)\right)} \cdot \left(\left(alphay \cdot \left(alphax \cdot alphax\right)\right) \cdot alphay\right) \]
9. Add Preprocessing

Alternative 10: 93.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right) \cdot u0\right) \cdot u0 + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (+ (* (* (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0) u0) u0)
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (((fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f) * u0) * u0) + u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(Float32(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)) * u0) * u0) + u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot u0} + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), u0, 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) \cdot u0} + \frac{1}{2}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u0, u0, \frac{1}{2}\right)}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-fma.f32 92.9

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)}, u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 92.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

7. Applied rewrites 93.2%

   \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right) \cdot u0\right) \cdot u0 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
8. Add Preprocessing

Alternative 11: 93.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) (* u0 u0) u0)
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), (u0 * u0), u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), Float32(u0 * u0), u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot u0} + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), u0, 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) \cdot u0} + \frac{1}{2}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u0, u0, \frac{1}{2}\right)}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-fma.f32 92.9

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)}, u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 92.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

7. Applied rewrites 93.2%

   \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right) \cdot u0\right) \cdot u0 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
8. Step-by-step derivation

9. Applied rewrites 93.2%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), \color{blue}{u0 \cdot u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
10. Add Preprocessing

Alternative 12: 93.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right) \cdot u0, u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (fma (* (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0) u0 u0)
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return fmaf((fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f) * u0), u0, u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(fma(Float32(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)) * u0), u0, u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot u0} + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), u0, 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) \cdot u0} + \frac{1}{2}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u0, u0, \frac{1}{2}\right)}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-fma.f32 92.9

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)}, u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 92.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

7. Applied rewrites 93.2%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right) \cdot u0, \color{blue}{u0}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
8. Add Preprocessing

Alternative 13: 92.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot u0} + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), u0, 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) \cdot u0} + \frac{1}{2}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u0, u0, \frac{1}{2}\right)}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-fma.f32 92.9

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)}, u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 92.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Add Preprocessing

Alternative 14: 69.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0\right) \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.0000000843119176e-17)
   (/
    (*
     (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
     (* alphax alphax))
    cos2phi)
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.0000000843119176e-17f) {
		tmp = ((fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) * (alphax * alphax)) / cos2phi;
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(5.0000000843119176e-17))
		tmp = Float32(Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) * Float32(alphax * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000008e-17

   1. Initial program 58.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right) \cdot \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right)} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \frac{{alphax}^{2} \cdot \left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right) + {alphax}^{2} \cdot \left({u0}^{3} \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.0%

      \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot \left(\left(u0 \cdot u0\right) \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{cos2phi} \]
   10. Step-by-step derivation

   11. Applied rewrites 74.9%

       \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0\right)}{cos2phi} \]

   if 5.00000008e-17 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.7%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 70.7%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0\right) \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), 1\right) \cdot u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.0000000843119176e-17)
   (*
    (/
     (* (fma u0 (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) 1.0) u0)
     cos2phi)
    (* alphax alphax))
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.0000000843119176e-17f) {
		tmp = ((fmaf(u0, fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), 1.0f) * u0) / cos2phi) * (alphax * alphax);
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(5.0000000843119176e-17))
		tmp = Float32(Float32(Float32(fma(u0, fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), Float32(1.0)) * u0) / cos2phi) * Float32(alphax * alphax));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000008e-17

   1. Initial program 58.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right) \cdot \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right)} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \frac{{alphax}^{2} \cdot \left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right) + {alphax}^{2} \cdot \left({u0}^{3} \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.0%

      \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot \left(\left(u0 \cdot u0\right) \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]
   9. Step-by-step derivation

   10. Applied rewrites 74.7%

       \[\leadsto \frac{1}{\frac{cos2phi}{\color{blue}{\left(u0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right)\right)\right) \cdot \left(alphax \cdot alphax\right)}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 74.9%

       \[\leadsto \left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), 1\right) \cdot u0}{\color{blue}{cos2phi}} \]

   if 5.00000008e-17 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.7%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 70.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), 1\right) \cdot u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), 1\right) \cdot \left(alphax \cdot alphax\right)}{cos2phi} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.0000000843119176e-17)
   (*
    (/
     (*
      (fma u0 (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) 1.0)
      (* alphax alphax))
     cos2phi)
    u0)
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.0000000843119176e-17f) {
		tmp = ((fmaf(u0, fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), 1.0f) * (alphax * alphax)) / cos2phi) * u0;
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(5.0000000843119176e-17))
		tmp = Float32(Float32(Float32(fma(u0, fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), Float32(1.0)) * Float32(alphax * alphax)) / cos2phi) * u0);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000008e-17

   1. Initial program 58.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right) \cdot \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right)} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \frac{{alphax}^{2} \cdot \left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right) + {alphax}^{2} \cdot \left({u0}^{3} \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.0%

      \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot \left(\left(u0 \cdot u0\right) \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]
   9. Step-by-step derivation

   10. Applied rewrites 74.7%

       \[\leadsto \frac{1}{\frac{cos2phi}{\color{blue}{\left(u0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right)\right)\right) \cdot \left(alphax \cdot alphax\right)}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 74.7%

       \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), 1\right) \cdot \left(alphax \cdot alphax\right)}{\color{blue}{cos2phi}} \]

   if 5.00000008e-17 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.7%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 70.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), 1\right) \cdot \left(alphax \cdot alphax\right)}{cos2phi} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 89.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.003000000026077032:\\ \;\;\;\;\frac{\left(0.5 \cdot u0\right) \cdot u0 + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \left(\left(u0 \cdot u0\right) \cdot u0\right) \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.003000000026077032)
   (/
    (+ (* (* 0.5 u0) u0) u0)
    (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (/
    (*
     (fma
      (fma 0.5 u0 1.0)
      u0
      (* (* (* u0 u0) u0) (fma 0.25 u0 0.3333333333333333)))
     (* alphay alphay))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.003000000026077032f) {
		tmp = (((0.5f * u0) * u0) + u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (fmaf(fmaf(0.5f, u0, 1.0f), u0, (((u0 * u0) * u0) * fmaf(0.25f, u0, 0.3333333333333333f))) * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.003000000026077032))
		tmp = Float32(Float32(Float32(Float32(Float32(0.5) * u0) * u0) + u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(fma(fma(Float32(0.5), u0, Float32(1.0)), u0, Float32(Float32(Float32(u0 * u0) * u0) * fma(Float32(0.25), u0, Float32(0.3333333333333333)))) * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00300000003

   1. Initial program 56.7%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot u0} + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-fma.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), u0, 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) \cdot u0} + \frac{1}{2}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. lower-fma.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u0, u0, \frac{1}{2}\right)}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. lower-fma.f32 93.2

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)}, u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Applied rewrites 93.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.4%

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right) \cdot u0\right) \cdot u0 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. Taylor expanded in u0 around 0

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

   10. Applied rewrites 87.9%

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

   if 0.00300000003 < sin2phi

   1. Initial program 67.3%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

   5. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right) \cdot \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right)} \]

   6. Taylor expanded in sin2phi around inf

      \[\leadsto \frac{{alphay}^{2} \cdot \left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right) + {alphay}^{2} \cdot \left({u0}^{3} \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{\color{blue}{sin2phi}} \]
   7. Step-by-step derivation

   8. Applied rewrites 93.7%

      \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot \left(\left(u0 \cdot u0\right) \cdot u0\right)\right)}{\color{blue}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.003000000026077032:\\ \;\;\;\;\frac{\left(0.5 \cdot u0\right) \cdot u0 + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \left(\left(u0 \cdot u0\right) \cdot u0\right) \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 91.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right) \cdot u0\right) \cdot u0 + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (+ (* (* (fma 0.3333333333333333 u0 0.5) u0) u0) u0)
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (((fmaf(0.3333333333333333f, u0, 0.5f) * u0) * u0) + u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(Float32(fma(Float32(0.3333333333333333), u0, Float32(0.5)) * u0) * u0) + u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) \cdot u0} + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), u0, 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) \cdot u0} + \frac{1}{2}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u0, u0, \frac{1}{2}\right)}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, u0, \frac{1}{2}\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-fma.f32 92.9

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)}, u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 92.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

7. Applied rewrites 93.2%

   \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right) \cdot u0\right) \cdot u0 + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Taylor expanded in u0 around 0

   \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{1}{3}, u0, \frac{1}{2}\right) \cdot u0\right) \cdot u0 + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
9. Step-by-step derivation

10. Applied rewrites 91.1%

    \[\leadsto \frac{\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right) \cdot u0\right) \cdot u0 + u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
11. Add Preprocessing

Alternative 19: 89.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.003000000026077032:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \cdot \mathsf{fma}\left(0.5, u0, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \left(\left(u0 \cdot u0\right) \cdot u0\right) \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.003000000026077032)
   (*
    (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
    (fma 0.5 u0 1.0))
   (/
    (*
     (fma
      (fma 0.5 u0 1.0)
      u0
      (* (* (* u0 u0) u0) (fma 0.25 u0 0.3333333333333333)))
     (* alphay alphay))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.003000000026077032f) {
		tmp = (u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))) * fmaf(0.5f, u0, 1.0f);
	} else {
		tmp = (fmaf(fmaf(0.5f, u0, 1.0f), u0, (((u0 * u0) * u0) * fmaf(0.25f, u0, 0.3333333333333333f))) * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.003000000026077032))
		tmp = Float32(Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))) * fma(Float32(0.5), u0, Float32(1.0)));
	else
		tmp = Float32(Float32(fma(fma(Float32(0.5), u0, Float32(1.0)), u0, Float32(Float32(Float32(u0 * u0) * u0) * fma(Float32(0.25), u0, Float32(0.3333333333333333)))) * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00300000003

   1. Initial program 56.7%

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

   3. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

      18. unpow2 N/A

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

   5. Applied rewrites 87.7%

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

   if 0.00300000003 < sin2phi

   1. Initial program 67.3%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

   5. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right) \cdot \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right)} \]

   6. Taylor expanded in sin2phi around inf

      \[\leadsto \frac{{alphay}^{2} \cdot \left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right) + {alphay}^{2} \cdot \left({u0}^{3} \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{\color{blue}{sin2phi}} \]
   7. Step-by-step derivation

   8. Applied rewrites 93.7%

      \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot \left(\left(u0 \cdot u0\right) \cdot u0\right)\right)}{\color{blue}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.003000000026077032:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \cdot \mathsf{fma}\left(0.5, u0, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \left(\left(u0 \cdot u0\right) \cdot u0\right) \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 91.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* (fma (fma 0.3333333333333333 u0 0.5) u0 1.0) u0)
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (fmaf(fmaf(0.3333333333333333f, u0, 0.5f), u0, 1.0f) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(fma(fma(Float32(0.3333333333333333), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right) + 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{3} \cdot u0\right) \cdot u0} + 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{3} \cdot u0, u0, 1\right)} \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot u0 + \frac{1}{2}}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. lower-fma.f32 90.8

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right)}, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 90.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Add Preprocessing

Alternative 21: 68.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, 1\right) \cdot u0\right) \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.0000000843119176e-17)
   (/
    (* (* (fma (fma 0.3333333333333333 u0 0.5) u0 1.0) u0) (* alphax alphax))
    cos2phi)
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.0000000843119176e-17f) {
		tmp = ((fmaf(fmaf(0.3333333333333333f, u0, 0.5f), u0, 1.0f) * u0) * (alphax * alphax)) / cos2phi;
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(5.0000000843119176e-17))
		tmp = Float32(Float32(Float32(fma(fma(Float32(0.3333333333333333), u0, Float32(0.5)), u0, Float32(1.0)) * u0) * Float32(alphax * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000008e-17

   1. Initial program 58.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right) \cdot \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right)} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \frac{{alphax}^{2} \cdot \left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right) + {alphax}^{2} \cdot \left({u0}^{3} \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.0%

      \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot \left(\left(u0 \cdot u0\right) \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)}{cos2phi} \]
   10. Step-by-step derivation

   11. Applied rewrites 74.0%

       \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, 1\right) \cdot u0\right)}{cos2phi} \]

   if 5.00000008e-17 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.7%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 70.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, 1\right) \cdot u0\right) \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 68.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \left(alphax \cdot alphax\right) \cdot u0, alphax \cdot alphax\right) \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.0000000843119176e-17)
   (/ (* (fma 0.5 (* (* alphax alphax) u0) (* 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 / (alphay * alphay)) <= 5.0000000843119176e-17f) {
		tmp = (fmaf(0.5f, ((alphax * alphax) * u0), (alphax * alphax)) * u0) / cos2phi;
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(5.0000000843119176e-17))
		tmp = Float32(Float32(fma(Float32(0.5), Float32(Float32(alphax * alphax) * u0), Float32(alphax * alphax)) * u0) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000008e-17

   1. Initial program 58.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right) \cdot \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right)} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \frac{{alphax}^{2} \cdot \left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right) + {alphax}^{2} \cdot \left({u0}^{3} \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.0%

      \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot \left(\left(u0 \cdot u0\right) \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) + {alphax}^{2}\right)}{cos2phi} \]
   10. Step-by-step derivation

   11. Applied rewrites 71.7%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, u0 \cdot \left(alphax \cdot alphax\right), alphax \cdot alphax\right) \cdot u0}{cos2phi} \]

   if 5.00000008e-17 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.7%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 70.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \left(alphax \cdot alphax\right) \cdot u0, alphax \cdot alphax\right) \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 68.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.0000000843119176e-17)
   (/ (* (* (fma 0.5 u0 1.0) u0) (* alphax alphax)) cos2phi)
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.0000000843119176e-17f) {
		tmp = ((fmaf(0.5f, u0, 1.0f) * u0) * (alphax * alphax)) / cos2phi;
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(5.0000000843119176e-17))
		tmp = Float32(Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) * Float32(alphax * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000008e-17

   1. Initial program 58.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right) \cdot \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right)} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \frac{{alphax}^{2} \cdot \left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right) + {alphax}^{2} \cdot \left({u0}^{3} \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.0%

      \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u0, 1\right), u0, \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot \left(\left(u0 \cdot u0\right) \cdot u0\right)\right)}{\color{blue}{cos2phi}} \]

   9. Taylor expanded in u0 around 0

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

   11. Applied rewrites 71.7%

       \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot \left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right)}{cos2phi} \]

   if 5.00000008e-17 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.7%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 70.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 75.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.4%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-add N/A

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

   6. associate-/r/ N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f32 76.2

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

7. Applied rewrites 76.2%

   \[\leadsto \color{blue}{\frac{\left(\left(alphay \cdot alphay\right) \cdot u0\right) \cdot \left(alphax \cdot alphax\right)}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, sin2phi \cdot \left(alphax \cdot alphax\right)\right)}} \]
8. Add Preprocessing

Alternative 25: 75.5% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 75.5

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

5. Applied rewrites 75.5%

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

6. Final simplification 75.5%

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

Alternative 26: 66.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{alphax}{cos2phi} \cdot \left(alphax \cdot u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.0000000843119176e-17)
   (* (/ alphax cos2phi) (* alphax u0))
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.0000000843119176e-17f) {
		tmp = (alphax / cos2phi) * (alphax * u0);
	} 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 / (alphay * alphay)) <= 5.0000000843119176e-17) then
        tmp = (alphax / cos2phi) * (alphax * u0)
    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 (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(5.0000000843119176e-17))
		tmp = Float32(Float32(alphax / cos2phi) * Float32(alphax * u0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(5.0000000843119176e-17))
		tmp = (alphax / cos2phi) * (alphax * u0);
	else
		tmp = ((alphay * alphay) * u0) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000008e-17

   1. Initial program 58.6%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 61.0%

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

   10. Applied rewrites 61.1%

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

   if 5.00000008e-17 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.7%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 70.7%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{alphax}{cos2phi} \cdot \left(alphax \cdot u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 24.2% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 75.5

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

5. Applied rewrites 75.5%

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

6. Taylor expanded in alphax around 0

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

8. Applied rewrites 24.3%

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

10. Applied rewrites 24.3%

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

11. Final simplification 24.3%

    \[\leadsto \frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(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)))))