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

Percentage Accurate: 60.9% → 98.2%

Time: 19.4s

Alternatives: 26

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: 60.9% 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: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \frac{-1}{\frac{1}{alphax \cdot \left(alphax \cdot \left(alphay \cdot alphay\right)\right)}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  (/
   (log1p (- u0))
   (fma (* alphax alphax) sin2phi (* cos2phi (* alphay alphay))))
  (/ -1.0 (/ 1.0 (* alphax (* alphax (* alphay alphay)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (log1pf(-u0) / fmaf((alphax * alphax), sin2phi, (cos2phi * (alphay * alphay)))) * (-1.0f / (1.0f / (alphax * (alphax * (alphay * alphay)))));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(log1p(Float32(-u0)) / fma(Float32(alphax * alphax), sin2phi, Float32(cos2phi * Float32(alphay * alphay)))) * Float32(Float32(-1.0) / Float32(Float32(1.0) / Float32(alphax * Float32(alphax * Float32(alphay * alphay))))))
end

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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-neg.f32 N/A

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

   3. neg-mul-1 N/A

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

   4. *-commutative N/A

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

   5. lift-+.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. frac-add N/A

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

   9. div-inv N/A

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

   10. times-frac N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites 98.3%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

4. Applied rewrites 98.3%

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

5. Final simplification 98.3%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 98.3%

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

5. Final simplification 98.3%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 98.2

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

4. Applied rewrites 98.2%

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

5. Final simplification 98.2%

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

Alternative 5: 93.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.4%

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

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 92.4

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

5. Applied rewrites 92.4%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. div-inv N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 92.5

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

7. Applied rewrites 92.5%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

      \[\leadsto \frac{\mathsf{Rewrite<=}\left(lift-fma.f32, \left(\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u0\right)\right)\right)}{\color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{Rewrite<=}\left(lift-fma.f32, \left(\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u0\right)\right)\right)}{\frac{sin2phi}{alphay} \cdot \color{blue}{\frac{1}{alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   4. un-div-inv N/A

      \[\leadsto \frac{\mathsf{Rewrite<=}\left(lift-fma.f32, \left(\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u0\right)\right)\right)}{\color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{Rewrite<=}\left(lift-fma.f32, \left(\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u0\right)\right)\right)}{\frac{\color{blue}{\frac{sin2phi}{alphay}}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{\mathsf{Rewrite<=}\left(lift-fma.f32, \left(\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u0\right)\right)\right)}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{Rewrite<=}\left(lift-fma.f32, \left(\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u0\right)\right)\right)}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   8. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{Rewrite<=}\left(lift-fma.f32, \left(\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u0\right)\right)\right)}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{Rewrite<=}\left(lift-fma.f32, \left(\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u0\right)\right)\right)}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   10. lift-/.f32 N/A

       \[\leadsto \frac{\mathsf{Rewrite<=}\left(lift-fma.f32, \left(\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u0\right)\right)\right)}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

   11. lift-/.f32 N/A

       \[\leadsto \frac{\mathsf{Rewrite<=}\left(lift-fma.f32, \left(\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

9. Applied rewrites 92.9%

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

10. Final simplification 92.9%

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

Alternative 6: 93.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.4%

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

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 92.4

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

5. Applied rewrites 92.4%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 92.4

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

7. Applied rewrites 92.4%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-add N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f32 N/A

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

9. Applied rewrites 92.7%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. associate-*l* N/A

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

    5. associate-/l* N/A

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

    6. lower-*.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. lower-*.f32 92.8

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

11. Applied rewrites 92.8%

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

Alternative 7: 93.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.4%

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

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 92.4

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

5. Applied rewrites 92.4%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 92.4

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

7. Applied rewrites 92.4%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/r* N/A

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

   7. lift-*.f32 N/A

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

   8. frac-add N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f32 N/A

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

9. Applied rewrites 92.6%

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

10. Final simplification 92.6%

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

Alternative 8: 81.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 10000:\\ \;\;\;\;\frac{u0}{\mathsf{fma}\left(\frac{1}{alphay \cdot alphay}, sin2phi, \frac{cos2phi}{alphax \cdot alphax}\right)}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 10000.0)
   (/ u0 (fma (/ 1.0 (* alphay alphay)) sin2phi (/ cos2phi (* alphax alphax))))
   (* u0 (/ (fma alphay alphay (* 0.5 (* u0 (* alphay alphay)))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 10000.0f) {
		tmp = u0 / fmaf((1.0f / (alphay * alphay)), sin2phi, (cos2phi / (alphax * alphax)));
	} else {
		tmp = u0 * (fmaf(alphay, alphay, (0.5f * (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(10000.0))
		tmp = Float32(u0 / fma(Float32(Float32(1.0) / Float32(alphay * alphay)), sin2phi, Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(u0 * Float32(fma(alphay, alphay, Float32(Float32(0.5) * Float32(u0 * Float32(alphay * alphay)))) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1e4

   1. Initial program 52.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 75.7%

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

   if 1e4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.8%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-+.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f32 88.6

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in sin2phi around inf

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

   11. Applied rewrites 89.6%

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

Alternative 9: 81.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 10000:\\ \;\;\;\;\frac{u0}{\mathsf{fma}\left(\frac{1}{alphax \cdot alphax}, cos2phi, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 10000.0)
     (/ u0 (fma (/ 1.0 (* alphax alphax)) cos2phi t_0))
     (* u0 (/ (fma alphay alphay (* 0.5 (* u0 (* alphay alphay)))) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 10000.0f) {
		tmp = u0 / fmaf((1.0f / (alphax * alphax)), cos2phi, t_0);
	} else {
		tmp = u0 * (fmaf(alphay, alphay, (0.5f * (u0 * (alphay * alphay)))) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(10000.0))
		tmp = Float32(u0 / fma(Float32(Float32(1.0) / Float32(alphax * alphax)), cos2phi, t_0));
	else
		tmp = Float32(u0 * Float32(fma(alphay, alphay, Float32(Float32(0.5) * Float32(u0 * Float32(alphay * alphay)))) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1e4

   1. Initial program 52.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 75.7%

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

   if 1e4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.8%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-+.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f32 88.6

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in sin2phi around inf

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

   11. Applied rewrites 89.6%

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

Alternative 10: 93.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.4%

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

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 92.4

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

5. Applied rewrites 92.4%

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

7. Applied rewrites 92.4%

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

Alternative 11: 93.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.4%

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

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 92.4

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

5. Applied rewrites 92.4%

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

Alternative 12: 81.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 10000:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 10000.0)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (* u0 (/ (fma alphay alphay (* 0.5 (* u0 (* alphay alphay)))) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 10000.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = u0 * (fmaf(alphay, alphay, (0.5f * (u0 * (alphay * alphay)))) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(10000.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(u0 * Float32(fma(alphay, alphay, Float32(Float32(0.5) * Float32(u0 * Float32(alphay * alphay)))) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1e4

   1. Initial program 52.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 75.7

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

   5. Applied rewrites 75.7%

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

   if 1e4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.8%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-+.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f32 88.6

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in sin2phi around inf

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

   11. Applied rewrites 89.6%

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

Alternative 13: 91.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.4%

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

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 90.6

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

5. Applied rewrites 90.6%

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

Alternative 14: 84.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\left(alphax \cdot \left(alphax \cdot alphay\right)\right) \cdot \frac{u0}{\mathsf{fma}\left(alphax \cdot alphax, \frac{sin2phi}{alphay}, cos2phi \cdot alphay\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.999999987845058e-8)
   (*
    (* alphax (* alphax alphay))
    (/ u0 (fma (* alphax alphax) (/ sin2phi alphay) (* cos2phi alphay))))
   (/
    (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0)
    (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.999999987845058e-8f) {
		tmp = (alphax * (alphax * alphay)) * (u0 / fmaf((alphax * alphax), (sin2phi / alphay), (cos2phi * alphay)));
	} else {
		tmp = fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0) / (sin2phi / (alphay * alphay));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.999999987845058e-8))
		tmp = Float32(Float32(alphax * Float32(alphax * alphay)) * Float32(u0 / fma(Float32(alphax * alphax), Float32(sin2phi / alphay), Float32(cos2phi * alphay))));
	else
		tmp = Float32(fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0) / Float32(sin2phi / Float32(alphay * alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999999e-8

   1. Initial program 52.2%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 92.0

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

   5. Applied rewrites 92.0%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 92.0

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

   7. Applied rewrites 92.0%

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

      1. lift-/.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. frac-add N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f32 N/A

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

   9. Applied rewrites 91.7%

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

   10. Taylor expanded in u0 around 0

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

       1. lower-/.f32 N/A

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

       2. +-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. lower-/.f32 N/A

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

       8. lower-*.f32 75.7

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

   12. Applied rewrites 75.7%

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

   if 1.99999999e-8 < sin2phi

   1. Initial program 64.4%

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

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 92.7

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in cos2phi around 0

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

      1. lower-/.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 91.4

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

   8. Applied rewrites 91.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\left(alphax \cdot \left(alphax \cdot alphay\right)\right) \cdot \frac{u0}{\mathsf{fma}\left(alphax \cdot alphax, \frac{sin2phi}{alphay}, cos2phi \cdot alphay\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.559999959072078 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.559999959072078e-13)
   (/ (* (* alphax alphax) (* u0 (fma u0 0.5 1.0))) cos2phi)
   (* u0 (/ (fma alphay alphay (* 0.5 (* u0 (* alphay alphay)))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.559999959072078e-13f) {
		tmp = ((alphax * alphax) * (u0 * fmaf(u0, 0.5f, 1.0f))) / cos2phi;
	} else {
		tmp = u0 * (fmaf(alphay, alphay, (0.5f * (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(1.559999959072078e-13))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(u0 * fma(u0, Float32(0.5), Float32(1.0)))) / cos2phi);
	else
		tmp = Float32(u0 * Float32(fma(alphay, alphay, Float32(Float32(0.5) * Float32(u0 * Float32(alphay * alphay)))) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.56e-13

   1. Initial program 53.8%

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

   3. Taylor expanded in u0 around 0

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

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 66.4%

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

   if 1.56e-13 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-+.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f32 88.7

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

   8. Applied rewrites 88.7%

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

   9. Taylor expanded in sin2phi around inf

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

   11. Applied rewrites 83.3%

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

Alternative 16: 76.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.559999959072078 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(0.5, u0 \cdot \left(alphay \cdot alphay\right), alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.559999959072078e-13)
   (/ (* (* alphax alphax) (* u0 (fma u0 0.5 1.0))) cos2phi)
   (* u0 (/ (fma 0.5 (* u0 (* alphay alphay)) (* alphay alphay)) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.559999959072078e-13f) {
		tmp = ((alphax * alphax) * (u0 * fmaf(u0, 0.5f, 1.0f))) / cos2phi;
	} else {
		tmp = u0 * (fmaf(0.5f, (u0 * (alphay * alphay)), (alphay * alphay)) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.559999959072078e-13))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(u0 * fma(u0, Float32(0.5), Float32(1.0)))) / cos2phi);
	else
		tmp = Float32(u0 * Float32(fma(Float32(0.5), Float32(u0 * Float32(alphay * alphay)), Float32(alphay * alphay)) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.56e-13

   1. Initial program 53.8%

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

   3. Taylor expanded in u0 around 0

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

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 66.4%

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

   if 1.56e-13 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-+.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f32 88.7

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

   8. Applied rewrites 88.7%

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

   9. Taylor expanded in alphax around 0

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

   11. Applied rewrites 10.9%

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

   12. Taylor expanded in sin2phi around inf

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

   14. Applied rewrites 83.3%

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

Alternative 17: 87.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in u0 around 0

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

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

   11. lower-fma.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-+.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f32 N/A

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

   17. lower-/.f32 N/A

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

5. Applied rewrites 87.7%

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

Alternative 18: 84.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;sin2phi \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{u0}{\mathsf{fma}\left(\frac{1}{alphax \cdot alphax}, cos2phi, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= sin2phi 1.999999987845058e-8)
     (/ u0 (fma (/ 1.0 (* alphax alphax)) cos2phi t_0))
     (/
      (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0)
      t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (sin2phi <= 1.999999987845058e-8f) {
		tmp = u0 / fmaf((1.0f / (alphax * alphax)), cos2phi, t_0);
	} else {
		tmp = fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.999999987845058e-8))
		tmp = Float32(u0 / fma(Float32(Float32(1.0) / Float32(alphax * alphax)), cos2phi, t_0));
	else
		tmp = Float32(fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999999e-8

   1. Initial program 52.2%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 75.5

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

   5. Applied rewrites 75.5%

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

   7. Applied rewrites 75.6%

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

   if 1.99999999e-8 < sin2phi

   1. Initial program 64.4%

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

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 92.7

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in cos2phi around 0

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

      1. lower-/.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 91.4

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

   8. Applied rewrites 91.4%

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

Alternative 19: 75.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.559999959072078 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot t\_0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot t\_0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* u0 (fma u0 0.5 1.0))))
   (if (<= (/ sin2phi (* alphay alphay)) 1.559999959072078e-13)
     (/ (* (* alphax alphax) t_0) cos2phi)
     (/ (* (* alphay alphay) t_0) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = u0 * fmaf(u0, 0.5f, 1.0f);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.559999959072078e-13f) {
		tmp = ((alphax * alphax) * t_0) / cos2phi;
	} else {
		tmp = ((alphay * alphay) * t_0) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(u0 * fma(u0, Float32(0.5), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.559999959072078e-13))
		tmp = Float32(Float32(Float32(alphax * alphax) * t_0) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * t_0) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.56e-13

   1. Initial program 53.8%

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

   3. Taylor expanded in u0 around 0

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

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 66.4%

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

   if 1.56e-13 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in cos2phi around 0

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

   8. Applied rewrites 83.1%

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

Alternative 20: 68.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.559999959072078 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.559999959072078e-13)
   (/ (* (* alphax alphax) (* u0 (fma u0 0.5 1.0))) cos2phi)
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.559999959072078e-13f) {
		tmp = ((alphax * alphax) * (u0 * fmaf(u0, 0.5f, 1.0f))) / cos2phi;
	} else {
		tmp = (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(1.559999959072078e-13))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(u0 * fma(u0, Float32(0.5), Float32(1.0)))) / cos2phi);
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.56e-13

   1. Initial program 53.8%

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

   3. Taylor expanded in u0 around 0

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

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 66.4%

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

   if 1.56e-13 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in cos2phi around 0

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

   8. Applied rewrites 74.7%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.559999959072078 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 68.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.559999959072078 \cdot 10^{-13}:\\ \;\;\;\;\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \frac{\mathsf{fma}\left(u0, 0.5, 1\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.559999959072078e-13)
   (* (* u0 (* alphax alphax)) (/ (fma u0 0.5 1.0) cos2phi))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.559999959072078e-13f) {
		tmp = (u0 * (alphax * alphax)) * (fmaf(u0, 0.5f, 1.0f) / cos2phi);
	} else {
		tmp = (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(1.559999959072078e-13))
		tmp = Float32(Float32(u0 * Float32(alphax * alphax)) * Float32(fma(u0, Float32(0.5), Float32(1.0)) / cos2phi));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.56e-13

   1. Initial program 53.8%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-+.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f32 84.9

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

   8. Applied rewrites 84.9%

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

   9. Taylor expanded in alphax around 0

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

   11. Applied rewrites 66.3%

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

   12. Taylor expanded in cos2phi around 0

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

   14. Applied rewrites 66.4%

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

   if 1.56e-13 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in cos2phi around 0

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

   8. Applied rewrites 74.7%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.559999959072078 \cdot 10^{-13}:\\ \;\;\;\;\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \frac{\mathsf{fma}\left(u0, 0.5, 1\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 66.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.559999959072078 \cdot 10^{-13}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.559999959072078e-13)
   (/ (* alphax (* u0 alphax)) cos2phi)
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.559999959072078e-13f) {
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	} else {
		tmp = (u0 * (alphay * alphay)) / 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)) <= 1.559999959072078e-13) then
        tmp = (alphax * (u0 * alphax)) / cos2phi
    else
        tmp = (u0 * (alphay * alphay)) / 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(1.559999959072078e-13))
		tmp = Float32(Float32(alphax * Float32(u0 * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.559999959072078e-13))
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.56e-13

   1. Initial program 53.8%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 57.9%

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

   10. Applied rewrites 58.0%

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

   if 1.56e-13 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in cos2phi around 0

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

   8. Applied rewrites 74.7%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.559999959072078 \cdot 10^{-13}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 66.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.559999959072078 \cdot 10^{-13}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.559999959072078e-13)
   (* (* u0 alphax) (/ alphax cos2phi))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.559999959072078e-13f) {
		tmp = (u0 * alphax) * (alphax / cos2phi);
	} else {
		tmp = (u0 * (alphay * alphay)) / 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)) <= 1.559999959072078e-13) then
        tmp = (u0 * alphax) * (alphax / cos2phi)
    else
        tmp = (u0 * (alphay * alphay)) / 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(1.559999959072078e-13))
		tmp = Float32(Float32(u0 * alphax) * Float32(alphax / cos2phi));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.559999959072078e-13))
		tmp = (u0 * alphax) * (alphax / cos2phi);
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.56e-13

   1. Initial program 53.8%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 57.9%

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

   10. Applied rewrites 57.9%

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

   12. Applied rewrites 57.9%

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

   if 1.56e-13 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in cos2phi around 0

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

   8. Applied rewrites 74.7%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.559999959072078 \cdot 10^{-13}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 23.7% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 77.6

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

5. Applied rewrites 77.6%

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

6. Taylor expanded in cos2phi around inf

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

8. Applied rewrites 24.2%

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

10. Applied rewrites 24.2%

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

12. Applied rewrites 24.2%

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

Alternative 25: 23.7% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 77.6

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

5. Applied rewrites 77.6%

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

6. Taylor expanded in cos2phi around inf

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

8. Applied rewrites 24.2%

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

10. Applied rewrites 24.2%

    \[\leadsto u0 \cdot \frac{alphax \cdot alphax}{\color{blue}{cos2phi}} \]
11. Add Preprocessing

Alternative 26: 23.7% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 77.6

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

5. Applied rewrites 77.6%

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

6. Taylor expanded in cos2phi around inf

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

8. Applied rewrites 24.2%

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

10. Applied rewrites 24.2%

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

Reproduce

herbie shell --seed 2024232 
(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)))))