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

Percentage Accurate: 61.0% → 98.5%

Time: 19.0s

Alternatives: 25

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: 61.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(alphax \cdot alphax\right) \cdot \left(\left(alphay \cdot alphay\right) \cdot \frac{\mathsf{log1p}\left(-u0\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)}\right) \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (alphax * alphax) * ((alphay * alphay) * (log1pf(-u0) / (0.0f - fmaf((alphax * alphax), sin2phi, (cos2phi * (alphay * alphay))))));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(alphax * alphax) * Float32(Float32(alphay * alphay) * Float32(log1p(Float32(-u0)) / Float32(Float32(0.0) - fma(Float32(alphax * alphax), sin2phi, Float32(cos2phi * Float32(alphay * alphay)))))))
end

Derivation

1. Initial program 60.3%

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

   1. frac-add N/A

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

   2. associate-/r/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-lowering-*.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 egg-rr 98.6%

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

5. Final simplification 98.6%

   \[\leadsto \left(alphax \cdot alphax\right) \cdot \left(\left(alphay \cdot alphay\right) \cdot \frac{\mathsf{log1p}\left(-u0\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)}\right) \]
6. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ alphay \cdot \left(\frac{\mathsf{log1p}\left(-u0\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot \left(alphax \cdot alphay\right)\right)\right) \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return alphay * ((log1pf(-u0) / (0.0f - fmaf((alphax * alphax), sin2phi, (cos2phi * (alphay * alphay))))) * (alphax * (alphax * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(alphay * Float32(Float32(log1p(Float32(-u0)) / Float32(Float32(0.0) - fma(Float32(alphax * alphax), sin2phi, Float32(cos2phi * Float32(alphay * alphay))))) * Float32(alphax * Float32(alphax * alphay))))
end

Derivation

1. Initial program 60.3%

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

   1. frac-add N/A

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

   2. associate-/r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

   \[\leadsto alphay \cdot \left(\frac{\mathsf{log1p}\left(-u0\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot \left(alphax \cdot alphay\right)\right)\right) \]
6. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  (/
   (log1p (- u0))
   (fma (* alphax alphax) sin2phi (* cos2phi (* alphay alphay))))
  (- (* 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)))) * -(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(alphax * Float32(alphax * Float32(alphay * alphay)))))
end

Derivation

1. Initial program 60.3%

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

   1. distribute-frac-neg N/A

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

   2. frac-add N/A

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

   3. associate-/r/ N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. distribute-lft-neg-in N/A

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

   6. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.3%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return log1pf(-u0) / ((0.0f - (cos2phi / (alphax * alphax))) - (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(0.0) - Float32(cos2phi / Float32(alphax * alphax))) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 60.3%

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

   1. sub-neg N/A

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

   2. accelerator-lowering-log1p.f32 N/A

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

   3. neg-lowering-neg.f32 97.9

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

4. Applied egg-rr 97.9%

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

5. Final simplification 97.9%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]
6. Add Preprocessing

Alternative 5: 92.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ u0 \cdot \left(\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{\mathsf{fma}\left(alphax, alphax \cdot sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(\left(alphax \cdot alphax\right) \cdot \left(0 - alphay \cdot alphay\right)\right)\right) \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  u0
  (*
   (/
    (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0)
    (fma alphax (* alphax sin2phi) (* cos2phi (* alphay alphay))))
   (* (* alphax alphax) (- 0.0 (* alphay alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 * ((fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f) / fmaf(alphax, (alphax * sin2phi), (cos2phi * (alphay * alphay)))) * ((alphax * alphax) * (0.0f - (alphay * alphay))));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(u0 * Float32(Float32(fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)) / fma(alphax, Float32(alphax * sin2phi), Float32(cos2phi * Float32(alphay * alphay)))) * Float32(Float32(alphax * alphax) * Float32(Float32(0.0) - Float32(alphay * alphay)))))
end

Derivation

1. Initial program 60.3%

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

   1. frac-add N/A

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

   2. associate-/r/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-lowering-*.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 egg-rr 98.6%

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

5. Taylor expanded in u0 around 0

   \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   2. sub-neg N/A

      \[\leadsto \left(\frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   11. accelerator-lowering-fma.f32 93.6

       \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

7. Simplified 93.6%

   \[\leadsto \left(\frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]
8. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(alphax \cdot alphax\right)\right)} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\left(u0 \cdot \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)}\right)} \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(alphax \cdot alphax\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(alphax \cdot alphax\right)\right)\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(alphax \cdot alphax\right)\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto u0 \cdot \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \color{blue}{\left(\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)\right)}\right) \]

   6. associate-*l* N/A

      \[\leadsto u0 \cdot \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \color{blue}{\left(\left(\left(alphax \cdot alphax\right) \cdot alphay\right) \cdot alphay\right)}\right) \]

   7. associate-*r* N/A

      \[\leadsto u0 \cdot \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(\color{blue}{\left(alphax \cdot \left(alphax \cdot alphay\right)\right)} \cdot alphay\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto u0 \cdot \color{blue}{\left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(\left(alphax \cdot \left(alphax \cdot alphay\right)\right) \cdot alphay\right)\right)} \]

9. Applied egg-rr 93.6%

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

10. Final simplification 93.6%

    \[\leadsto u0 \cdot \left(\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{\mathsf{fma}\left(alphax, alphax \cdot sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(\left(alphax \cdot alphax\right) \cdot \left(0 - alphay \cdot alphay\right)\right)\right) \]
11. Add Preprocessing

Alternative 6: 92.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.3%

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

   1. frac-add N/A

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

   2. associate-/r/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-lowering-*.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 egg-rr 98.6%

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

5. Taylor expanded in u0 around 0

   \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   2. sub-neg N/A

      \[\leadsto \left(\frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   11. accelerator-lowering-fma.f32 93.6

       \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

7. Simplified 93.6%

   \[\leadsto \left(\frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)}\right)} \cdot \left(alphax \cdot alphax\right) \]

   2. associate-/l* N/A

      \[\leadsto \left(\left(alphay \cdot alphay\right) \cdot \color{blue}{\left(u0 \cdot \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)}\right)}\right) \cdot \left(alphax \cdot alphax\right) \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\left(alphay \cdot alphay\right) \cdot u0\right) \cdot \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)}\right)} \cdot \left(alphax \cdot alphax\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(\left(alphay \cdot alphay\right) \cdot u0\right) \cdot \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)}\right)} \cdot \left(alphax \cdot alphax\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \left(\color{blue}{\left(\left(alphay \cdot alphay\right) \cdot u0\right)} \cdot \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)}\right) \cdot \left(alphax \cdot alphax\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\color{blue}{\left(alphay \cdot alphay\right)} \cdot u0\right) \cdot \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1}{0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)}\right) \cdot \left(alphax \cdot alphax\right) \]

   7. frac-2neg N/A

      \[\leadsto \left(\left(\left(alphay \cdot alphay\right) \cdot u0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)\right)}{\mathsf{neg}\left(\left(0 - \left(\left(alphax \cdot alphax\right) \cdot sin2phi + cos2phi \cdot \left(alphay \cdot alphay\right)\right)\right)\right)}}\right) \cdot \left(alphax \cdot alphax\right) \]

   8. sub0-neg N/A

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

   9. remove-double-neg N/A

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

   10. /-lowering-/.f32 N/A

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

9. Applied egg-rr 93.6%

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

10. Final simplification 93.6%

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

Alternative 7: 93.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in u0 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 93.0

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

5. Simplified 93.0%

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

Alternative 8: 81.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{\left(-u0\right) \cdot t\_0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot t\_0}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0)))
   (if (<= (/ sin2phi (* alphay alphay)) 4.99999991225835e-14)
     (* (* alphax alphax) (/ (* (- u0) t_0) cos2phi))
     (/ (* (* u0 (* alphay alphay)) t_0) (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.99999991225835e-14f) {
		tmp = (alphax * alphax) * ((-u0 * t_0) / cos2phi);
	} else {
		tmp = ((u0 * (alphay * alphay)) * t_0) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.99999991225835e-14))
		tmp = Float32(Float32(alphax * alphax) * Float32(Float32(Float32(-u0) * t_0) / cos2phi));
	else
		tmp = Float32(Float32(Float32(u0 * Float32(alphay * alphay)) * t_0) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999991e-14

   1. Initial program 52.6%

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

      1. frac-add N/A

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

      2. associate-/r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.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 egg-rr 98.4%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      8. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      11. accelerator-lowering-fma.f32 94.7

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   7. Simplified 94.7%

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   8. Taylor expanded in alphax around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f32 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f32 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(u0\right)\right)} \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}{cos2phi} \cdot \left(alphax \cdot alphax\right) \]

      6. neg-lowering-neg.f32 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(u0\right)\right)} \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}{cos2phi} \cdot \left(alphax \cdot alphax\right) \]

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f32 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f32 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f32 76.1

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

   10. Simplified 76.1%

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

   if 4.99999991e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.8%

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

      1. frac-add N/A

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

      2. associate-/r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.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 egg-rr 98.7%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      8. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      11. accelerator-lowering-fma.f32 93.2

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   7. Simplified 93.2%

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   8. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

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

      3. /-lowering-/.f32 N/A

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

   10. Simplified 84.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{t\_0 \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot t\_0}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0)))
   (if (<= (/ sin2phi (* alphay alphay)) 4.99999991225835e-14)
     (/ (* t_0 (* u0 (* alphax alphax))) (- cos2phi))
     (/ (* (* u0 (* alphay alphay)) t_0) (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.99999991225835e-14f) {
		tmp = (t_0 * (u0 * (alphax * alphax))) / -cos2phi;
	} else {
		tmp = ((u0 * (alphay * alphay)) * t_0) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.99999991225835e-14))
		tmp = Float32(Float32(t_0 * Float32(u0 * Float32(alphax * alphax))) / Float32(-cos2phi));
	else
		tmp = Float32(Float32(Float32(u0 * Float32(alphay * alphay)) * t_0) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999991e-14

   1. Initial program 52.6%

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

      1. frac-add N/A

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

      2. associate-/r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.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 egg-rr 98.4%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      8. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      11. accelerator-lowering-fma.f32 94.7

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   7. Simplified 94.7%

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   8. Taylor expanded in alphax around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphax}^{2} \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}{cos2phi}\right)} \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphax}^{2} \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}{cos2phi}\right)} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{alphax}^{2} \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}{cos2phi}}\right) \]

   10. Simplified 76.0%

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

   if 4.99999991e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.8%

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

      1. frac-add N/A

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

      2. associate-/r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.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 egg-rr 98.7%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      8. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      11. accelerator-lowering-fma.f32 93.2

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   7. Simplified 93.2%

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   8. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

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

      3. /-lowering-/.f32 N/A

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

   10. Simplified 84.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.999999980020986e-13)
   (/
    (*
     (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0)
     (* u0 (* alphax alphax)))
    (- cos2phi))
   (*
    u0
    (/
     (fma
      (* alphay alphay)
      (* u0 (fma u0 0.3333333333333333 0.5))
      (* alphay alphay))
     sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.999999980020986e-13f) {
		tmp = (fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f) * (u0 * (alphax * alphax))) / -cos2phi;
	} else {
		tmp = u0 * (fmaf((alphay * alphay), (u0 * fmaf(u0, 0.3333333333333333f, 0.5f)), (alphay * alphay)) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.999999980020986e-13))
		tmp = Float32(Float32(fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)) * Float32(u0 * Float32(alphax * alphax))) / Float32(-cos2phi));
	else
		tmp = Float32(u0 * Float32(fma(Float32(alphay * alphay), Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5))), Float32(alphay * alphay)) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999998e-13

   1. Initial program 53.9%

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

      1. frac-add N/A

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

      2. associate-/r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.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 egg-rr 98.4%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      8. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      11. accelerator-lowering-fma.f32 94.3

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   7. Simplified 94.3%

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   8. Taylor expanded in alphax around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphax}^{2} \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}{cos2phi}\right)} \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphax}^{2} \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}{cos2phi}\right)} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{alphax}^{2} \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}{cos2phi}}\right) \]

   10. Simplified 73.1%

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

   if 4.99999998e-13 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.7%

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

   3. Taylor expanded in u0 around 0

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

      1. *-lowering-*.f32 N/A

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

   5. Simplified 90.8%

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

   6. Taylor expanded in sin2phi around inf

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

      1. /-lowering-/.f32 N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f32 84.8

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

   8. Simplified 84.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(alphax \cdot alphax, t\_0, alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphay \cdot alphay, t\_0, alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* u0 (fma u0 0.3333333333333333 0.5))))
   (if (<= (/ sin2phi (* alphay alphay)) 4.99999991225835e-14)
     (/ (* u0 (fma (* alphax alphax) t_0 (* alphax alphax))) cos2phi)
     (* u0 (/ (fma (* alphay alphay) t_0 (* alphay alphay)) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = u0 * fmaf(u0, 0.3333333333333333f, 0.5f);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.99999991225835e-14f) {
		tmp = (u0 * fmaf((alphax * alphax), t_0, (alphax * alphax))) / cos2phi;
	} else {
		tmp = u0 * (fmaf((alphay * alphay), t_0, (alphay * alphay)) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5)))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.99999991225835e-14))
		tmp = Float32(Float32(u0 * fma(Float32(alphax * alphax), t_0, Float32(alphax * alphax))) / cos2phi);
	else
		tmp = Float32(u0 * Float32(fma(Float32(alphay * alphay), t_0, Float32(alphay * alphay)) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999991e-14

   1. Initial program 52.6%

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

   3. Taylor expanded in u0 around 0

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

      1. *-lowering-*.f32 N/A

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

   5. Simplified 93.5%

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

   6. Taylor expanded in sin2phi around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 N/A

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

      5. associate-/l* N/A

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

      6. *-lowering-*.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f32 N/A

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

      11. /-lowering-/.f32 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 75.5

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

   8. Simplified 75.5%

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

   9. Taylor expanded in cos2phi around 0

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

       1. /-lowering-/.f32 N/A

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

       2. *-lowering-*.f32 N/A

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

       3. accelerator-lowering-fma.f32 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f32 N/A

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

       6. *-lowering-*.f32 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. accelerator-lowering-fma.f32 N/A

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

       10. unpow2 N/A

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

       11. *-lowering-*.f32 75.5

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

   11. Simplified 75.5%

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

   if 4.99999991e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.8%

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

   3. Taylor expanded in u0 around 0

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

      1. *-lowering-*.f32 N/A

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

   5. Simplified 90.7%

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

   6. Taylor expanded in sin2phi around inf

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

      1. /-lowering-/.f32 N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f32 83.4

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

   8. Simplified 83.4%

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

Alternative 12: 79.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphax \cdot alphax, t\_0, alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphay \cdot alphay, t\_0, alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* u0 (fma u0 0.3333333333333333 0.5))))
   (if (<= (/ sin2phi (* alphay alphay)) 4.99999991225835e-14)
     (* u0 (/ (fma (* alphax alphax) t_0 (* alphax alphax)) cos2phi))
     (* u0 (/ (fma (* alphay alphay) t_0 (* alphay alphay)) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = u0 * fmaf(u0, 0.3333333333333333f, 0.5f);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.99999991225835e-14f) {
		tmp = u0 * (fmaf((alphax * alphax), t_0, (alphax * alphax)) / cos2phi);
	} else {
		tmp = u0 * (fmaf((alphay * alphay), t_0, (alphay * alphay)) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5)))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.99999991225835e-14))
		tmp = Float32(u0 * Float32(fma(Float32(alphax * alphax), t_0, Float32(alphax * alphax)) / cos2phi));
	else
		tmp = Float32(u0 * Float32(fma(Float32(alphay * alphay), t_0, Float32(alphay * alphay)) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999991e-14

   1. Initial program 52.6%

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

   3. Taylor expanded in u0 around 0

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

      1. *-lowering-*.f32 N/A

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

   5. Simplified 93.5%

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

   6. Taylor expanded in cos2phi around inf

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

      1. /-lowering-/.f32 N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f32 75.4

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

   8. Simplified 75.4%

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

   if 4.99999991e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.8%

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

   3. Taylor expanded in u0 around 0

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

      1. *-lowering-*.f32 N/A

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

   5. Simplified 90.7%

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

   6. Taylor expanded in sin2phi around inf

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

      1. /-lowering-/.f32 N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f32 83.4

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

   8. Simplified 83.4%

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

Alternative 13: 79.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphax \cdot alphax, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.999999980020986e-13)
   (*
    u0
    (/
     (fma
      (* alphax alphax)
      (* u0 (fma u0 0.3333333333333333 0.5))
      (* alphax alphax))
     cos2phi))
   (/
    (* (* alphay alphay) (* u0 (fma u0 (fma u0 0.3333333333333333 0.5) 1.0)))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.999999980020986e-13f) {
		tmp = u0 * (fmaf((alphax * alphax), (u0 * fmaf(u0, 0.3333333333333333f, 0.5f)), (alphax * alphax)) / cos2phi);
	} else {
		tmp = ((alphay * alphay) * (u0 * fmaf(u0, fmaf(u0, 0.3333333333333333f, 0.5f), 1.0f))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.999999980020986e-13))
		tmp = Float32(u0 * Float32(fma(Float32(alphax * alphax), Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5))), Float32(alphax * alphax)) / cos2phi));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * fma(u0, fma(u0, Float32(0.3333333333333333), Float32(0.5)), Float32(1.0)))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999998e-13

   1. Initial program 53.9%

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

   3. Taylor expanded in u0 around 0

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

      1. *-lowering-*.f32 N/A

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

   5. Simplified 92.9%

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

   6. Taylor expanded in cos2phi around inf

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

      1. /-lowering-/.f32 N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f32 72.5

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

   8. Simplified 72.5%

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

   if 4.99999998e-13 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.7%

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

   3. Taylor expanded in cos2phi around 0

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

      1. /-lowering-/.f32 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 57.7

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

   5. Simplified 57.7%

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

   6. Taylor expanded in u0 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f32 83.5

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

   8. Simplified 83.5%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f32 N/A

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

      4. *-lowering-*.f32 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. *-lowering-*.f32 84.7

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

   10. Applied egg-rr 84.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphax \cdot alphax, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.0001500000071246177:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.0001500000071246177)
   (/
    (* u0 (fma u0 0.5 1.0))
    (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (/
    (*
     (* u0 (* alphay alphay))
     (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0))
    (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.0001500000071246177f) {
		tmp = (u0 * fmaf(u0, 0.5f, 1.0f)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = ((u0 * (alphay * alphay)) * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.0001500000071246177))
		tmp = Float32(Float32(u0 * fma(u0, Float32(0.5), Float32(1.0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(Float32(u0 * Float32(alphay * alphay)) * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0))) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.50000007e-4

   1. Initial program 57.0%

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

   3. Taylor expanded in u0 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f32 87.4

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

   5. Simplified 87.4%

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

   if 1.50000007e-4 < sin2phi

   1. Initial program 62.8%

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

      1. frac-add N/A

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

      2. associate-/r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.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 egg-rr 98.7%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      8. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      11. accelerator-lowering-fma.f32 93.8

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   7. Simplified 93.8%

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   8. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

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

      3. /-lowering-/.f32 N/A

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

   10. Simplified 93.4%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.0001500000071246177:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.0001500000071246177:\\ \;\;\;\;\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.0001500000071246177)
   (*
    (fma u0 0.5 1.0)
    (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))
   (/
    (*
     (* u0 (* alphay alphay))
     (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0))
    (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.0001500000071246177f) {
		tmp = fmaf(u0, 0.5f, 1.0f) * (u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay))));
	} else {
		tmp = ((u0 * (alphay * alphay)) * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.0001500000071246177))
		tmp = Float32(fma(u0, Float32(0.5), Float32(1.0)) * Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))));
	else
		tmp = Float32(Float32(Float32(u0 * Float32(alphay * alphay)) * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0))) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.50000007e-4

   1. Initial program 57.0%

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

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

      14. +-lowering-+.f32 N/A

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

   5. Simplified 87.3%

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

   if 1.50000007e-4 < sin2phi

   1. Initial program 62.8%

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

      1. frac-add N/A

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

      2. associate-/r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.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 egg-rr 98.7%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      8. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      11. accelerator-lowering-fma.f32 93.8

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   7. Simplified 93.8%

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   8. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

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

      3. /-lowering-/.f32 N/A

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

   10. Simplified 93.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.0001500000071246177:\\ \;\;\;\;\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 91.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in u0 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f32 91.4

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

5. Simplified 91.4%

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

Alternative 17: 77.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.0000000036274937e-15)
   (* alphax (/ (* u0 alphax) cos2phi))
   (/
    (* (* alphay alphay) (* u0 (fma u0 (fma u0 0.3333333333333333 0.5) 1.0)))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.0000000036274937e-15f) {
		tmp = alphax * ((u0 * alphax) / cos2phi);
	} else {
		tmp = ((alphay * alphay) * (u0 * fmaf(u0, fmaf(u0, 0.3333333333333333f, 0.5f), 1.0f))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.0000000036274937e-15))
		tmp = Float32(alphax * Float32(Float32(u0 * alphax) / cos2phi));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * fma(u0, fma(u0, Float32(0.3333333333333333), Float32(0.5)), Float32(1.0)))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1e-15

   1. Initial program 50.7%

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 78.6

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

   5. Simplified 78.6%

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

   6. Taylor expanded in sin2phi around 0

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

      1. /-lowering-/.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 67.5

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

   8. Simplified 67.5%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 67.9

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

   10. Applied egg-rr 67.9%

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

   if 1e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.0%

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

   3. Taylor expanded in cos2phi around 0

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

      1. /-lowering-/.f32 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 56.0

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

   5. Simplified 56.0%

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

   6. Taylor expanded in u0 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f32 81.0

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

   8. Simplified 81.0%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f32 N/A

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

      4. *-lowering-*.f32 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. *-lowering-*.f32 82.1

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

   10. Applied egg-rr 82.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 77.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right) \cdot \frac{u0}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.0000000036274937e-15)
   (* alphax (/ (* u0 alphax) cos2phi))
   (*
    (* alphay alphay)
    (* (fma u0 (fma u0 0.3333333333333333 0.5) 1.0) (/ u0 sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.0000000036274937e-15f) {
		tmp = alphax * ((u0 * alphax) / cos2phi);
	} else {
		tmp = (alphay * alphay) * (fmaf(u0, fmaf(u0, 0.3333333333333333f, 0.5f), 1.0f) * (u0 / sin2phi));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.0000000036274937e-15))
		tmp = Float32(alphax * Float32(Float32(u0 * alphax) / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(fma(u0, fma(u0, Float32(0.3333333333333333), Float32(0.5)), Float32(1.0)) * Float32(u0 / sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1e-15

   1. Initial program 50.7%

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 78.6

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

   5. Simplified 78.6%

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

   6. Taylor expanded in sin2phi around 0

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

      1. /-lowering-/.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 67.5

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

   8. Simplified 67.5%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 67.9

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

   10. Applied egg-rr 67.9%

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

   if 1e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.0%

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

   3. Taylor expanded in cos2phi around 0

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

      1. /-lowering-/.f32 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 56.0

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

   5. Simplified 56.0%

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

   6. Taylor expanded in u0 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f32 81.0

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

   8. Simplified 81.0%

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

      1. associate-/r/ N/A

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

      2. *-lowering-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-lowering-*.f32 N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. /-lowering-/.f32 N/A

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

      9. *-lowering-*.f32 82.1

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

   10. Applied egg-rr 82.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right) \cdot \frac{u0}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 76.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right) \cdot \frac{u0}{sin2phi}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.0000000036274937e-15)
   (* alphax (/ (* u0 alphax) cos2phi))
   (*
    alphay
    (*
     alphay
     (* (fma u0 (fma u0 0.3333333333333333 0.5) 1.0) (/ u0 sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.0000000036274937e-15f) {
		tmp = alphax * ((u0 * alphax) / cos2phi);
	} else {
		tmp = alphay * (alphay * (fmaf(u0, fmaf(u0, 0.3333333333333333f, 0.5f), 1.0f) * (u0 / sin2phi)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.0000000036274937e-15))
		tmp = Float32(alphax * Float32(Float32(u0 * alphax) / cos2phi));
	else
		tmp = Float32(alphay * Float32(alphay * Float32(fma(u0, fma(u0, Float32(0.3333333333333333), Float32(0.5)), Float32(1.0)) * Float32(u0 / sin2phi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1e-15

   1. Initial program 50.7%

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 78.6

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

   5. Simplified 78.6%

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

   6. Taylor expanded in sin2phi around 0

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

      1. /-lowering-/.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 67.5

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

   8. Simplified 67.5%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 67.9

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

   10. Applied egg-rr 67.9%

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

   if 1e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.0%

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

   3. Taylor expanded in cos2phi around 0

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

      1. /-lowering-/.f32 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 56.0

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

   5. Simplified 56.0%

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

   6. Taylor expanded in u0 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f32 81.0

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

   8. Simplified 81.0%

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

      1. associate-/r/ N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f32 N/A

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

      4. *-lowering-*.f32 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-lowering-*.f32 N/A

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

      8. accelerator-lowering-fma.f32 N/A

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

      9. accelerator-lowering-fma.f32 N/A

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

      10. /-lowering-/.f32 82.1

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

   10. Applied egg-rr 82.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right) \cdot \frac{u0}{sin2phi}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 84.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\mathsf{fma}\left(\frac{1}{alphax \cdot alphax}, cos2phi, \frac{sin2phi}{alphay \cdot alphay}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.9999999949504854e-6)
   (/ u0 (fma (/ 1.0 (* alphax alphax)) cos2phi (/ sin2phi (* alphay alphay))))
   (/
    (*
     (* u0 (* alphay alphay))
     (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0))
    (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.9999999949504854e-6f) {
		tmp = u0 / fmaf((1.0f / (alphax * alphax)), cos2phi, (sin2phi / (alphay * alphay)));
	} else {
		tmp = ((u0 * (alphay * alphay)) * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.9999999949504854e-6))
		tmp = Float32(u0 / fma(Float32(Float32(1.0) / Float32(alphax * alphax)), cos2phi, Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(Float32(u0 * Float32(alphay * alphay)) * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0))) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999999e-6

   1. Initial program 56.1%

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 75.6

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

   5. Simplified 75.6%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. /-lowering-/.f32 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. *-lowering-*.f32 75.7

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

   7. Applied egg-rr 75.7%

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

   if 1.99999999e-6 < sin2phi

   1. Initial program 63.0%

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

      1. frac-add N/A

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

      2. associate-/r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.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 egg-rr 98.7%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      8. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      11. accelerator-lowering-fma.f32 93.6

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   7. Simplified 93.6%

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   8. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

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

      3. /-lowering-/.f32 N/A

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

   10. Simplified 92.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\mathsf{fma}\left(\frac{1}{alphax \cdot alphax}, cos2phi, \frac{sin2phi}{alphay \cdot alphay}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 84.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.9999999949504854e-6)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (/
    (*
     (* u0 (* alphay alphay))
     (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0))
    (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.9999999949504854e-6f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = ((u0 * (alphay * alphay)) * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.9999999949504854e-6))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(Float32(u0 * Float32(alphay * alphay)) * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0))) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999999e-6

   1. Initial program 56.1%

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 75.6

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

   5. Simplified 75.6%

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

   if 1.99999999e-6 < sin2phi

   1. Initial program 63.0%

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

      1. frac-add N/A

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

      2. associate-/r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.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 egg-rr 98.7%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      8. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

      11. accelerator-lowering-fma.f32 93.6

          \[\leadsto \left(\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   7. Simplified 93.6%

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot alphax\right) \]

   8. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

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

      3. /-lowering-/.f32 N/A

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

   10. Simplified 92.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 67.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.999999980020986e-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)) <= 4.999999980020986e-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)) <= 4.999999980020986e-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(4.999999980020986e-13))
		tmp = Float32(alphax * Float32(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(4.999999980020986e-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)) < 4.99999998e-13

   1. Initial program 53.9%

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 76.4

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

   5. Simplified 76.4%

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

   6. Taylor expanded in sin2phi around 0

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

      1. /-lowering-/.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 62.3

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

   8. Simplified 62.3%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 62.6

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

   10. Applied egg-rr 62.6%

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

   if 4.99999998e-13 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.7%

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 76.2

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

   5. Simplified 76.2%

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

   6. Taylor expanded in sin2phi around inf

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

      1. /-lowering-/.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 71.8

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

   8. Simplified 71.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 67.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.999999980020986e-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)) <= 4.999999980020986e-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)) <= 4.999999980020986e-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(4.999999980020986e-13))
		tmp = Float32(alphax * Float32(Float32(u0 * alphax) / cos2phi));
	else
		tmp = Float32(u0 * Float32(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(4.999999980020986e-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)) < 4.99999998e-13

   1. Initial program 53.9%

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 76.4

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

   5. Simplified 76.4%

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

   6. Taylor expanded in sin2phi around 0

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

      1. /-lowering-/.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 62.3

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

   8. Simplified 62.3%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 62.6

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

   10. Applied egg-rr 62.6%

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

   if 4.99999998e-13 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.7%

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 76.2

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

   5. Simplified 76.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. /-lowering-/.f32 N/A

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

      7. *-lowering-*.f32 76.3

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

   7. Applied egg-rr 76.3%

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

   8. Taylor expanded in alphay around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-lowering-*.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 71.7

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

   10. Simplified 71.7%

       \[\leadsto \color{blue}{u0 \cdot \frac{alphay \cdot alphay}{sin2phi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 23.5% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in u0 around 0

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

   1. /-lowering-/.f32 N/A

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

   2. +-commutative N/A

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

   3. +-lowering-+.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. unpow2 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. /-lowering-/.f32 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f32 76.3

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

5. Simplified 76.3%

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

6. Taylor expanded in sin2phi around 0

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

   1. /-lowering-/.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 26.1

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

8. Simplified 26.1%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f32 26.2

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

10. Applied egg-rr 26.2%

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

Alternative 25: 23.5% 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 60.3%

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

3. Taylor expanded in u0 around 0

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

   1. /-lowering-/.f32 N/A

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

   2. +-commutative N/A

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

   3. +-lowering-+.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. unpow2 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. /-lowering-/.f32 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f32 76.3

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

5. Simplified 76.3%

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

6. Taylor expanded in sin2phi around 0

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

   1. /-lowering-/.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 26.1

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

8. Simplified 26.1%

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

   1. associate-/l* N/A

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

   2. associate-*l* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. /-lowering-/.f32 26.2

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

10. Applied egg-rr 26.2%

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

Reproduce

herbie shell --seed 2024195 
(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)))))