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

Percentage Accurate: 60.8% → 98.4%

Time: 17.0s

Alternatives: 22

Speedup: 5.4×

Specification

\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32(cos2phi / Float32(alphax * Float32(-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 98.3

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

4. Applied egg-rr 98.3%

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

5. Final simplification 98.3%

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

Alternative 2: 96.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 20:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay}{sin2phi}\right) \cdot \left(-alphay\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 20.0)
   (/
    (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0)
    (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (* (* (log1p (- u0)) (/ alphay sin2phi)) (- alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 20.0f) {
		tmp = fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (log1pf(-u0) * (alphay / sin2phi)) * -alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(20.0))
		tmp = Float32(fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) * Float32(alphay / sin2phi)) * Float32(-alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 20

   1. Initial program 56.2%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 95.5

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

   5. Simplified 95.5%

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

   if 20 < sin2phi

   1. Initial program 64.5%

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

   3. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 98.3

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

   5. Simplified 98.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      8. neg-lowering-neg.f32 98.8

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

   7. Applied egg-rr 98.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 20:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay}{sin2phi}\right) \cdot \left(-alphay\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f32 93.8

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

5. Simplified 93.8%

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

Alternative 4: 93.1% 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. 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 98.3

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. 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 93.6

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

7. Simplified 93.6%

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

Alternative 5: 91.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f32 92.4

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

5. Simplified 92.4%

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

Alternative 6: 87.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(fma(Float32(u0 * u0), Float32(0.5), u0) / 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. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f32 92.4

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

5. Simplified 92.4%

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

6. Taylor expanded in u0 around 0

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

8. Simplified 89.3%

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

Alternative 7: 87.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f32 89.3

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

5. Simplified 89.3%

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

Alternative 8: 87.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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

   \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
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. +-lowering-+.f32 N/A

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

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

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

   15. unpow2 N/A

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

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

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

   17. /-lowering-/.f32 N/A

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

5. Simplified 89.1%

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

Alternative 9: 83.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 20:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 20.0)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (/
    (*
     (* alphay alphay)
     (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 20.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = ((alphay * alphay) * fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(20.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 20

   1. Initial program 56.2%

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

   3. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 76.1

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

   5. Simplified 76.1%

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

   if 20 < sin2phi

   1. Initial program 64.5%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 92.0

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

   5. Simplified 92.0%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 92.4

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

   8. Simplified 92.4%

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

Alternative 10: 80.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)\\ \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot t\_0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot t\_0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0)))
   (if (<= sin2phi 3.250000038259134e-16)
     (/ (* (* alphax alphax) t_0) cos2phi)
     (/ (* (* alphay alphay) t_0) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0);
	float tmp;
	if (sin2phi <= 3.250000038259134e-16f) {
		tmp = ((alphax * alphax) * t_0) / cos2phi;
	} else {
		tmp = ((alphay * alphay) * t_0) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.250000038259134e-16))
		tmp = Float32(Float32(Float32(alphax * alphax) * t_0) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * t_0) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.25000004e-16

   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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 96.0

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

   5. Simplified 96.0%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 80.3

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

   8. Simplified 80.3%

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

   if 3.25000004e-16 < sin2phi

   1. Initial program 61.4%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 93.0

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

   5. Simplified 93.0%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 84.7

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

   8. Simplified 84.7%

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

Alternative 11: 79.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.250000038259134e-16)
   (/
    (*
     (* alphax alphax)
     (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0))
    cos2phi)
   (*
    alphay
    (*
     alphay
     (/ (fma u0 (* u0 (fma u0 0.3333333333333333 0.5)) u0) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.250000038259134e-16f) {
		tmp = ((alphax * alphax) * fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0)) / cos2phi;
	} else {
		tmp = alphay * (alphay * (fmaf(u0, (u0 * fmaf(u0, 0.3333333333333333f, 0.5f)), u0) / sin2phi));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.250000038259134e-16))
		tmp = Float32(Float32(Float32(alphax * alphax) * fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0)) / cos2phi);
	else
		tmp = Float32(alphay * Float32(alphay * Float32(fma(u0, Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5))), u0) / sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.25000004e-16

   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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 96.0

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

   5. Simplified 96.0%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 80.3

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

   8. Simplified 80.3%

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

   if 3.25000004e-16 < sin2phi

   1. Initial program 61.4%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 91.6

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

   5. Simplified 91.6%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 83.6

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

   8. Simplified 83.6%

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

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. accelerator-lowering-fma.f32 83.7

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

   10. Applied egg-rr 83.7%

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

Alternative 12: 79.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.250000038259134e-16)
   (/
    (* (* alphax alphax) (fma (* u0 u0) (fma u0 0.3333333333333333 0.5) u0))
    cos2phi)
   (*
    alphay
    (*
     alphay
     (/ (fma u0 (* u0 (fma u0 0.3333333333333333 0.5)) u0) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.250000038259134e-16f) {
		tmp = ((alphax * alphax) * fmaf((u0 * u0), fmaf(u0, 0.3333333333333333f, 0.5f), u0)) / cos2phi;
	} else {
		tmp = alphay * (alphay * (fmaf(u0, (u0 * fmaf(u0, 0.3333333333333333f, 0.5f)), u0) / sin2phi));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.250000038259134e-16))
		tmp = Float32(Float32(Float32(alphax * alphax) * fma(Float32(u0 * u0), fma(u0, Float32(0.3333333333333333), Float32(0.5)), u0)) / cos2phi);
	else
		tmp = Float32(alphay * Float32(alphay * Float32(fma(u0, Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5))), u0) / sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.25000004e-16

   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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 94.6

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

   5. Simplified 94.6%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 79.0

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

   8. Simplified 79.0%

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

   if 3.25000004e-16 < sin2phi

   1. Initial program 61.4%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 91.6

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

   5. Simplified 91.6%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 83.6

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

   8. Simplified 83.6%

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

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. accelerator-lowering-fma.f32 83.7

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

   10. Applied egg-rr 83.7%

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

Alternative 13: 76.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.250000038259134e-16)
   (/ (* alphax (* u0 alphax)) cos2phi)
   (*
    alphay
    (*
     alphay
     (/ (fma u0 (* u0 (fma u0 0.3333333333333333 0.5)) u0) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.250000038259134e-16f) {
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	} else {
		tmp = alphay * (alphay * (fmaf(u0, (u0 * fmaf(u0, 0.3333333333333333f, 0.5f)), u0) / sin2phi));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.250000038259134e-16))
		tmp = Float32(Float32(alphax * Float32(u0 * alphax)) / cos2phi);
	else
		tmp = Float32(alphay * Float32(alphay * Float32(fma(u0, Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5))), u0) / sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.25000004e-16

   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

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 74.8

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

   5. Simplified 74.8%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 65.8

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

   8. Simplified 65.8%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f32 65.9

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

   10. Applied egg-rr 65.9%

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

   if 3.25000004e-16 < sin2phi

   1. Initial program 61.4%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 91.6

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

   5. Simplified 91.6%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 83.6

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

   8. Simplified 83.6%

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

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. accelerator-lowering-fma.f32 83.7

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

   10. Applied egg-rr 83.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.250000038259134e-16)
   (/ (* alphax (* u0 alphax)) cos2phi)
   (/ (* u0 (fma alphay alphay (* 0.5 (* u0 (* alphay alphay))))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.250000038259134e-16f) {
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	} else {
		tmp = (u0 * fmaf(alphay, alphay, (0.5f * (u0 * (alphay * alphay))))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.250000038259134e-16))
		tmp = Float32(Float32(alphax * Float32(u0 * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(u0 * fma(alphay, alphay, Float32(Float32(0.5) * Float32(u0 * Float32(alphay * alphay))))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.25000004e-16

   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

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 74.8

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

   5. Simplified 74.8%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 65.8

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

   8. Simplified 65.8%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f32 65.9

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

   10. Applied egg-rr 65.9%

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

   if 3.25000004e-16 < sin2phi

   1. Initial program 61.4%

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

   3. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 89.3

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

   5. Simplified 89.3%

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

   6. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 80.9

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

   8. Simplified 80.9%

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

   9. Taylor expanded in sin2phi around 0

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

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

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

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

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

       3. +-commutative N/A

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

       4. unpow2 N/A

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

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

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

       6. *-commutative N/A

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

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

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f32 80.9

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

   11. Simplified 80.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.250000038259134e-16)
   (/ (* alphax (* u0 alphax)) cos2phi)
   (/ (* (* alphay alphay) (fma u0 (* u0 0.5) u0)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.250000038259134e-16f) {
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	} else {
		tmp = ((alphay * alphay) * fmaf(u0, (u0 * 0.5f), u0)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.250000038259134e-16))
		tmp = Float32(Float32(alphax * Float32(u0 * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * fma(u0, Float32(u0 * Float32(0.5)), u0)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.25000004e-16

   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

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 74.8

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

   5. Simplified 74.8%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 65.8

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

   8. Simplified 65.8%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f32 65.9

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

   10. Applied egg-rr 65.9%

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

   if 3.25000004e-16 < sin2phi

   1. Initial program 61.4%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 91.6

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

   5. Simplified 91.6%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 83.6

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

   8. Simplified 83.6%

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

   9. Taylor expanded in u0 around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f32 80.9

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

   11. Simplified 80.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.250000038259134e-16)
   (/ (* alphax (* u0 alphax)) cos2phi)
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.250000038259134e-16f) {
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 3.250000038259134e-16) then
        tmp = (alphax * (u0 * alphax)) / cos2phi
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.250000038259134e-16))
		tmp = Float32(Float32(alphax * Float32(u0 * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(3.250000038259134e-16))
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.25000004e-16

   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

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 74.8

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

   5. Simplified 74.8%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 65.8

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

   8. Simplified 65.8%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f32 65.9

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

   10. Applied egg-rr 65.9%

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

   if 3.25000004e-16 < sin2phi

   1. Initial program 61.4%

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

      1. 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

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

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

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 77.2

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

   7. Simplified 77.2%

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

   8. Taylor expanded in alphay around 0

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

      1. /-lowering-/.f32 70.7

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

   10. Simplified 70.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 66.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.250000038259134e-16)
   (/ (* u0 (* alphax alphax)) cos2phi)
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.250000038259134e-16f) {
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 3.250000038259134e-16) then
        tmp = (u0 * (alphax * alphax)) / cos2phi
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.250000038259134e-16))
		tmp = Float32(Float32(u0 * Float32(alphax * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(3.250000038259134e-16))
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.25000004e-16

   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

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 74.8

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

   5. Simplified 74.8%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 65.8

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

   8. Simplified 65.8%

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

   if 3.25000004e-16 < sin2phi

   1. Initial program 61.4%

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

      1. 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

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

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

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 77.2

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

   7. Simplified 77.2%

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

   8. Taylor expanded in alphay around 0

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

      1. /-lowering-/.f32 70.7

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

   10. Simplified 70.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 66.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.250000038259134e-16)
   (* (* u0 alphax) (/ alphax cos2phi))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.250000038259134e-16f) {
		tmp = (u0 * alphax) * (alphax / cos2phi);
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 3.250000038259134e-16) then
        tmp = (u0 * alphax) * (alphax / cos2phi)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.250000038259134e-16))
		tmp = Float32(Float32(u0 * alphax) * Float32(alphax / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(3.250000038259134e-16))
		tmp = (u0 * alphax) * (alphax / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.25000004e-16

   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

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

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      8. *-lowering-*.f32 74.8

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. Simplified 74.8%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{u0 \cdot {alphax}^{2}}}{cos2phi} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{u0 \cdot {alphax}^{2}}}{cos2phi} \]

      4. unpow2 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}}{cos2phi} \]

      5. *-lowering-*.f32 65.8

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}}{cos2phi} \]

   8. Simplified 65.8%

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(u0 \cdot alphax\right) \cdot alphax}}{cos2phi} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u0 \cdot alphax\right)} \cdot \frac{alphax}{cos2phi} \]

      5. /-lowering-/.f32 65.8

         \[\leadsto \left(u0 \cdot alphax\right) \cdot \color{blue}{\frac{alphax}{cos2phi}} \]

   10. Applied egg-rr 65.8%

       \[\leadsto \color{blue}{\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}} \]

   if 3.25000004e-16 < sin2phi

   1. Initial program 61.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. 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

         \[\leadsto \color{blue}{\left(\frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

   4. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{log1p}\left(-u0\right)}{-\mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\color{blue}{\frac{u0}{{alphax}^{2} \cdot sin2phi + {alphay}^{2} \cdot cos2phi}} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \left(\color{blue}{\frac{u0}{{alphax}^{2} \cdot sin2phi + {alphay}^{2} \cdot cos2phi}} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{u0}{\color{blue}{{alphay}^{2} \cdot cos2phi + {alphax}^{2} \cdot sin2phi}} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      3. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0}{\color{blue}{\mathsf{fma}\left({alphay}^{2}, cos2phi, {alphax}^{2} \cdot sin2phi\right)}} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{u0}{\mathsf{fma}\left(\color{blue}{alphay \cdot alphay}, cos2phi, {alphax}^{2} \cdot sin2phi\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{u0}{\mathsf{fma}\left(\color{blue}{alphay \cdot alphay}, cos2phi, {alphax}^{2} \cdot sin2phi\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{u0}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, \color{blue}{{alphax}^{2} \cdot sin2phi}\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{u0}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, \color{blue}{\left(alphax \cdot alphax\right)} \cdot sin2phi\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      8. *-lowering-*.f32 77.2

         \[\leadsto \left(\frac{u0}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, \color{blue}{\left(alphax \cdot alphax\right)} \cdot sin2phi\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

   7. Simplified 77.2%

      \[\leadsto \left(\color{blue}{\frac{u0}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, \left(alphax \cdot alphax\right) \cdot sin2phi\right)}} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

   8. Taylor expanded in alphay around 0

      \[\leadsto \color{blue}{\frac{u0}{sin2phi}} \cdot \left(alphay \cdot alphay\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 70.7

         \[\leadsto \color{blue}{\frac{u0}{sin2phi}} \cdot \left(alphay \cdot alphay\right) \]

   10. Simplified 70.7%

       \[\leadsto \color{blue}{\frac{u0}{sin2phi}} \cdot \left(alphay \cdot alphay\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 66.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \left(u0 \cdot alphay\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.250000038259134e-16)
   (* (* u0 alphax) (/ alphax cos2phi))
   (* (/ alphay sin2phi) (* u0 alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.250000038259134e-16f) {
		tmp = (u0 * alphax) * (alphax / cos2phi);
	} else {
		tmp = (alphay / sin2phi) * (u0 * alphay);
	}
	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 <= 3.250000038259134e-16) then
        tmp = (u0 * alphax) * (alphax / cos2phi)
    else
        tmp = (alphay / sin2phi) * (u0 * alphay)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.250000038259134e-16))
		tmp = Float32(Float32(u0 * alphax) * Float32(alphax / cos2phi));
	else
		tmp = Float32(Float32(alphay / sin2phi) * Float32(u0 * alphay));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(3.250000038259134e-16))
		tmp = (u0 * alphax) * (alphax / cos2phi);
	else
		tmp = (alphay / sin2phi) * (u0 * alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.25000004e-16

   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

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      8. *-lowering-*.f32 74.8

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. Simplified 74.8%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{u0 \cdot {alphax}^{2}}}{cos2phi} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{u0 \cdot {alphax}^{2}}}{cos2phi} \]

      4. unpow2 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}}{cos2phi} \]

      5. *-lowering-*.f32 65.8

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}}{cos2phi} \]

   8. Simplified 65.8%

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(u0 \cdot alphax\right) \cdot alphax}}{cos2phi} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u0 \cdot alphax\right)} \cdot \frac{alphax}{cos2phi} \]

      5. /-lowering-/.f32 65.8

         \[\leadsto \left(u0 \cdot alphax\right) \cdot \color{blue}{\frac{alphax}{cos2phi}} \]

   10. Applied egg-rr 65.8%

       \[\leadsto \color{blue}{\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}} \]

   if 3.25000004e-16 < sin2phi

   1. Initial program 61.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\log \left(1 - u0\right) \cdot {alphay}^{2}}}{sin2phi}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\log \left(1 + \color{blue}{-1 \cdot u0}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      8. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot u0\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      10. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

      13. *-lowering-*.f32 89.3

          \[\leadsto -\mathsf{log1p}\left(-u0\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

   6. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} - -1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} - -1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)} \]

      2. sub-neg N/A

         \[\leadsto u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \left(\mathsf{neg}\left(-1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{{alphay}^{2}}{sin2phi}\right)\right)}\right)\right)\right) \]

      4. remove-double-neg N/A

         \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

      5. accelerator-lowering-fma.f32 N/A

         \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{{alphay}^{2} \cdot u0}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right)} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}}, \frac{{alphay}^{2}}{sin2phi}\right) \]

      7. *-commutative N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{u0 \cdot {alphay}^{2}}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{u0 \cdot {alphay}^{2}}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

      9. unpow2 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \color{blue}{\left(alphay \cdot alphay\right)}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \color{blue}{\left(alphay \cdot alphay\right)}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

      12. unpow2 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

      13. *-lowering-*.f32 80.9

          \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

   8. Simplified 80.9%

      \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{alphay \cdot alphay}{sin2phi}\right)} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto u0 \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}} \]

       2. unpow2 N/A

          \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

       3. *-lowering-*.f32 70.6

          \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   11. Simplified 70.6%

       \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto u0 \cdot \color{blue}{\left(alphay \cdot \frac{alphay}{sin2phi}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(u0 \cdot alphay\right) \cdot \frac{alphay}{sin2phi}} \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \color{blue}{\left(u0 \cdot alphay\right) \cdot \frac{alphay}{sin2phi}} \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \color{blue}{\left(u0 \cdot alphay\right)} \cdot \frac{alphay}{sin2phi} \]

       5. /-lowering-/.f32 70.6

          \[\leadsto \left(u0 \cdot alphay\right) \cdot \color{blue}{\frac{alphay}{sin2phi}} \]

   13. Applied egg-rr 70.6%

       \[\leadsto \color{blue}{\left(u0 \cdot alphay\right) \cdot \frac{alphay}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \left(u0 \cdot alphay\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 66.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.250000038259134e-16)
   (* (* u0 alphax) (/ alphax cos2phi))
   (* u0 (* alphay (/ alphay sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.250000038259134e-16f) {
		tmp = (u0 * alphax) * (alphax / cos2phi);
	} else {
		tmp = u0 * (alphay * (alphay / sin2phi));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 3.250000038259134e-16) then
        tmp = (u0 * alphax) * (alphax / cos2phi)
    else
        tmp = u0 * (alphay * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.250000038259134e-16))
		tmp = Float32(Float32(u0 * alphax) * Float32(alphax / cos2phi));
	else
		tmp = Float32(u0 * Float32(alphay * Float32(alphay / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(3.250000038259134e-16))
		tmp = (u0 * alphax) * (alphax / cos2phi);
	else
		tmp = u0 * (alphay * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.25000004e-16

   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

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      8. *-lowering-*.f32 74.8

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. Simplified 74.8%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{u0 \cdot {alphax}^{2}}}{cos2phi} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{u0 \cdot {alphax}^{2}}}{cos2phi} \]

      4. unpow2 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}}{cos2phi} \]

      5. *-lowering-*.f32 65.8

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}}{cos2phi} \]

   8. Simplified 65.8%

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(u0 \cdot alphax\right) \cdot alphax}}{cos2phi} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u0 \cdot alphax\right)} \cdot \frac{alphax}{cos2phi} \]

      5. /-lowering-/.f32 65.8

         \[\leadsto \left(u0 \cdot alphax\right) \cdot \color{blue}{\frac{alphax}{cos2phi}} \]

   10. Applied egg-rr 65.8%

       \[\leadsto \color{blue}{\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}} \]

   if 3.25000004e-16 < sin2phi

   1. Initial program 61.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\log \left(1 - u0\right) \cdot {alphay}^{2}}}{sin2phi}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\log \left(1 + \color{blue}{-1 \cdot u0}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      8. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot u0\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      10. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

      13. *-lowering-*.f32 89.3

          \[\leadsto -\mathsf{log1p}\left(-u0\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

   6. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} - -1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} - -1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)} \]

      2. sub-neg N/A

         \[\leadsto u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \left(\mathsf{neg}\left(-1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{{alphay}^{2}}{sin2phi}\right)\right)}\right)\right)\right) \]

      4. remove-double-neg N/A

         \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

      5. accelerator-lowering-fma.f32 N/A

         \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{{alphay}^{2} \cdot u0}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right)} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}}, \frac{{alphay}^{2}}{sin2phi}\right) \]

      7. *-commutative N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{u0 \cdot {alphay}^{2}}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{u0 \cdot {alphay}^{2}}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

      9. unpow2 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \color{blue}{\left(alphay \cdot alphay\right)}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \color{blue}{\left(alphay \cdot alphay\right)}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

      12. unpow2 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

      13. *-lowering-*.f32 80.9

          \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

   8. Simplified 80.9%

      \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{alphay \cdot alphay}{sin2phi}\right)} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto u0 \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}} \]

       2. unpow2 N/A

          \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

       3. *-lowering-*.f32 70.6

          \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   11. Simplified 70.6%

       \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto u0 \cdot \color{blue}{\left(alphay \cdot \frac{alphay}{sin2phi}\right)} \]

       2. *-commutative N/A

          \[\leadsto u0 \cdot \color{blue}{\left(\frac{alphay}{sin2phi} \cdot alphay\right)} \]

       3. *-lowering-*.f32 N/A

          \[\leadsto u0 \cdot \color{blue}{\left(\frac{alphay}{sin2phi} \cdot alphay\right)} \]

       4. /-lowering-/.f32 70.6

          \[\leadsto u0 \cdot \left(\color{blue}{\frac{alphay}{sin2phi}} \cdot alphay\right) \]

   13. Applied egg-rr 70.6%

       \[\leadsto u0 \cdot \color{blue}{\left(\frac{alphay}{sin2phi} \cdot alphay\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.250000038259134 \cdot 10^{-16}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 59.4% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right) \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (* u0 (* alphay (/ alphay sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 * (alphay * (alphay / sin2phi));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 * (alphay * (alphay / sin2phi))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(u0 * Float32(alphay * Float32(alphay / sin2phi)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = u0 * (alphay * (alphay / sin2phi));
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 cos2phi around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

   2. neg-lowering-neg.f32 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\log \left(1 - u0\right) \cdot {alphay}^{2}}}{sin2phi}\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\log \left(1 + \color{blue}{-1 \cdot u0}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

   8. accelerator-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot u0\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

   10. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

   11. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

   13. *-lowering-*.f32 70.7

       \[\leadsto -\mathsf{log1p}\left(-u0\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

5. Simplified 70.7%

   \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} - -1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} - -1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)} \]

   2. sub-neg N/A

      \[\leadsto u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \left(\mathsf{neg}\left(-1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right)\right)} \]

   3. mul-1-neg N/A

      \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{{alphay}^{2}}{sin2phi}\right)\right)}\right)\right)\right) \]

   4. remove-double-neg N/A

      \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

   5. accelerator-lowering-fma.f32 N/A

      \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{{alphay}^{2} \cdot u0}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right)} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}}, \frac{{alphay}^{2}}{sin2phi}\right) \]

   7. *-commutative N/A

      \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{u0 \cdot {alphay}^{2}}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{u0 \cdot {alphay}^{2}}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

   9. unpow2 N/A

      \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \color{blue}{\left(alphay \cdot alphay\right)}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \color{blue}{\left(alphay \cdot alphay\right)}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

   11. /-lowering-/.f32 N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

   12. unpow2 N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

   13. *-lowering-*.f32 64.2

       \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

8. Simplified 64.2%

   \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{alphay \cdot alphay}{sin2phi}\right)} \]

9. Taylor expanded in u0 around 0

   \[\leadsto u0 \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}} \]
10. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto u0 \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}} \]

    2. unpow2 N/A

       \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

    3. *-lowering-*.f32 56.5

       \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

11. Simplified 56.5%

    \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]
12. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto u0 \cdot \color{blue}{\left(alphay \cdot \frac{alphay}{sin2phi}\right)} \]

    2. *-commutative N/A

       \[\leadsto u0 \cdot \color{blue}{\left(\frac{alphay}{sin2phi} \cdot alphay\right)} \]

    3. *-lowering-*.f32 N/A

       \[\leadsto u0 \cdot \color{blue}{\left(\frac{alphay}{sin2phi} \cdot alphay\right)} \]

    4. /-lowering-/.f32 56.5

       \[\leadsto u0 \cdot \left(\color{blue}{\frac{alphay}{sin2phi}} \cdot alphay\right) \]

13. Applied egg-rr 56.5%

    \[\leadsto u0 \cdot \color{blue}{\left(\frac{alphay}{sin2phi} \cdot alphay\right)} \]

14. Final simplification 56.5%

    \[\leadsto u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right) \]
15. Add Preprocessing

Alternative 22: 59.4% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right) \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (* alphay (* u0 (/ alphay sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return alphay * (u0 * (alphay / sin2phi));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = alphay * (u0 * (alphay / sin2phi))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(alphay * Float32(u0 * Float32(alphay / sin2phi)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = alphay * (u0 * (alphay / sin2phi));
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 cos2phi around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

   2. neg-lowering-neg.f32 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\log \left(1 - u0\right) \cdot {alphay}^{2}}}{sin2phi}\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\log \left(1 + \color{blue}{-1 \cdot u0}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

   8. accelerator-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot u0\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

   10. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

   11. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

   13. *-lowering-*.f32 70.7

       \[\leadsto -\mathsf{log1p}\left(-u0\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

5. Simplified 70.7%

   \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} - -1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} - -1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)} \]

   2. sub-neg N/A

      \[\leadsto u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \left(\mathsf{neg}\left(-1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right)\right)} \]

   3. mul-1-neg N/A

      \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{{alphay}^{2}}{sin2phi}\right)\right)}\right)\right)\right) \]

   4. remove-double-neg N/A

      \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

   5. accelerator-lowering-fma.f32 N/A

      \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{{alphay}^{2} \cdot u0}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right)} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}}, \frac{{alphay}^{2}}{sin2phi}\right) \]

   7. *-commutative N/A

      \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{u0 \cdot {alphay}^{2}}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{u0 \cdot {alphay}^{2}}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

   9. unpow2 N/A

      \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \color{blue}{\left(alphay \cdot alphay\right)}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \color{blue}{\left(alphay \cdot alphay\right)}}{sin2phi}, \frac{{alphay}^{2}}{sin2phi}\right) \]

   11. /-lowering-/.f32 N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

   12. unpow2 N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

   13. *-lowering-*.f32 64.2

       \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

8. Simplified 64.2%

   \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{alphay \cdot alphay}{sin2phi}\right)} \]

9. Taylor expanded in u0 around 0

   \[\leadsto u0 \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}} \]
10. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto u0 \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}} \]

    2. unpow2 N/A

       \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

    3. *-lowering-*.f32 56.5

       \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

11. Simplified 56.5%

    \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{\frac{alphay \cdot alphay}{sin2phi} \cdot u0} \]

    2. associate-/l* N/A

       \[\leadsto \color{blue}{\left(alphay \cdot \frac{alphay}{sin2phi}\right)} \cdot u0 \]

    3. associate-*l* N/A

       \[\leadsto \color{blue}{alphay \cdot \left(\frac{alphay}{sin2phi} \cdot u0\right)} \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{alphay \cdot \left(\frac{alphay}{sin2phi} \cdot u0\right)} \]

    5. *-lowering-*.f32 N/A

       \[\leadsto alphay \cdot \color{blue}{\left(\frac{alphay}{sin2phi} \cdot u0\right)} \]

    6. /-lowering-/.f32 56.5

       \[\leadsto alphay \cdot \left(\color{blue}{\frac{alphay}{sin2phi}} \cdot u0\right) \]

13. Applied egg-rr 56.5%

    \[\leadsto \color{blue}{alphay \cdot \left(\frac{alphay}{sin2phi} \cdot u0\right)} \]

14. Final simplification 56.5%

    \[\leadsto alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024201 
(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)))))