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

Percentage Accurate: 60.8% → 98.4%

Time: 17.1s

Alternatives: 26

Speedup: 3.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.8% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

   1. 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 98.5

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 95.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.8100000023841858:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot \left(-alphay\right)}{sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (- 1.0 u0) 0.8100000023841858)
   (* (log1p (- u0)) (/ (* alphay (- alphay)) sin2phi))
   (/
    (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) {
	float tmp;
	if ((1.0f - u0) <= 0.8100000023841858f) {
		tmp = log1pf(-u0) * ((alphay * -alphay) / sin2phi);
	} else {
		tmp = fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.8100000023841858))
		tmp = Float32(log1p(Float32(-u0)) * Float32(Float32(alphay * Float32(-alphay)) / sin2phi));
	else
		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))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.810000002

   1. Initial program 97.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 87.1

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

   5. Simplified 87.1%

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

   if 0.810000002 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 53.7%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      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 96.9

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

   5. Simplified 96.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.8100000023841858:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot \left(-alphay\right)}{sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% 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 57.9%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 92.1

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

5. Simplified 92.1%

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

Alternative 4: 80.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 8.00000036650964e-18)
   (/
    (fma u0 (* u0 (fma u0 0.3333333333333333 0.5)) u0)
    (/ cos2phi (* alphax alphax)))
   (-
    (*
     (/ (* alphay alphay) sin2phi)
     (* u0 (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.00000036650964e-18f) {
		tmp = fmaf(u0, (u0 * fmaf(u0, 0.3333333333333333f, 0.5f)), u0) / (cos2phi / (alphax * alphax));
	} else {
		tmp = -(((alphay * alphay) / sin2phi) * (u0 * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.00000036650964e-18))
		tmp = Float32(fma(u0, Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5))), u0) / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(-Float32(Float32(Float32(alphay * alphay) / sin2phi) * Float32(u0 * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000037e-18

   1. Initial program 52.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.9

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

   4. Applied egg-rr 98.9%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f32 98.8

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 90.5

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

   9. Simplified 90.5%

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

   10. Taylor expanded in alphay around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f32 72.3

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

   12. Simplified 72.3%

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

   if 8.00000037e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. 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 92.1

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

   5. Simplified 92.1%

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

   6. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f32 85.9

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

   8. Simplified 85.9%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)\right) \cdot \frac{alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 8.00000036650964e-18)
   (/
    (fma u0 (* u0 (fma u0 0.3333333333333333 0.5)) u0)
    (/ cos2phi (* alphax alphax)))
   (*
    (* alphay (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0))
    (/ alphay sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.00000036650964e-18f) {
		tmp = fmaf(u0, (u0 * fmaf(u0, 0.3333333333333333f, 0.5f)), u0) / (cos2phi / (alphax * alphax));
	} else {
		tmp = (alphay * fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0)) * (alphay / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.00000036650964e-18))
		tmp = Float32(fma(u0, Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5))), u0) / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(alphay * fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0)) * Float32(alphay / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000037e-18

   1. Initial program 52.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.9

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

   4. Applied egg-rr 98.9%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f32 98.8

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 90.5

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

   9. Simplified 90.5%

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

   10. Taylor expanded in alphay around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f32 72.3

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

   12. Simplified 72.3%

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

   if 8.00000037e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. 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 92.1

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

   5. Simplified 92.1%

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

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f32 91.9

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

   7. Applied egg-rr 91.9%

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

   8. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 85.8

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

   10. Simplified 85.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)\right) \cdot \frac{alphay}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)\right) \cdot \frac{alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 8.00000036650964e-18)
   (/
    (*
     (* u0 (* alphax alphax))
     (fma u0 (fma u0 -0.3333333333333333 -0.5) -1.0))
    (- cos2phi))
   (*
    (* alphay (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0))
    (/ alphay sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.00000036650964e-18f) {
		tmp = ((u0 * (alphax * alphax)) * fmaf(u0, fmaf(u0, -0.3333333333333333f, -0.5f), -1.0f)) / -cos2phi;
	} else {
		tmp = (alphay * fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0)) * (alphay / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.00000036650964e-18))
		tmp = Float32(Float32(Float32(u0 * Float32(alphax * alphax)) * fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))) / Float32(-cos2phi));
	else
		tmp = Float32(Float32(alphay * fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0)) * Float32(alphay / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000037e-18

   1. Initial program 52.7%

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

   3. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f32 90.3

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

   5. Simplified 90.3%

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

   6. Taylor expanded in cos2phi around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

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

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

      14. sub-neg N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

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

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

      18. mul-1-neg N/A

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

      19. neg-lowering-neg.f32 72.2

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

   8. Simplified 72.2%

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

   if 8.00000037e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. 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 92.1

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

   5. Simplified 92.1%

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

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f32 91.9

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

   7. Applied egg-rr 91.9%

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

   8. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 85.8

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

   10. Simplified 85.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)\right) \cdot \frac{alphay}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(alphay, 0.5, alphay \cdot \left(u0 \cdot 0.3333333333333333\right)\right), alphay\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 8.00000036650964e-18)
   (/
    (*
     (* u0 (* alphax alphax))
     (fma u0 (fma u0 -0.3333333333333333 -0.5) -1.0))
    (- cos2phi))
   (*
    (/ alphay sin2phi)
    (*
     u0
     (fma u0 (fma alphay 0.5 (* alphay (* u0 0.3333333333333333))) alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.00000036650964e-18f) {
		tmp = ((u0 * (alphax * alphax)) * fmaf(u0, fmaf(u0, -0.3333333333333333f, -0.5f), -1.0f)) / -cos2phi;
	} else {
		tmp = (alphay / sin2phi) * (u0 * fmaf(u0, fmaf(alphay, 0.5f, (alphay * (u0 * 0.3333333333333333f))), alphay));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.00000036650964e-18))
		tmp = Float32(Float32(Float32(u0 * Float32(alphax * alphax)) * fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))) / Float32(-cos2phi));
	else
		tmp = Float32(Float32(alphay / sin2phi) * Float32(u0 * fma(u0, fma(alphay, Float32(0.5), Float32(alphay * Float32(u0 * Float32(0.3333333333333333)))), alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000037e-18

   1. Initial program 52.7%

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

   3. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f32 90.3

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

   5. Simplified 90.3%

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

   6. Taylor expanded in cos2phi around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

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

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

      14. sub-neg N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

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

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

      18. mul-1-neg N/A

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

      19. neg-lowering-neg.f32 72.2

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

   8. Simplified 72.2%

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

   if 8.00000037e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. 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 92.1

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

   5. Simplified 92.1%

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

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f32 91.9

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

   7. Applied egg-rr 91.9%

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

   8. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f32 84.2

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

   10. Simplified 84.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(alphay, 0.5, alphay \cdot \left(u0 \cdot 0.3333333333333333\right)\right), alphay\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.2% 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 57.9%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 90.1

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

5. Simplified 90.1%

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

Alternative 9: 79.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot t\_0}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(\left(-u0\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (fma u0 (fma u0 -0.3333333333333333 -0.5) -1.0)))
   (if (<= (/ sin2phi (* alphay alphay)) 8.00000036650964e-18)
     (/ (* (* u0 (* alphax alphax)) t_0) (- cos2phi))
     (* (/ (* alphay alphay) sin2phi) (* (- u0) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = fmaf(u0, fmaf(u0, -0.3333333333333333f, -0.5f), -1.0f);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.00000036650964e-18f) {
		tmp = ((u0 * (alphax * alphax)) * t_0) / -cos2phi;
	} else {
		tmp = ((alphay * alphay) / sin2phi) * (-u0 * t_0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.00000036650964e-18))
		tmp = Float32(Float32(Float32(u0 * Float32(alphax * alphax)) * t_0) / Float32(-cos2phi));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * Float32(Float32(-u0) * t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000037e-18

   1. Initial program 52.7%

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

   3. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f32 90.3

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

   5. Simplified 90.3%

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

   6. Taylor expanded in cos2phi around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

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

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

      14. sub-neg N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

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

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

      18. mul-1-neg N/A

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

      19. neg-lowering-neg.f32 72.2

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

   8. Simplified 72.2%

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

   if 8.00000037e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. 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 92.1

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

   5. Simplified 92.1%

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

   6. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f32 84.1

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

   8. Simplified 84.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 8.00000036650964e-18)
   (/
    (* (* alphax alphax) (fma (* u0 u0) (fma u0 0.3333333333333333 0.5) u0))
    cos2phi)
   (*
    (/ (* alphay alphay) sin2phi)
    (* (- u0) (fma u0 (fma u0 -0.3333333333333333 -0.5) -1.0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.00000036650964e-18f) {
		tmp = ((alphax * alphax) * fmaf((u0 * u0), fmaf(u0, 0.3333333333333333f, 0.5f), u0)) / cos2phi;
	} else {
		tmp = ((alphay * alphay) / sin2phi) * (-u0 * fmaf(u0, fmaf(u0, -0.3333333333333333f, -0.5f), -1.0f));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.00000036650964e-18))
		tmp = Float32(Float32(Float32(alphax * alphax) * fma(Float32(u0 * u0), fma(u0, Float32(0.3333333333333333), Float32(0.5)), u0)) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * Float32(Float32(-u0) * fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000037e-18

   1. Initial program 52.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.9

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

   4. Applied egg-rr 98.9%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f32 98.8

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 90.5

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

   9. Simplified 90.5%

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

   10. Taylor expanded in alphay around inf

       \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot \left(u0 + {u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)}{cos2phi}} \]
   11. 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 72.1

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

   12. Simplified 72.1%

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

   if 8.00000037e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. 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 92.1

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

   5. Simplified 92.1%

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

   6. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f32 84.1

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

   8. Simplified 84.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999999e-8

   1. Initial program 54.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 74.5

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

   5. Simplified 74.5%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f32 74.6

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

   7. Applied egg-rr 74.6%

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

   if 1.99999999e-8 < sin2phi

   1. Initial program 60.7%

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

   3. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   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 97.8

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

   5. Simplified 97.8%

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

   6. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f32 91.8

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

   8. Simplified 91.8%

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

4. Final simplification 84.8%

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

Alternative 12: 79.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot t\_0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot t\_0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (fma (* u0 u0) (fma u0 0.3333333333333333 0.5) u0)))
   (if (<= (/ sin2phi (* alphay alphay)) 8.00000036650964e-18)
     (/ (* (* 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, 0.3333333333333333f, 0.5f), u0);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.00000036650964e-18f) {
		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, Float32(0.3333333333333333), Float32(0.5)), u0)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.00000036650964e-18))
		tmp = Float32(Float32(Float32(alphax * alphax) * t_0) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * t_0) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000037e-18

   1. Initial program 52.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.9

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

   4. Applied egg-rr 98.9%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f32 98.8

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 90.5

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

   9. Simplified 90.5%

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

   10. Taylor expanded in alphay around inf

       \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot \left(u0 + {u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)}{cos2phi}} \]
   11. 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 72.1

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

   12. Simplified 72.1%

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

   if 8.00000037e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f32 98.3

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 89.9

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

   9. Simplified 89.9%

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

   10. Taylor expanded in alphay around 0

       \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot \left(u0 + {u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)}{sin2phi}} \]
   11. 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 84.1

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

   12. Simplified 84.1%

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

Alternative 13: 79.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \left(alphay \cdot \mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 8.00000036650964e-18)
   (/
    (* (* alphax alphax) (fma (* u0 u0) (fma u0 0.3333333333333333 0.5) u0))
    cos2phi)
   (*
    (/ alphay sin2phi)
    (* alphay (fma u0 (* u0 (fma u0 0.3333333333333333 0.5)) u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.00000036650964e-18f) {
		tmp = ((alphax * alphax) * fmaf((u0 * u0), fmaf(u0, 0.3333333333333333f, 0.5f), u0)) / cos2phi;
	} else {
		tmp = (alphay / sin2phi) * (alphay * fmaf(u0, (u0 * fmaf(u0, 0.3333333333333333f, 0.5f)), u0));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.00000036650964e-18))
		tmp = Float32(Float32(Float32(alphax * alphax) * fma(Float32(u0 * u0), fma(u0, Float32(0.3333333333333333), Float32(0.5)), u0)) / cos2phi);
	else
		tmp = Float32(Float32(alphay / sin2phi) * Float32(alphay * fma(u0, Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5))), u0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000037e-18

   1. Initial program 52.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.9

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

   4. Applied egg-rr 98.9%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f32 98.8

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 90.5

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

   9. Simplified 90.5%

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

   10. Taylor expanded in alphay around inf

       \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot \left(u0 + {u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)}{cos2phi}} \]
   11. 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 72.1

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

   12. Simplified 72.1%

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

   if 8.00000037e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. 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 92.1

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

   5. Simplified 92.1%

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

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f32 91.9

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

   7. Applied egg-rr 91.9%

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

   8. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 84.1

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

   10. Simplified 84.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \left(alphay \cdot \mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \left(alphay \cdot \mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 8.00000036650964e-18)
   (* alphax (* alphax (/ (fma u0 (* u0 0.5) u0) cos2phi)))
   (*
    (/ alphay sin2phi)
    (* alphay (fma u0 (* u0 (fma u0 0.3333333333333333 0.5)) u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.00000036650964e-18f) {
		tmp = alphax * (alphax * (fmaf(u0, (u0 * 0.5f), u0) / cos2phi));
	} else {
		tmp = (alphay / sin2phi) * (alphay * fmaf(u0, (u0 * fmaf(u0, 0.3333333333333333f, 0.5f)), u0));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.00000036650964e-18))
		tmp = Float32(alphax * Float32(alphax * Float32(fma(u0, Float32(u0 * Float32(0.5)), u0) / cos2phi)));
	else
		tmp = Float32(Float32(alphay / sin2phi) * Float32(alphay * fma(u0, Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5))), u0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000037e-18

   1. Initial program 52.7%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 86.7

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

   5. Simplified 86.7%

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

   6. Taylor expanded in cos2phi around inf

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{u0 + \frac{1}{2} \cdot {u0}^{2}}{cos2phi}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f32 69.5

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

   8. Simplified 69.5%

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

   if 8.00000037e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. 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 92.1

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

   5. Simplified 92.1%

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

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f32 91.9

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

   7. Applied egg-rr 91.9%

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

   8. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 84.1

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

   10. Simplified 84.1%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \left(alphay \cdot \mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 84.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999999e-8

   1. Initial program 54.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 74.5

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

   5. Simplified 74.5%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 74.6

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

   7. Applied egg-rr 74.6%

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

   if 1.99999999e-8 < sin2phi

   1. Initial program 60.7%

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

   3. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   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 97.8

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

   5. Simplified 97.8%

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

   6. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f32 91.8

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

   8. Simplified 91.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 84.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999999e-8

   1. Initial program 54.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 74.5

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

   5. Simplified 74.5%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 74.5

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

   7. Applied egg-rr 74.5%

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

   if 1.99999999e-8 < sin2phi

   1. Initial program 60.7%

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

   3. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   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 97.8

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

   5. Simplified 97.8%

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

   6. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f32 91.8

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

   8. Simplified 91.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 87.4% 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 57.9%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 86.6

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

5. Simplified 86.6%

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

Alternative 18: 87.2% 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 57.9%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
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 86.6%

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

Alternative 19: 76.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(\frac{alphay}{sin2phi} \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, -0.5, -1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 8.00000036650964e-18)
   (* alphax (* alphax (/ (fma u0 (* u0 0.5) u0) cos2phi)))
   (* alphay (* (/ alphay sin2phi) (* (- u0) (fma u0 -0.5 -1.0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.00000036650964e-18f) {
		tmp = alphax * (alphax * (fmaf(u0, (u0 * 0.5f), u0) / cos2phi));
	} else {
		tmp = alphay * ((alphay / sin2phi) * (-u0 * fmaf(u0, -0.5f, -1.0f)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.00000036650964e-18))
		tmp = Float32(alphax * Float32(alphax * Float32(fma(u0, Float32(u0 * Float32(0.5)), u0) / cos2phi)));
	else
		tmp = Float32(alphay * Float32(Float32(alphay / sin2phi) * Float32(Float32(-u0) * fma(u0, Float32(-0.5), Float32(-1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000037e-18

   1. Initial program 52.7%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 86.7

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

   5. Simplified 86.7%

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

   6. Taylor expanded in cos2phi around inf

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{u0 + \frac{1}{2} \cdot {u0}^{2}}{cos2phi}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f32 69.5

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

   8. Simplified 69.5%

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

   if 8.00000037e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. 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 92.1

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

   5. Simplified 92.1%

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

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f32 91.9

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

   7. Applied egg-rr 91.9%

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

   8. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f32 80.9

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

   10. Simplified 80.9%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

       8. accelerator-lowering-fma.f32 81.0

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

   12. Applied egg-rr 81.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(\frac{alphay}{sin2phi} \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, -0.5, -1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 76.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)\right) \cdot \frac{alphay \cdot \left(-alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 8.00000036650964e-18)
   (* alphax (* alphax (/ (fma u0 (* u0 0.5) u0) cos2phi)))
   (* (* u0 (fma u0 -0.5 -1.0)) (/ (* alphay (- alphay)) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.00000036650964e-18f) {
		tmp = alphax * (alphax * (fmaf(u0, (u0 * 0.5f), u0) / cos2phi));
	} else {
		tmp = (u0 * fmaf(u0, -0.5f, -1.0f)) * ((alphay * -alphay) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.00000036650964e-18))
		tmp = Float32(alphax * Float32(alphax * Float32(fma(u0, Float32(u0 * Float32(0.5)), u0) / cos2phi)));
	else
		tmp = Float32(Float32(u0 * fma(u0, Float32(-0.5), Float32(-1.0))) * Float32(Float32(alphay * Float32(-alphay)) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000037e-18

   1. Initial program 52.7%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 86.7

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

   5. Simplified 86.7%

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

   6. Taylor expanded in cos2phi around inf

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{u0 + \frac{1}{2} \cdot {u0}^{2}}{cos2phi}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f32 69.5

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

   8. Simplified 69.5%

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

   if 8.00000037e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. 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 92.1

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

   5. Simplified 92.1%

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

   6. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f32 81.0

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

   8. Simplified 81.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)\right) \cdot \frac{alphay \cdot \left(-alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 76.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \left(u0 \cdot \mathsf{fma}\left(alphay, u0 \cdot 0.5, alphay\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 8.00000036650964e-18)
   (* alphax (* alphax (/ (fma u0 (* u0 0.5) u0) cos2phi)))
   (* (/ alphay sin2phi) (* u0 (fma alphay (* u0 0.5) alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.00000036650964e-18f) {
		tmp = alphax * (alphax * (fmaf(u0, (u0 * 0.5f), u0) / cos2phi));
	} else {
		tmp = (alphay / sin2phi) * (u0 * fmaf(alphay, (u0 * 0.5f), alphay));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.00000036650964e-18))
		tmp = Float32(alphax * Float32(alphax * Float32(fma(u0, Float32(u0 * Float32(0.5)), u0) / cos2phi)));
	else
		tmp = Float32(Float32(alphay / sin2phi) * Float32(u0 * fma(alphay, Float32(u0 * Float32(0.5)), alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000037e-18

   1. Initial program 52.7%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 86.7

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

   5. Simplified 86.7%

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

   6. Taylor expanded in cos2phi around inf

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{u0 + \frac{1}{2} \cdot {u0}^{2}}{cos2phi}} \]

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f32 69.5

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

   8. Simplified 69.5%

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

   if 8.00000037e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. 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 92.1

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

   5. Simplified 92.1%

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

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f32 91.9

         \[\leadsto \left(\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot alphay\right) \cdot \color{blue}{\frac{alphay}{sin2phi}} \]

   7. Applied egg-rr 91.9%

      \[\leadsto \color{blue}{\left(\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot alphay\right) \cdot \frac{alphay}{sin2phi}} \]

   8. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\left(u0 \cdot \left(alphay + \frac{1}{2} \cdot \left(alphay \cdot u0\right)\right)\right)} \cdot \frac{alphay}{sin2phi} \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u0 \cdot \left(alphay + \frac{1}{2} \cdot \left(alphay \cdot u0\right)\right)\right)} \cdot \frac{alphay}{sin2phi} \]

      2. +-commutative N/A

         \[\leadsto \left(u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(alphay \cdot u0\right) + alphay\right)}\right) \cdot \frac{alphay}{sin2phi} \]

      3. *-commutative N/A

         \[\leadsto \left(u0 \cdot \left(\color{blue}{\left(alphay \cdot u0\right) \cdot \frac{1}{2}} + alphay\right)\right) \cdot \frac{alphay}{sin2phi} \]

      4. associate-*l* N/A

         \[\leadsto \left(u0 \cdot \left(\color{blue}{alphay \cdot \left(u0 \cdot \frac{1}{2}\right)} + alphay\right)\right) \cdot \frac{alphay}{sin2phi} \]

      5. *-commutative N/A

         \[\leadsto \left(u0 \cdot \left(alphay \cdot \color{blue}{\left(\frac{1}{2} \cdot u0\right)} + alphay\right)\right) \cdot \frac{alphay}{sin2phi} \]

      6. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(u0 \cdot \color{blue}{\mathsf{fma}\left(alphay, \frac{1}{2} \cdot u0, alphay\right)}\right) \cdot \frac{alphay}{sin2phi} \]

      7. *-commutative N/A

         \[\leadsto \left(u0 \cdot \mathsf{fma}\left(alphay, \color{blue}{u0 \cdot \frac{1}{2}}, alphay\right)\right) \cdot \frac{alphay}{sin2phi} \]

      8. *-lowering-*.f32 81.0

         \[\leadsto \left(u0 \cdot \mathsf{fma}\left(alphay, \color{blue}{u0 \cdot 0.5}, alphay\right)\right) \cdot \frac{alphay}{sin2phi} \]

   10. Simplified 81.0%

       \[\leadsto \color{blue}{\left(u0 \cdot \mathsf{fma}\left(alphay, u0 \cdot 0.5, alphay\right)\right)} \cdot \frac{alphay}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \left(u0 \cdot \mathsf{fma}\left(alphay, u0 \cdot 0.5, alphay\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 69.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 8.00000036650964e-18)
   (* alphax (* alphax (/ (fma u0 (* u0 0.5) u0) cos2phi)))
   (* alphay (* u0 (/ alphay sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.00000036650964e-18f) {
		tmp = alphax * (alphax * (fmaf(u0, (u0 * 0.5f), u0) / cos2phi));
	} else {
		tmp = alphay * (u0 * (alphay / sin2phi));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.00000036650964e-18))
		tmp = Float32(alphax * Float32(alphax * Float32(fma(u0, Float32(u0 * Float32(0.5)), u0) / cos2phi)));
	else
		tmp = Float32(alphay * Float32(u0 * Float32(alphay / sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000037e-18

   1. Initial program 52.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 86.7

         \[\leadsto \frac{\mathsf{fma}\left(u0, \color{blue}{u0 \cdot 0.5}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Simplified 86.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot \left(u0 + \frac{1}{2} \cdot {u0}^{2}\right)}{cos2phi}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{u0 + \frac{1}{2} \cdot {u0}^{2}}{cos2phi}} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(alphax \cdot alphax\right)} \cdot \frac{u0 + \frac{1}{2} \cdot {u0}^{2}}{cos2phi} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{alphax \cdot \left(alphax \cdot \frac{u0 + \frac{1}{2} \cdot {u0}^{2}}{cos2phi}\right)} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{alphax \cdot \left(alphax \cdot \frac{u0 + \frac{1}{2} \cdot {u0}^{2}}{cos2phi}\right)} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto alphax \cdot \color{blue}{\left(alphax \cdot \frac{u0 + \frac{1}{2} \cdot {u0}^{2}}{cos2phi}\right)} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto alphax \cdot \left(alphax \cdot \color{blue}{\frac{u0 + \frac{1}{2} \cdot {u0}^{2}}{cos2phi}}\right) \]

      7. +-commutative N/A

         \[\leadsto alphax \cdot \left(alphax \cdot \frac{\color{blue}{\frac{1}{2} \cdot {u0}^{2} + u0}}{cos2phi}\right) \]

      8. *-commutative N/A

         \[\leadsto alphax \cdot \left(alphax \cdot \frac{\color{blue}{{u0}^{2} \cdot \frac{1}{2}} + u0}{cos2phi}\right) \]

      9. unpow2 N/A

         \[\leadsto alphax \cdot \left(alphax \cdot \frac{\color{blue}{\left(u0 \cdot u0\right)} \cdot \frac{1}{2} + u0}{cos2phi}\right) \]

      10. associate-*r* N/A

          \[\leadsto alphax \cdot \left(alphax \cdot \frac{\color{blue}{u0 \cdot \left(u0 \cdot \frac{1}{2}\right)} + u0}{cos2phi}\right) \]

      11. *-commutative N/A

          \[\leadsto alphax \cdot \left(alphax \cdot \frac{u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot u0\right)} + u0}{cos2phi}\right) \]

      12. accelerator-lowering-fma.f32 N/A

          \[\leadsto alphax \cdot \left(alphax \cdot \frac{\color{blue}{\mathsf{fma}\left(u0, \frac{1}{2} \cdot u0, u0\right)}}{cos2phi}\right) \]

      13. *-commutative N/A

          \[\leadsto alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{1}{2}}, u0\right)}{cos2phi}\right) \]

      14. *-lowering-*.f32 69.5

          \[\leadsto alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, \color{blue}{u0 \cdot 0.5}, u0\right)}{cos2phi}\right) \]

   8. Simplified 69.5%

      \[\leadsto \color{blue}{alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{cos2phi}\right)} \]

   if 8.00000037e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. 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 92.1

          \[\leadsto -\mathsf{log1p}\left(-u0\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   5. Simplified 92.1%

      \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

   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. cancel-sign-sub-inv N/A

         \[\leadsto u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)} \]

      3. metadata-eval N/A

         \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \color{blue}{1} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      4. *-lft-identity 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 81.0

          \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

   8. Simplified 81.0%

      \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{alphay \cdot alphay}{sin2phi}\right)} \]

   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 72.0

          \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   11. Simplified 72.0%

       \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]
   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 72.2

          \[\leadsto alphay \cdot \left(\color{blue}{\frac{alphay}{sin2phi}} \cdot u0\right) \]

   13. Applied egg-rr 72.2%

       \[\leadsto \color{blue}{alphay \cdot \left(\frac{alphay}{sin2phi} \cdot u0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.00000036650964 \cdot 10^{-18}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 84.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.999999987845058e-8)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (-
    (*
     (/ (* alphay alphay) sin2phi)
     (* u0 (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.999999987845058e-8f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = -(((alphay * alphay) / sin2phi) * (u0 * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.999999987845058e-8))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(-Float32(Float32(Float32(alphay * alphay) / sin2phi) * Float32(u0 * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999999e-8

   1. Initial program 54.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 74.5

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. Simplified 74.5%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   if 1.99999999e-8 < sin2phi

   1. Initial program 60.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   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 97.8

          \[\leadsto -\mathsf{log1p}\left(-u0\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   5. Simplified 97.8%

      \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

   6. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \mathsf{neg}\left(\left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \mathsf{neg}\left(\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      11. accelerator-lowering-fma.f32 91.8

          \[\leadsto -\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi} \]

   8. Simplified 91.8%

      \[\leadsto -\color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \cdot \frac{alphay \cdot alphay}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 66.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.5199999517635943 \cdot 10^{-19}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.5199999517635943e-19)
   (* (* u0 alphax) (/ alphax cos2phi))
   (* alphay (* u0 (/ alphay sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.5199999517635943e-19f) {
		tmp = (u0 * alphax) * (alphax / cos2phi);
	} else {
		tmp = alphay * (u0 * (alphay / sin2phi));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.5199999517635943e-19) then
        tmp = (u0 * alphax) * (alphax / cos2phi)
    else
        tmp = alphay * (u0 * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.5199999517635943e-19))
		tmp = Float32(Float32(u0 * alphax) * Float32(alphax / cos2phi));
	else
		tmp = Float32(alphay * Float32(u0 * Float32(alphay / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.5199999517635943e-19))
		tmp = (u0 * alphax) * (alphax / cos2phi);
	else
		tmp = alphay * (u0 * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.51999995e-19

   1. Initial program 52.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. /-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 75.3

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. Simplified 75.3%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   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 63.7

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}}{cos2phi} \]

   8. Simplified 63.7%

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}} \]
   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. *-commutative N/A

         \[\leadsto \color{blue}{\left(alphax \cdot u0\right)} \cdot \frac{alphax}{cos2phi} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(alphax \cdot u0\right)} \cdot \frac{alphax}{cos2phi} \]

      6. /-lowering-/.f32 63.8

         \[\leadsto \left(alphax \cdot u0\right) \cdot \color{blue}{\frac{alphax}{cos2phi}} \]

   10. Applied egg-rr 63.8%

       \[\leadsto \color{blue}{\left(alphax \cdot u0\right) \cdot \frac{alphax}{cos2phi}} \]

   if 1.51999995e-19 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 90.5

          \[\leadsto -\mathsf{log1p}\left(-u0\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   5. Simplified 90.5%

      \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

   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. cancel-sign-sub-inv N/A

         \[\leadsto u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)} \]

      3. metadata-eval N/A

         \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \color{blue}{1} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      4. *-lft-identity 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 79.8

          \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

   8. Simplified 79.8%

      \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{alphay \cdot alphay}{sin2phi}\right)} \]

   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 71.0

          \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   11. Simplified 71.0%

       \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]
   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 71.2

          \[\leadsto alphay \cdot \left(\color{blue}{\frac{alphay}{sin2phi}} \cdot u0\right) \]

   13. Applied egg-rr 71.2%

       \[\leadsto \color{blue}{alphay \cdot \left(\frac{alphay}{sin2phi} \cdot u0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.5199999517635943 \cdot 10^{-19}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 66.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.5199999517635943 \cdot 10^{-19}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.5199999517635943e-19)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* alphay (* u0 (/ alphay sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.5199999517635943e-19f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = alphay * (u0 * (alphay / sin2phi));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.5199999517635943e-19) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = alphay * (u0 * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.5199999517635943e-19))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(alphay * Float32(u0 * Float32(alphay / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.5199999517635943e-19))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = alphay * (u0 * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.51999995e-19

   1. Initial program 52.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. /-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 75.3

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. Simplified 75.3%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   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 63.7

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}}{cos2phi} \]

   8. Simplified 63.7%

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(alphax \cdot alphax\right) \cdot u0}}{cos2phi} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(alphax \cdot alphax\right)} \cdot \frac{u0}{cos2phi} \]

      5. /-lowering-/.f32 63.8

         \[\leadsto \left(alphax \cdot alphax\right) \cdot \color{blue}{\frac{u0}{cos2phi}} \]

   10. Applied egg-rr 63.8%

       \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]

   if 1.51999995e-19 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 90.5

          \[\leadsto -\mathsf{log1p}\left(-u0\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   5. Simplified 90.5%

      \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

   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. cancel-sign-sub-inv N/A

         \[\leadsto u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)} \]

      3. metadata-eval N/A

         \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \color{blue}{1} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      4. *-lft-identity 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 79.8

          \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

   8. Simplified 79.8%

      \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{alphay \cdot alphay}{sin2phi}\right)} \]

   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 71.0

          \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   11. Simplified 71.0%

       \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]
   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 71.2

          \[\leadsto alphay \cdot \left(\color{blue}{\frac{alphay}{sin2phi}} \cdot u0\right) \]

   13. Applied egg-rr 71.2%

       \[\leadsto \color{blue}{alphay \cdot \left(\frac{alphay}{sin2phi} \cdot u0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.5199999517635943 \cdot 10^{-19}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 59.0% 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 57.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 74.2

       \[\leadsto -\mathsf{log1p}\left(-u0\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

5. Simplified 74.2%

   \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

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. cancel-sign-sub-inv N/A

      \[\leadsto u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)} \]

   3. metadata-eval N/A

      \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \color{blue}{1} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

   4. *-lft-identity 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 65.5

       \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

8. Simplified 65.5%

   \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}, \frac{alphay \cdot alphay}{sin2phi}\right)} \]

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 58.5

       \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

11. Simplified 58.5%

    \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]
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 58.6

       \[\leadsto alphay \cdot \left(\color{blue}{\frac{alphay}{sin2phi}} \cdot u0\right) \]

13. Applied egg-rr 58.6%

    \[\leadsto \color{blue}{alphay \cdot \left(\frac{alphay}{sin2phi} \cdot u0\right)} \]

14. Final simplification 58.6%

    \[\leadsto alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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)))))