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

Percentage Accurate: 60.8% → 98.4%

Time: 15.9s

Alternatives: 23

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 63.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.6

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

4. Applied rewrites 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 84.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 6.000000212225132 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\mathsf{fma}\left(\frac{1}{alphay \cdot alphay}, sin2phi, \frac{cos2phi}{alphax \cdot alphax}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 6.000000212225132e-7)
     (/
      u0
      (fma (/ 1.0 (* alphay alphay)) sin2phi (/ cos2phi (* alphax alphax))))
     (/
      (fma u0 (* u0 (fma u0 (fma u0 0.25 0.3333333333333333) 0.5)) u0)
      t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 6.000000212225132e-7f) {
		tmp = u0 / fmaf((1.0f / (alphay * alphay)), sin2phi, (cos2phi / (alphax * alphax)));
	} else {
		tmp = fmaf(u0, (u0 * fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f)), u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(6.000000212225132e-7))
		tmp = Float32(u0 / fma(Float32(Float32(1.0) / Float32(alphay * alphay)), sin2phi, Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(fma(u0, Float32(u0 * fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5))), u0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 6.0000002e-7

   1. Initial program 61.0%

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

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 71.2%

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

   if 6.0000002e-7 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-*.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. lower-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. lower-fma.f32 92.6

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

   5. Applied rewrites 92.6%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-/r* N/A

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

      7. frac-add N/A

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

      8. lower-/.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in alphax around inf

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

      1. lower-/.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 90.4

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

   10. Applied rewrites 90.4%

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

   11. Taylor expanded in u0 around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f32 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f32 92.3

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

   13. Applied rewrites 92.3%

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

Alternative 3: 84.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 6.000000212225132 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\mathsf{fma}\left(\frac{1}{alphay \cdot alphay}, sin2phi, \frac{cos2phi}{alphax \cdot alphax}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 6.000000212225132e-7)
   (/ u0 (fma (/ 1.0 (* alphay alphay)) sin2phi (/ cos2phi (* alphax alphax))))
   (*
    (* u0 (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -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)) <= 6.000000212225132e-7f) {
		tmp = u0 / fmaf((1.0f / (alphay * alphay)), sin2phi, (cos2phi / (alphax * alphax)));
	} else {
		tmp = (u0 * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -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(6.000000212225132e-7))
		tmp = Float32(u0 / fma(Float32(Float32(1.0) / Float32(alphay * alphay)), sin2phi, Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(u0 * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0))) * Float32(Float32(alphay * alphay) / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 6.0000002e-7

   1. Initial program 61.0%

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

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 71.2%

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

   if 6.0000002e-7 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-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. lower-*.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. lower-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. lower-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. lower-/.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. lower-*.f32 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 92.3%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 6.000000212225132 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\mathsf{fma}\left(\frac{1}{alphay \cdot alphay}, sin2phi, \frac{cos2phi}{alphax \cdot alphax}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in u0 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 94.1

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

5. Applied rewrites 94.1%

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

Alternative 5: 84.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 6.000000212225132 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 6.000000212225132e-7)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (*
      (* u0 (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0))
      (/ (* alphay alphay) (- sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 6.000000212225132e-7f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (u0 * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)) * ((alphay * alphay) / -sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(6.000000212225132e-7))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(u0 * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0))) * Float32(Float32(alphay * alphay) / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 6.0000002e-7

   1. Initial program 61.0%

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

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 71.1

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

   5. Applied rewrites 71.1%

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

   if 6.0000002e-7 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lower-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. lower-*.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. lower-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. lower-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. lower-/.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. lower-*.f32 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 92.3%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 6.000000212225132 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;-\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.99999991225835e-15)
   (- (/ (* u0 (fma u0 -0.5 -1.0)) (/ cos2phi (* alphax alphax))))
   (*
    (* u0 (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -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)) <= 4.99999991225835e-15f) {
		tmp = -((u0 * fmaf(u0, -0.5f, -1.0f)) / (cos2phi / (alphax * alphax)));
	} else {
		tmp = (u0 * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -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(4.99999991225835e-15))
		tmp = Float32(-Float32(Float32(u0 * fma(u0, Float32(-0.5), Float32(-1.0))) / Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(u0 * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0))) * Float32(Float32(alphay * alphay) / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999991e-15

   1. Initial program 60.9%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-*.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. lower-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. lower-fma.f32 91.7

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

   5. Applied rewrites 91.7%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-/r* N/A

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

      7. frac-add N/A

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

      8. lower-/.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 91.4

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

   7. Applied rewrites 91.4%

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

   8. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 85.6

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

   10. Applied rewrites 85.6%

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

   11. Taylor expanded in alphax around 0

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

       1. lower-/.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 63.4

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

   13. Applied rewrites 63.4%

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

   if 4.99999991e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.4%

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

   3. Taylor expanded in 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. lower-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. lower-*.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. lower-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. lower-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. lower-/.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. lower-*.f32 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 89.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;-\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;-\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 4.99999991225835e-15)
     (- (/ (* u0 (fma u0 -0.5 -1.0)) (/ cos2phi (* alphax alphax))))
     (/ (fma u0 (* u0 (fma u0 0.3333333333333333 0.5)) u0) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 4.99999991225835e-15f) {
		tmp = -((u0 * fmaf(u0, -0.5f, -1.0f)) / (cos2phi / (alphax * alphax)));
	} else {
		tmp = fmaf(u0, (u0 * fmaf(u0, 0.3333333333333333f, 0.5f)), u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(4.99999991225835e-15))
		tmp = Float32(-Float32(Float32(u0 * fma(u0, Float32(-0.5), Float32(-1.0))) / Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(fma(u0, Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5))), u0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999991e-15

   1. Initial program 60.9%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-*.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. lower-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. lower-fma.f32 91.7

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

   5. Applied rewrites 91.7%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-/r* N/A

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

      7. frac-add N/A

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

      8. lower-/.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 91.4

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

   7. Applied rewrites 91.4%

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

   8. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 85.6

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

   10. Applied rewrites 85.6%

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

   11. Taylor expanded in alphax around 0

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

       1. lower-/.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 63.4

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

   13. Applied rewrites 63.4%

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

   if 4.99999991e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.4%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-*.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. lower-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. lower-fma.f32 92.3

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

   5. Applied rewrites 92.3%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-/r* N/A

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

      7. frac-add N/A

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

      8. lower-/.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 92.3

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in alphax around inf

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

      1. lower-/.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 87.9

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

   10. Applied rewrites 87.9%

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

   11. Taylor expanded in u0 around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f32 88.2

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

   13. Applied rewrites 88.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;-\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.5:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.5)
   (/
    (* u0 (fma u0 0.5 1.0))
    (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (*
    (* u0 (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0))
    (/ (* alphay alphay) (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.5f) {
		tmp = (u0 * fmaf(u0, 0.5f, 1.0f)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (u0 * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)) * ((alphay * alphay) / -sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.5))
		tmp = Float32(Float32(u0 * fma(u0, Float32(0.5), Float32(1.0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(u0 * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0))) * Float32(Float32(alphay * alphay) / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.5

   1. Initial program 58.8%

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

   3. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

   5. Applied rewrites 87.8%

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

   7. Applied rewrites 87.9%

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

   if 1.5 < sin2phi

   1. Initial program 67.4%

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

   3. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

      2. lower-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. lower-*.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. lower-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. lower-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. lower-/.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. lower-*.f32 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 93.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.5:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.550000011920929:\\ \;\;\;\;\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.550000011920929)
   (*
    (fma u0 0.5 1.0)
    (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))
   (*
    (* u0 (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0))
    (/ (* alphay alphay) (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.550000011920929f) {
		tmp = fmaf(u0, 0.5f, 1.0f) * (u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay))));
	} else {
		tmp = (u0 * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)) * ((alphay * alphay) / -sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.550000011920929))
		tmp = Float32(fma(u0, Float32(0.5), Float32(1.0)) * Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))));
	else
		tmp = Float32(Float32(u0 * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0))) * Float32(Float32(alphay * alphay) / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.550000012

   1. Initial program 58.7%

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

   3. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

   5. Applied rewrites 87.6%

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

   if 0.550000012 < sin2phi

   1. Initial program 67.3%

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

   3. Taylor expanded in 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. lower-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. lower-*.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. lower-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. lower-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. lower-/.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. lower-*.f32 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 93.3%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.550000011920929:\\ \;\;\;\;\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in u0 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 92.3

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

5. Applied rewrites 92.3%

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

Alternative 11: 78.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;-\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.99999991225835e-15)
   (- (/ (* u0 (fma u0 -0.5 -1.0)) (/ cos2phi (* alphax alphax))))
   (*
    (* u0 (fma u0 (fma u0 -0.3333333333333333 -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)) <= 4.99999991225835e-15f) {
		tmp = -((u0 * fmaf(u0, -0.5f, -1.0f)) / (cos2phi / (alphax * alphax)));
	} else {
		tmp = (u0 * fmaf(u0, fmaf(u0, -0.3333333333333333f, -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(4.99999991225835e-15))
		tmp = Float32(-Float32(Float32(u0 * fma(u0, Float32(-0.5), Float32(-1.0))) / Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(u0 * fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))) * Float32(Float32(alphay * alphay) / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999991e-15

   1. Initial program 60.9%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-*.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. lower-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. lower-fma.f32 91.7

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

   5. Applied rewrites 91.7%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-/r* N/A

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

      7. frac-add N/A

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

      8. lower-/.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 91.4

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

   7. Applied rewrites 91.4%

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

   8. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 85.6

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

   10. Applied rewrites 85.6%

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

   11. Taylor expanded in alphax around 0

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

       1. lower-/.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 63.4

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

   13. Applied rewrites 63.4%

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

   if 4.99999991e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.4%

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

   3. Taylor expanded in 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. lower-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. lower-*.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. lower-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. lower-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. lower-/.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. lower-*.f32 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 88.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;-\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.99999991225835e-15)
   (* (fma u0 0.5 1.0) (/ (* u0 (* alphax alphax)) cos2phi))
   (*
    (* u0 (fma u0 (fma u0 -0.3333333333333333 -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)) <= 4.99999991225835e-15f) {
		tmp = fmaf(u0, 0.5f, 1.0f) * ((u0 * (alphax * alphax)) / cos2phi);
	} else {
		tmp = (u0 * fmaf(u0, fmaf(u0, -0.3333333333333333f, -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(4.99999991225835e-15))
		tmp = Float32(fma(u0, Float32(0.5), Float32(1.0)) * Float32(Float32(u0 * Float32(alphax * alphax)) / cos2phi));
	else
		tmp = Float32(Float32(u0 * fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))) * Float32(Float32(alphay * alphay) / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999991e-15

   1. Initial program 60.9%

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 63.3%

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

   if 4.99999991e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.4%

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

   3. Taylor expanded in 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. lower-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. lower-*.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. lower-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. lower-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. lower-/.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. lower-*.f32 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 88.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. lower-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. lower-*.f32 87.8

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

5. Applied rewrites 87.8%

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

Alternative 14: 76.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.99999991225835e-15)
   (* (fma u0 0.5 1.0) (/ (* u0 (* alphax alphax)) cos2phi))
   (/ (* (* alphay alphay) (fma u0 (* u0 0.5) u0)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.99999991225835e-15f) {
		tmp = fmaf(u0, 0.5f, 1.0f) * ((u0 * (alphax * alphax)) / cos2phi);
	} else {
		tmp = ((alphay * alphay) * fmaf(u0, (u0 * 0.5f), u0)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.99999991225835e-15))
		tmp = Float32(fma(u0, Float32(0.5), Float32(1.0)) * Float32(Float32(u0 * Float32(alphax * alphax)) / cos2phi));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * fma(u0, Float32(u0 * Float32(0.5)), u0)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999991e-15

   1. Initial program 60.9%

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 63.3%

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

   if 4.99999991e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.4%

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

   3. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in cos2phi around 0

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

   8. Applied rewrites 84.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{t\_0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot t\_0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (fma u0 (* u0 0.5) u0)))
   (if (<= (/ sin2phi (* alphay alphay)) 4.99999991225835e-15)
     (* (* 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 * 0.5f), u0);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.99999991225835e-15f) {
		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(u0, Float32(u0 * Float32(0.5)), u0)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.99999991225835e-15))
		tmp = Float32(Float32(alphax * alphax) * Float32(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)) < 4.99999991e-15

   1. Initial program 60.9%

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 63.2%

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

   if 4.99999991e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.4%

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

   3. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in cos2phi around 0

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

   8. Applied rewrites 84.6%

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

Alternative 16: 68.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(-u0\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.99999991225835e-15)
   (* (* alphax alphax) (/ (fma u0 (* u0 0.5) u0) cos2phi))
   (* (- u0) (/ (* alphay alphay) (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.99999991225835e-15f) {
		tmp = (alphax * alphax) * (fmaf(u0, (u0 * 0.5f), u0) / cos2phi);
	} else {
		tmp = -u0 * ((alphay * alphay) / -sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.99999991225835e-15))
		tmp = Float32(Float32(alphax * alphax) * Float32(fma(u0, Float32(u0 * Float32(0.5)), u0) / cos2phi));
	else
		tmp = Float32(Float32(-u0) * Float32(Float32(alphay * alphay) / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999991e-15

   1. Initial program 60.9%

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 63.2%

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

   if 4.99999991e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.4%

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

   3. Taylor expanded in 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. lower-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. lower-*.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. lower-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. lower-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. lower-/.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. lower-*.f32 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 73.1%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(-u0\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 66.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(-u0\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.99999991225835e-15)
   (/ (* alphax (* u0 alphax)) cos2phi)
   (* (- u0) (/ (* alphay alphay) (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.99999991225835e-15f) {
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	} else {
		tmp = -u0 * ((alphay * alphay) / -sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.99999991225835e-15) then
        tmp = (alphax * (u0 * alphax)) / cos2phi
    else
        tmp = -u0 * ((alphay * alphay) / -sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.99999991225835e-15))
		tmp = Float32(Float32(alphax * Float32(u0 * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(-u0) * Float32(Float32(alphay * alphay) / Float32(-sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.99999991225835e-15))
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	else
		tmp = -u0 * ((alphay * alphay) / -sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999991e-15

   1. Initial program 60.9%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 54.1%

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

   10. Applied rewrites 54.2%

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

   if 4.99999991e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.4%

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

   3. Taylor expanded in 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. lower-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. lower-*.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. lower-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. lower-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. lower-/.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. lower-*.f32 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 73.1%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(-u0\right) \cdot \frac{alphay \cdot alphay}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 66.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.99999991225835e-15)
   (/ (* alphax (* u0 alphax)) cos2phi)
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.99999991225835e-15f) {
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.99999991225835e-15) then
        tmp = (alphax * (u0 * alphax)) / cos2phi
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.99999991225835e-15))
		tmp = Float32(Float32(alphax * Float32(u0 * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.99999991225835e-15))
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999991e-15

   1. Initial program 60.9%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 54.1%

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

   10. Applied rewrites 54.2%

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

   if 4.99999991e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.4%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in cos2phi around 0

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

   8. Applied rewrites 73.1%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 66.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.99999991225835e-15)
   (* alphax (/ (* u0 alphax) cos2phi))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.99999991225835e-15f) {
		tmp = alphax * ((u0 * alphax) / cos2phi);
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.99999991225835e-15) then
        tmp = alphax * ((u0 * alphax) / cos2phi)
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.99999991225835e-15))
		tmp = Float32(alphax * Float32(Float32(u0 * alphax) / cos2phi));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.99999991225835e-15))
		tmp = alphax * ((u0 * alphax) / cos2phi);
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999991e-15

   1. Initial program 60.9%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 54.1%

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

   10. Applied rewrites 54.0%

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

   12. Applied rewrites 54.1%

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

   if 4.99999991e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.4%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in cos2phi around 0

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

   8. Applied rewrites 73.1%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 23.9% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 74.4

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

5. Applied rewrites 74.4%

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

6. Taylor expanded in cos2phi around inf

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

8. Applied rewrites 21.2%

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

10. Applied rewrites 21.2%

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

12. Applied rewrites 21.3%

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

13. Final simplification 21.3%

    \[\leadsto alphax \cdot \frac{u0 \cdot alphax}{cos2phi} \]
14. Add Preprocessing

Alternative 21: 23.9% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 74.4

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

5. Applied rewrites 74.4%

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

6. Taylor expanded in cos2phi around inf

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

8. Applied rewrites 21.2%

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

10. Applied rewrites 21.3%

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

11. Final simplification 21.3%

    \[\leadsto u0 \cdot \frac{alphax \cdot alphax}{cos2phi} \]
12. Add Preprocessing

Alternative 22: 23.9% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 74.4

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

5. Applied rewrites 74.4%

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

6. Taylor expanded in cos2phi around inf

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

8. Applied rewrites 21.2%

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

10. Applied rewrites 21.3%

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

Alternative 23: 23.9% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 74.4

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

5. Applied rewrites 74.4%

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

6. Taylor expanded in cos2phi around inf

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

8. Applied rewrites 21.2%

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

10. Applied rewrites 21.2%

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

Reproduce

herbie shell --seed 2024233 
(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)))))