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

Percentage Accurate: 61.2% → 98.4%

Time: 15.8s

Alternatives: 19

Speedup: 3.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 61.2% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.6%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-add N/A

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

   6. associate-/r/ N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.6%

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

   1. 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 98.3

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

4. Applied rewrites 98.3%

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

5. Final simplification 98.3%

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

Alternative 3: 82.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\frac{u0}{\mathsf{fma}\left(\frac{cos2phi}{alphax}, \frac{1}{alphax}, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 5.00000006675716e-11)
     (/ u0 (fma (/ cos2phi alphax) (/ 1.0 alphax) t_0))
     (/
      (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0)
      t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 5.00000006675716e-11f) {
		tmp = u0 / fmaf((cos2phi / alphax), (1.0f / alphax), t_0);
	} else {
		tmp = fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(5.00000006675716e-11))
		tmp = Float32(u0 / fma(Float32(cos2phi / alphax), Float32(Float32(1.0) / alphax), t_0));
	else
		tmp = Float32(fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000007e-11

   1. Initial program 57.7%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 73.3%

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

   if 5.00000007e-11 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.1%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in cos2phi around 0

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

      1. lower-/.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 89.0

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

   8. Applied rewrites 89.0%

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

Alternative 4: 92.8% accurate, 2.0× speedup

Math

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

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(Float32(sin2phi / alphay) / alphay)))
end

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 93.2

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

5. Applied rewrites 93.2%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 93.2

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

7. Applied rewrites 93.2%

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

Alternative 5: 92.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 93.2

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

5. Applied rewrites 93.2%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-add N/A

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

   6. lift-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lift-fma.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-/r/ N/A

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

   14. lower-*.f32 N/A

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

7. Applied rewrites 93.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

9. Applied rewrites 93.2%

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

10. Final simplification 93.2%

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

Alternative 6: 82.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 5.00000006675716e-11)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
     (/
      (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 <= 5.00000006675716e-11f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} 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(5.00000006675716e-11))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000007e-11

   1. Initial program 57.7%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 73.3%

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

   if 5.00000007e-11 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.1%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in cos2phi around 0

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

      1. lower-/.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 89.0

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

   8. Applied rewrites 89.0%

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

Alternative 7: 82.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 5.00000006675716e-11)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/
      (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0)
      t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 5.00000006675716e-11f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(5.00000006675716e-11))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000007e-11

   1. Initial program 57.7%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 73.3

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

   5. Applied rewrites 73.3%

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

   if 5.00000007e-11 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.1%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in cos2phi around 0

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

      1. lower-/.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 89.0

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

   8. Applied rewrites 89.0%

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

Alternative 8: 92.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 93.2

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

5. Applied rewrites 93.2%

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

Alternative 9: 90.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

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(sin2phi / Float32(alphay * alphay)) + Float32(Float32(cos2phi / alphax) / alphax)))
end

Derivation

1. Initial program 60.6%

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

   1. 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 98.3

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

4. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 98.3

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in u0 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 91.0

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

9. Applied rewrites 91.0%

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

10. Final simplification 91.0%

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

Alternative 10: 90.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 91.0

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

5. Applied rewrites 91.0%

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

Alternative 11: 87.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 87.0

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

5. Applied rewrites 87.0%

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

Alternative 12: 86.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
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 86.9%

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

Alternative 13: 66.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.600000073301928 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{cos2phi} \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.600000073301928e-17)
   (* (/ 1.0 cos2phi) (* u0 (* alphax alphax)))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.600000073301928e-17f) {
		tmp = (1.0f / cos2phi) * (u0 * (alphax * alphax));
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.600000073301928e-17) then
        tmp = (1.0e0 / cos2phi) * (u0 * (alphax * alphax))
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.600000073301928e-17))
		tmp = Float32(Float32(Float32(1.0) / cos2phi) * Float32(u0 * Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.600000073301928e-17))
		tmp = (single(1.0) / cos2phi) * (u0 * (alphax * alphax));
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.60000007e-17

   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 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 57.2%

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

   10. Applied rewrites 57.3%

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

   if 1.60000007e-17 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in cos2phi around 0

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

   8. Applied rewrites 71.3%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.600000073301928 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{cos2phi} \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return 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 = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
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 74.1

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

5. Applied rewrites 74.1%

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

Alternative 15: 66.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.600000073301928 \cdot 10^{-17}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.600000073301928e-17)
   (/ (* u0 (* alphax alphax)) cos2phi)
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.600000073301928e-17f) {
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.600000073301928e-17) then
        tmp = (u0 * (alphax * alphax)) / cos2phi
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.600000073301928e-17))
		tmp = Float32(Float32(u0 * Float32(alphax * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.600000073301928e-17))
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.60000007e-17

   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 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in cos2phi around inf

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

   8. Applied rewrites 57.2%

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

   if 1.60000007e-17 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in cos2phi around 0

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

   8. Applied rewrites 71.3%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.600000073301928 \cdot 10^{-17}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 23.6% accurate, 6.9× speedup

Math

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

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 * Float32(alphax * alphax)) / cos2phi)
end

MATLAB

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

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
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 74.1

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

5. Applied rewrites 74.1%

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

6. Taylor expanded in cos2phi around inf

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

8. Applied rewrites 23.1%

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

9. Final simplification 23.1%

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

Alternative 17: 23.6% 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 60.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
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 74.1

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

5. Applied rewrites 74.1%

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

6. Taylor expanded in cos2phi around inf

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

8. Applied rewrites 23.1%

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

10. Applied rewrites 23.1%

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

Alternative 18: 23.6% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
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 74.1

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

5. Applied rewrites 74.1%

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

6. Taylor expanded in cos2phi around inf

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

8. Applied rewrites 23.1%

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

10. Applied rewrites 23.1%

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

11. Final simplification 23.1%

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

Alternative 19: 23.6% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
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 74.1

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

5. Applied rewrites 74.1%

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

6. Taylor expanded in cos2phi around inf

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

8. Applied rewrites 23.1%

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

10. Applied rewrites 23.1%

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

Reproduce

herbie shell --seed 2024234 
(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)))))