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

Percentage Accurate: 61.0% → 98.4%

Time: 19.2s

Alternatives: 23

Speedup: 5.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 61.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. frac-add N/A

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

   2. associate-/r/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

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

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. frac-add N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l* N/A

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

   4. associate-*r* N/A

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

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

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

4. Applied egg-rr 98.3%

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

5. Final simplification 98.3%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.5%

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

   1. sub-neg N/A

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

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

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

   3. neg-lowering-neg.f32 98.1

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

4. Applied egg-rr 98.1%

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

5. Final simplification 98.1%

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

Alternative 4: 89.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 4000:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 4000.0)
     (/ (fma u0 (* u0 0.5) u0) (+ (/ cos2phi (* alphax alphax)) t_0))
     (/
      (*
       (* alphay alphay)
       (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0))
      sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 4000.0f) {
		tmp = fmaf(u0, (u0 * 0.5f), u0) / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = ((alphay * alphay) * fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(4000.0))
		tmp = Float32(fma(u0, Float32(u0 * Float32(0.5)), u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4e3

   1. Initial program 55.7%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 88.3

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

   5. Simplified 88.3%

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

   if 4e3 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.9%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 94.0

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

   8. Simplified 94.0%

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

Alternative 5: 89.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 4000:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0} \cdot \mathsf{fma}\left(u0, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 4000.0)
     (* (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0)) (fma u0 0.5 1.0))
     (/
      (*
       (* alphay alphay)
       (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0))
      sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 4000.0f) {
		tmp = (u0 / ((cos2phi / (alphax * alphax)) + t_0)) * fmaf(u0, 0.5f, 1.0f);
	} else {
		tmp = ((alphay * alphay) * fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(4000.0))
		tmp = Float32(Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0)) * fma(u0, Float32(0.5), Float32(1.0)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4e3

   1. Initial program 55.7%

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

   3. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

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

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

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

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

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

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

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

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

      15. unpow2 N/A

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

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

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

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

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

   5. Simplified 88.2%

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

   if 4e3 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.9%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 94.0

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

   8. Simplified 94.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4000:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \cdot \mathsf{fma}\left(u0, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.8% accurate, 2.0× speedup

Math

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

FPCore

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

Julia

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

Derivation

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. Step-by-step derivation

   1. frac-add N/A

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

   2. associate-/r/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

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

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

4. Applied egg-rr 98.5%

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

   1. associate-*l* N/A

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

   2. distribute-frac-neg2 N/A

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

   3. distribute-frac-neg N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

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

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in u0 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

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

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

   6. +-commutative N/A

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

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

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. accelerator-lowering-fma.f32 93.8

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

9. Simplified 93.8%

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

Alternative 7: 92.7% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (*
   alphax
   (*
    (fma u0 (* u0 (fma u0 (fma u0 0.25 0.3333333333333333) 0.5)) u0)
    (* alphay (* alphax alphay))))
  (fma alphax (* alphax sin2phi) (* cos2phi (* 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) * (alphay * (alphax * alphay)))) / fmaf(alphax, (alphax * sin2phi), (cos2phi * (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) * Float32(alphay * Float32(alphax * alphay)))) / fma(alphax, Float32(alphax * sin2phi), Float32(cos2phi * Float32(alphay * alphay))))
end

Derivation

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. Step-by-step derivation

   1. frac-add N/A

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

   2. associate-/r/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

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

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

4. Applied egg-rr 98.5%

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

   1. associate-*l* N/A

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

   2. distribute-frac-neg2 N/A

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

   3. distribute-frac-neg N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

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

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

6. Applied egg-rr 98.4%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

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

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

   11. *-lowering-*.f32 98.3

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

8. Applied egg-rr 98.3%

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

9. Taylor expanded in u0 around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. *-rgt-identity N/A

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

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

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

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

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

    6. +-commutative N/A

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

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

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

    8. +-commutative N/A

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

    9. *-commutative N/A

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

    10. accelerator-lowering-fma.f32 93.7

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

11. Simplified 93.7%

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

12. Final simplification 93.7%

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

Alternative 8: 92.7% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  (* alphax (* alphax (* alphay alphay)))
  (/
   (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) 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))) * (fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0) / fmaf(alphax, (alphax * sin2phi), (cos2phi * (alphay * alphay))));
}

Julia

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

Derivation

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

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f32 93.6

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

5. Simplified 93.6%

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

   1. frac-add N/A

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

   2. +-commutative N/A

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

   3. associate-/r/ N/A

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

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

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

7. Applied egg-rr 93.7%

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

8. Final simplification 93.7%

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

Alternative 9: 92.7% accurate, 2.0× speedup

Math

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

FPCore

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

Julia

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

Derivation

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

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f32 93.6

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

5. Simplified 93.6%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

   7. *-lowering-*.f32 93.6

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

7. Applied egg-rr 93.6%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. frac-add N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. associate-/r/ N/A

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

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

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

9. Applied egg-rr 93.7%

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

10. Final simplification 93.7%

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

Alternative 10: 92.8% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0)
  (fma (/ 1.0 (* alphax alphax)) cos2phi (/ 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) / fmaf((1.0f / (alphax * alphax)), cos2phi, (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) / fma(Float32(Float32(1.0) / Float32(alphax * alphax)), cos2phi, Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

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

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f32 93.6

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

5. Simplified 93.6%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

   7. *-lowering-*.f32 93.6

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

7. Applied egg-rr 93.6%

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

Alternative 11: 83.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0}{\mathsf{fma}\left(\frac{1}{alphax \cdot alphax}, cos2phi, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.019999999552965164)
     (/ u0 (fma (/ 1.0 (* alphax alphax)) cos2phi t_0))
     (/
      (*
       (* alphay alphay)
       (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0))
      sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.019999999552965164f) {
		tmp = u0 / fmaf((1.0f / (alphax * alphax)), cos2phi, t_0);
	} else {
		tmp = ((alphay * alphay) * fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.019999999552965164))
		tmp = Float32(u0 / fma(Float32(Float32(1.0) / Float32(alphax * alphax)), cos2phi, t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.0199999996

   1. Initial program 54.4%

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

   3. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 76.4

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

   5. Simplified 76.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f32 76.4

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

   7. Applied egg-rr 76.4%

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

   if 0.0199999996 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.2%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 92.2

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

   5. Simplified 92.2%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 91.8

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

   8. Simplified 91.8%

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

Alternative 12: 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 60.5%

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

3. Taylor expanded in u0 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f32 93.6

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

5. Simplified 93.6%

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

Alternative 13: 92.7% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  (fma u0 (fma u0 (fma u0 0.25 0.3333333333333333) 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, fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), 1.0f) * (u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay))));
}

Julia

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in u0 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f32 93.6

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

5. Simplified 93.6%

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

6. Taylor expanded in u0 around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 93.4

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

8. Simplified 93.4%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f32 93.4

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

10. Applied egg-rr 93.4%

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

Alternative 14: 92.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in u0 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f32 93.6

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

5. Simplified 93.6%

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

6. Taylor expanded in u0 around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 93.4

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

8. Simplified 93.4%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f32 93.3

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

10. Applied egg-rr 93.3%

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

11. Final simplification 93.3%

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

Alternative 15: 83.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.019999999552965164)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/
      (*
       (* alphay alphay)
       (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0))
      sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.019999999552965164f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = ((alphay * alphay) * fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.019999999552965164))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.0199999996

   1. Initial program 54.4%

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

   3. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 76.4

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

   5. Simplified 76.4%

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

   if 0.0199999996 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.2%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 92.2

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

   5. Simplified 92.2%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 91.8

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

   8. Simplified 91.8%

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

Alternative 16: 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 60.5%

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

3. Taylor expanded in u0 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f32 91.8

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

5. Simplified 91.8%

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

Alternative 17: 90.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in u0 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f32 93.6

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

5. Simplified 93.6%

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

6. Taylor expanded in u0 around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f32 91.6

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

8. Simplified 91.6%

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

Alternative 18: 80.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)\\ \mathbf{if}\;sin2phi \leq 1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{t\_0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot t\_0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0)))
   (if (<= sin2phi 1.999999936531045e-19)
     (* alphax (* alphax (/ t_0 cos2phi)))
     (/ (* (* alphay alphay) t_0) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0);
	float tmp;
	if (sin2phi <= 1.999999936531045e-19f) {
		tmp = alphax * (alphax * (t_0 / cos2phi));
	} else {
		tmp = ((alphay * alphay) * t_0) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.999999936531045e-19))
		tmp = Float32(alphax * Float32(alphax * Float32(t_0 / cos2phi)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * t_0) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999994e-19

   1. Initial program 55.3%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 94.4

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

   5. Simplified 94.4%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 65.1

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

   8. Simplified 65.1%

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

      1. associate-/r/ N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. accelerator-lowering-fma.f32 65.3

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

   10. Applied egg-rr 65.3%

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

   if 1.99999994e-19 < sin2phi

   1. Initial program 62.0%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 93.4

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

   5. Simplified 93.4%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 83.7

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

   8. Simplified 83.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 69.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.0000000168623835e-16)
   (*
    alphax
    (*
     alphax
     (/
      (fma (* u0 u0) (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) u0)
      cos2phi)))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.0000000168623835e-16f) {
		tmp = alphax * (alphax * (fmaf((u0 * u0), fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), u0) / cos2phi));
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.0000000168623835e-16))
		tmp = Float32(alphax * Float32(alphax * Float32(fma(Float32(u0 * u0), fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u0) / cos2phi)));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.00000002e-16

   1. Initial program 55.0%

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

   3. Taylor expanded in u0 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 94.9

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

   5. Simplified 94.9%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 63.1

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

   8. Simplified 63.1%

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

      1. associate-/r/ N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. accelerator-lowering-fma.f32 63.4

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

   10. Applied egg-rr 63.4%

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

   if 1.00000002e-16 < sin2phi

   1. Initial program 62.5%

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 69.3

         \[\leadsto \frac{\color{blue}{\left(alphay \cdot alphay\right)} \cdot u0}{sin2phi} \]

   8. Simplified 69.3%

      \[\leadsto \color{blue}{\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 68.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0 \cdot u0, 0.5, u0\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.0000000168623835e-16)
   (/ (fma (* u0 u0) 0.5 u0) (/ cos2phi (* alphax alphax)))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.0000000168623835e-16f) {
		tmp = fmaf((u0 * u0), 0.5f, u0) / (cos2phi / (alphax * alphax));
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.0000000168623835e-16))
		tmp = Float32(fma(Float32(u0 * u0), Float32(0.5), u0) / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.00000002e-16

   1. Initial program 55.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0 \cdot 1}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(u0 \cdot u0\right) \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)} + u0 \cdot 1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\color{blue}{{u0}^{2}} \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + u0 \cdot 1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{{u0}^{2} \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({u0}^{2}, \frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{u0 \cdot u0}, \frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{u0 \cdot u0}, \frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(u0 \cdot u0, \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. accelerator-lowering-fma.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(u0 \cdot u0, \color{blue}{\mathsf{fma}\left(u0, \frac{1}{3} + \frac{1}{4} \cdot u0, \frac{1}{2}\right)}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, \frac{1}{2}\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{1}{4}} + \frac{1}{3}, \frac{1}{2}\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. accelerator-lowering-fma.f32 94.9

          \[\leadsto \frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right)}, 0.5\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Simplified 94.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u0\right)}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u0\right)}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u0\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

      3. *-lowering-*.f32 63.1

         \[\leadsto \frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   8. Simplified 63.1%

      \[\leadsto \frac{\mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{fma}\left(u0 \cdot u0, \color{blue}{\frac{1}{2}}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax}} \]
   10. Step-by-step derivation

   11. Simplified 58.5%

       \[\leadsto \frac{\mathsf{fma}\left(u0 \cdot u0, \color{blue}{0.5}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax}} \]

   if 1.00000002e-16 < sin2phi

   1. Initial program 62.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      8. *-lowering-*.f32 75.0

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   6. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{{alphay}^{2} \cdot u0}}{sin2phi} \]

      3. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(alphay \cdot alphay\right)} \cdot u0}{sin2phi} \]

      4. *-lowering-*.f32 69.3

         \[\leadsto \frac{\color{blue}{\left(alphay \cdot alphay\right)} \cdot u0}{sin2phi} \]

   8. Simplified 69.3%

      \[\leadsto \color{blue}{\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0 \cdot u0, 0.5, u0\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 66.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;alphax \cdot \frac{1}{\frac{cos2phi}{u0 \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.999999936531045e-19)
   (* alphax (/ 1.0 (/ cos2phi (* u0 alphax))))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.999999936531045e-19f) {
		tmp = alphax * (1.0f / (cos2phi / (u0 * 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 <= 1.999999936531045e-19) then
        tmp = alphax * (1.0e0 / (cos2phi / (u0 * 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 (sin2phi <= Float32(1.999999936531045e-19))
		tmp = Float32(alphax * Float32(Float32(1.0) / Float32(cos2phi / Float32(u0 * 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 <= single(1.999999936531045e-19))
		tmp = alphax * (single(1.0) / (cos2phi / (u0 * alphax)));
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999994e-19

   1. Initial program 55.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      8. *-lowering-*.f32 75.4

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{{alphax}^{2} \cdot u0}}{cos2phi} \]

      3. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(alphax \cdot alphax\right)} \cdot u0}{cos2phi} \]

      4. *-lowering-*.f32 52.5

         \[\leadsto \frac{\color{blue}{\left(alphax \cdot alphax\right)} \cdot u0}{cos2phi} \]

   8. Simplified 52.5%

      \[\leadsto \color{blue}{\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto alphax \cdot \color{blue}{\left(alphax \cdot \frac{u0}{cos2phi}\right)} \]

      5. /-lowering-/.f32 52.8

         \[\leadsto alphax \cdot \left(alphax \cdot \color{blue}{\frac{u0}{cos2phi}}\right) \]

   10. Applied egg-rr 52.8%

       \[\leadsto \color{blue}{alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto alphax \cdot \color{blue}{\frac{alphax \cdot u0}{cos2phi}} \]

       2. clear-num N/A

          \[\leadsto alphax \cdot \color{blue}{\frac{1}{\frac{cos2phi}{alphax \cdot u0}}} \]

       3. /-lowering-/.f32 N/A

          \[\leadsto alphax \cdot \color{blue}{\frac{1}{\frac{cos2phi}{alphax \cdot u0}}} \]

       4. /-lowering-/.f32 N/A

          \[\leadsto alphax \cdot \frac{1}{\color{blue}{\frac{cos2phi}{alphax \cdot u0}}} \]

       5. *-commutative N/A

          \[\leadsto alphax \cdot \frac{1}{\frac{cos2phi}{\color{blue}{u0 \cdot alphax}}} \]

       6. *-lowering-*.f32 52.8

          \[\leadsto alphax \cdot \frac{1}{\frac{cos2phi}{\color{blue}{u0 \cdot alphax}}} \]

   12. Applied egg-rr 52.8%

       \[\leadsto alphax \cdot \color{blue}{\frac{1}{\frac{cos2phi}{u0 \cdot alphax}}} \]

   if 1.99999994e-19 < sin2phi

   1. Initial program 62.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      8. *-lowering-*.f32 75.2

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. Simplified 75.2%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   6. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{{alphay}^{2} \cdot u0}}{sin2phi} \]

      3. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(alphay \cdot alphay\right)} \cdot u0}{sin2phi} \]

      4. *-lowering-*.f32 68.2

         \[\leadsto \frac{\color{blue}{\left(alphay \cdot alphay\right)} \cdot u0}{sin2phi} \]

   8. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;alphax \cdot \frac{1}{\frac{cos2phi}{u0 \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 66.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.999999936531045e-19)
   (* alphax (* alphax (/ u0 cos2phi)))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.999999936531045e-19f) {
		tmp = alphax * (alphax * (u0 / 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 <= 1.999999936531045e-19) then
        tmp = alphax * (alphax * (u0 / cos2phi))
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.999999936531045e-19))
		tmp = Float32(alphax * Float32(alphax * Float32(u0 / 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 <= single(1.999999936531045e-19))
		tmp = alphax * (alphax * (u0 / cos2phi));
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999994e-19

   1. Initial program 55.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      8. *-lowering-*.f32 75.4

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{{alphax}^{2} \cdot u0}}{cos2phi} \]

      3. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(alphax \cdot alphax\right)} \cdot u0}{cos2phi} \]

      4. *-lowering-*.f32 52.5

         \[\leadsto \frac{\color{blue}{\left(alphax \cdot alphax\right)} \cdot u0}{cos2phi} \]

   8. Simplified 52.5%

      \[\leadsto \color{blue}{\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto alphax \cdot \color{blue}{\left(alphax \cdot \frac{u0}{cos2phi}\right)} \]

      5. /-lowering-/.f32 52.8

         \[\leadsto alphax \cdot \left(alphax \cdot \color{blue}{\frac{u0}{cos2phi}}\right) \]

   10. Applied egg-rr 52.8%

       \[\leadsto \color{blue}{alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)} \]

   if 1.99999994e-19 < sin2phi

   1. Initial program 62.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      8. *-lowering-*.f32 75.2

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. Simplified 75.2%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   6. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{{alphay}^{2} \cdot u0}}{sin2phi} \]

      3. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(alphay \cdot alphay\right)} \cdot u0}{sin2phi} \]

      4. *-lowering-*.f32 68.2

         \[\leadsto \frac{\color{blue}{\left(alphay \cdot alphay\right)} \cdot u0}{sin2phi} \]

   8. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 23.5% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right) \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (* alphax (* alphax (/ u0 cos2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return alphax * (alphax * (u0 / cos2phi));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = alphax * (alphax * (u0 / cos2phi))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(alphax * Float32(alphax * Float32(u0 / cos2phi)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = alphax * (alphax * (u0 / cos2phi));
end

Derivation

1. Initial program 60.5%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}} \]

   4. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   8. *-lowering-*.f32 75.2

      \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

5. Simplified 75.2%

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

6. Taylor expanded in cos2phi around inf

   \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
7. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \frac{\color{blue}{{alphax}^{2} \cdot u0}}{cos2phi} \]

   3. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\left(alphax \cdot alphax\right)} \cdot u0}{cos2phi} \]

   4. *-lowering-*.f32 22.5

      \[\leadsto \frac{\color{blue}{\left(alphax \cdot alphax\right)} \cdot u0}{cos2phi} \]

8. Simplified 22.5%

   \[\leadsto \color{blue}{\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}} \]
9. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto alphax \cdot \color{blue}{\left(alphax \cdot \frac{u0}{cos2phi}\right)} \]

   5. /-lowering-/.f32 22.5

      \[\leadsto alphax \cdot \left(alphax \cdot \color{blue}{\frac{u0}{cos2phi}}\right) \]

10. Applied egg-rr 22.5%

    \[\leadsto \color{blue}{alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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)))))