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

Percentage Accurate: 60.8% → 98.5%

Time: 18.4s

Alternatives: 32

Speedup: 3.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.3%

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

   1. frac-add N/A

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

   2. associate-/r/ N/A

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

   3. associate-*r* N/A

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

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

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

4. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\left(\frac{\mathsf{log1p}\left(-u0\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

5. Final simplification 98.6%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.3%

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

   1. distribute-frac-neg N/A

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

   2. frac-add N/A

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

   3. associate-/r/ N/A

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

   4. distribute-rgt-neg-in N/A

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

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

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

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

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return log1pf(-u0) / ((0.0f - (cos2phi / (alphax * alphax))) - (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(0.0) - Float32(cos2phi / Float32(alphax * alphax))) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 60.3%

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

   1. sub-neg N/A

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

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

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

   3. neg-lowering-neg.f32 98.4

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]
6. Add Preprocessing

Alternative 4: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 5000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \left(0 - \frac{alphay \cdot alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 5000.0)
     (/
      (fma (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) (* u0 u0) u0)
      (+ (/ cos2phi (* alphax alphax)) t_0))
     (* (log1p (- u0)) (- 0.0 (/ (* alphay alphay) sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 5000.0f) {
		tmp = fmaf(fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), (u0 * u0), u0) / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = log1pf(-u0) * (0.0f - ((alphay * alphay) / sin2phi));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(5000.0))
		tmp = Float32(fma(fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(u0 * u0), u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(log1p(Float32(-u0)) * Float32(Float32(0.0) - Float32(Float32(alphay * alphay) / sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5e3

   1. Initial program 53.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. *-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 95.2

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

   5. Simplified 95.2%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqr-neg N/A

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

      5. *-lft-identity N/A

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

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

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

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

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

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

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

      9. sqr-neg N/A

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

      10. *-lowering-*.f32 95.4

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

   7. Applied egg-rr 95.4%

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

   if 5e3 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.6%

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

   3. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 98.7

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

   5. Simplified 98.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \left(0 - \frac{alphay \cdot alphay}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 10000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot alphay\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 10000.0)
     (/
      (fma (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) (* u0 u0) u0)
      (+ (/ cos2phi (* alphax alphax)) t_0))
     (/ (* (log1p (- u0)) (* alphay alphay)) (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 10000.0f) {
		tmp = fmaf(fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), (u0 * u0), u0) / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (log1pf(-u0) * (alphay * alphay)) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(10000.0))
		tmp = Float32(fma(fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(u0 * u0), u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) * Float32(alphay * alphay)) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1e4

   1. Initial program 53.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. *-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 95.2

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

   5. Simplified 95.2%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqr-neg N/A

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

      5. *-lft-identity N/A

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

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

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

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

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

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

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

      9. sqr-neg N/A

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

      10. *-lowering-*.f32 95.4

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

   7. Applied egg-rr 95.4%

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

   if 1e4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.8%

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

   3. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 98.7

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

   5. Simplified 98.7%

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

      1. associate-*r/ N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. neg-lowering-neg.f32 98.8

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

   7. Applied egg-rr 98.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 10000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0 \cdot u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot alphay\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in u0 around 0

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

   1. *-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 93.5

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

5. Simplified 93.5%

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

   1. associate-/r* N/A

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

   2. frac-add N/A

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

   10. *-lowering-*.f32 93.3

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

7. Applied egg-rr 93.3%

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

   1. associate-/r* N/A

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

   2. associate-/r/ N/A

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

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

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

9. Applied egg-rr 93.7%

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

10. Final simplification 93.7%

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

Alternative 7: 90.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 50:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 50.0)
     (/ (* u0 (fma u0 0.5 1.0)) (+ (/ cos2phi (* alphax alphax)) t_0))
     (/
      (*
       (fma u0 (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) 1.0)
       (* u0 (* alphay alphay)))
      sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 50.0f) {
		tmp = (u0 * fmaf(u0, 0.5f, 1.0f)) / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (fmaf(u0, fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), 1.0f) * (u0 * (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(50.0))
		tmp = Float32(Float32(u0 * fma(u0, Float32(0.5), Float32(1.0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(fma(u0, fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)) * Float32(u0 * Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 50

   1. Initial program 54.0%

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

   3. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f32 89.7

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

   5. Simplified 89.7%

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

   if 50 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.2%

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

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

   5. Simplified 92.1%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f32 92.7

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

   8. Simplified 92.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 50:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 85:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0} \cdot \mathsf{fma}\left(u0, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 85.0)
     (* (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0)) (fma u0 0.5 1.0))
     (/
      (*
       (fma u0 (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) 1.0)
       (* u0 (* alphay alphay)))
      sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 85.0f) {
		tmp = (u0 / ((cos2phi / (alphax * alphax)) + t_0)) * fmaf(u0, 0.5f, 1.0f);
	} else {
		tmp = (fmaf(u0, fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), 1.0f) * (u0 * (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(85.0))
		tmp = Float32(Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0)) * fma(u0, Float32(0.5), Float32(1.0)));
	else
		tmp = Float32(Float32(fma(u0, fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)) * Float32(u0 * Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 85

   1. Initial program 54.2%

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

   3. Taylor expanded in u0 around 0

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

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

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

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

   5. Simplified 89.7%

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

   if 85 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 92.1

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

   5. Simplified 92.1%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f32 92.7

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

   8. Simplified 92.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 85:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \cdot \mathsf{fma}\left(u0, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.1% accurate, 2.0× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in u0 around 0

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

   1. *-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 93.5

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

5. Simplified 93.5%

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

   1. frac-add N/A

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

   2. *-commutative N/A

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

   3. associate-/r/ N/A

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

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

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

7. Applied egg-rr 93.7%

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

8. Final simplification 93.7%

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

Alternative 10: 84.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.499999936844688 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot \left(\left(alphax \cdot alphax\right) \cdot alphay\right)\right)}{\mathsf{fma}\left(sin2phi, alphax \cdot alphax, alphay \cdot \left(cos2phi \cdot alphay\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 2.499999936844688e-5)
   (/
    (* u0 (* alphay (* (* alphax alphax) alphay)))
    (fma sin2phi (* alphax alphax) (* alphay (* cos2phi alphay))))
   (/
    (*
     (fma u0 (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) 1.0)
     (* u0 (* alphay alphay)))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.499999936844688e-5f) {
		tmp = (u0 * (alphay * ((alphax * alphax) * alphay))) / fmaf(sin2phi, (alphax * alphax), (alphay * (cos2phi * alphay)));
	} else {
		tmp = (fmaf(u0, fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), 1.0f) * (u0 * (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.499999936844688e-5))
		tmp = Float32(Float32(u0 * Float32(alphay * Float32(Float32(alphax * alphax) * alphay))) / fma(sin2phi, Float32(alphax * alphax), Float32(alphay * Float32(cos2phi * alphay))));
	else
		tmp = Float32(Float32(fma(u0, fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)) * Float32(u0 * Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.49999994e-5

   1. Initial program 53.3%

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

      1. frac-add N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

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

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

   4. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{log1p}\left(-u0\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

   5. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 76.6

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

   7. Simplified 76.6%

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

      1. associate-*l* N/A

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

      2. associate-*l/ N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

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

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

      14. associate-*l* N/A

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

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

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

      16. *-lowering-*.f32 76.9

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

   9. Applied egg-rr 76.9%

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

   if 2.49999994e-5 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.4%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 92.5

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

   5. Simplified 92.5%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f32 91.3

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

   8. Simplified 91.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.499999936844688 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot \left(\left(alphax \cdot alphax\right) \cdot alphay\right)\right)}{\mathsf{fma}\left(sin2phi, alphax \cdot alphax, alphay \cdot \left(cos2phi \cdot alphay\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.499999936844688 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\mathsf{fma}\left(\frac{1}{alphay \cdot alphay}, sin2phi, \frac{cos2phi}{alphax \cdot alphax}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 2.499999936844688e-5)
   (/ u0 (fma (/ 1.0 (* alphay alphay)) sin2phi (/ cos2phi (* alphax alphax))))
   (/
    (*
     (fma u0 (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) 1.0)
     (* u0 (* alphay alphay)))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.499999936844688e-5f) {
		tmp = u0 / fmaf((1.0f / (alphay * alphay)), sin2phi, (cos2phi / (alphax * alphax)));
	} else {
		tmp = (fmaf(u0, fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), 1.0f) * (u0 * (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.499999936844688e-5))
		tmp = Float32(u0 / fma(Float32(Float32(1.0) / Float32(alphay * alphay)), sin2phi, Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(fma(u0, fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)) * Float32(u0 * Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.49999994e-5

   1. Initial program 53.3%

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 76.8

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

   5. Simplified 76.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f32 76.8

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

   7. Applied egg-rr 76.8%

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

   if 2.49999994e-5 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.4%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 92.5

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

   5. Simplified 92.5%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f32 91.3

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

   8. Simplified 91.3%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.499999936844688 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\mathsf{fma}\left(\frac{1}{alphay \cdot alphay}, sin2phi, \frac{cos2phi}{alphax \cdot alphax}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in u0 around 0

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

   1. *-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 93.5

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

5. Simplified 93.5%

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqr-neg N/A

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

   5. *-lft-identity N/A

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

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

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

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

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

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

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

   9. sqr-neg N/A

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

   10. *-lowering-*.f32 93.7

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

7. Applied egg-rr 93.7%

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

Alternative 13: 93.0% 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 60.3%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. *-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 93.5

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

5. Simplified 93.5%

   \[\leadsto \frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. 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. +-commutative N/A

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

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

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

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

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

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

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

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

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

   13. *-lowering-*.f32 93.5

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

7. Applied egg-rr 93.5%

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

8. Final simplification 93.5%

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

Alternative 14: 93.0% 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 60.3%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. *-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 93.5

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

5. Simplified 93.5%

   \[\leadsto \frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. 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. +-commutative N/A

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

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

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

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

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

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

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

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

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

   13. *-lowering-*.f32 93.4

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

7. Applied egg-rr 93.4%

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

8. Final simplification 93.4%

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

Alternative 15: 84.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 2.499999936844688 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 2.499999936844688e-5)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/
      (*
       (fma u0 (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) 1.0)
       (* u0 (* alphay alphay)))
      sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 2.499999936844688e-5f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (fmaf(u0, fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), 1.0f) * (u0 * (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(2.499999936844688e-5))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(fma(u0, fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)) * Float32(u0 * Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.49999994e-5

   1. Initial program 53.3%

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 76.8

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

   5. Simplified 76.8%

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

   if 2.49999994e-5 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.4%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 92.5

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

   5. Simplified 92.5%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f32 91.3

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

   8. Simplified 91.3%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.499999936844688 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot \left(alphax \cdot t\_0\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (fma u0 (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) 1.0)))
   (if (<= (/ sin2phi (* alphay alphay)) 9.999999960041972e-12)
     (/ (* u0 (* alphax (* alphax t_0))) cos2phi)
     (/ (* t_0 (* u0 (* alphay alphay))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = fmaf(u0, fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), 1.0f);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.999999960041972e-12f) {
		tmp = (u0 * (alphax * (alphax * t_0))) / cos2phi;
	} else {
		tmp = (t_0 * (u0 * (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = fma(u0, fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.999999960041972e-12))
		tmp = Float32(Float32(u0 * Float32(alphax * Float32(alphax * t_0))) / cos2phi);
	else
		tmp = Float32(Float32(t_0 * Float32(u0 * Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999996e-12

   1. Initial program 56.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 93.8

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

   5. Simplified 93.8%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f32 66.7

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

   8. Simplified 66.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      9. accelerator-lowering-fma.f32 66.8

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

   10. Applied egg-rr 66.8%

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

   if 9.99999996e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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

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

   5. Simplified 93.4%

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

   6. Taylor expanded in cos2phi around 0

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

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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f32 87.5

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

   8. Simplified 87.5%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot \left(alphax \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 80.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot \left(alphax \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.999999960041972e-12)
   (/
    (*
     u0
     (*
      alphax
      (* alphax (fma u0 (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) 1.0))))
    cos2phi)
   (*
    (/ (* alphay alphay) sin2phi)
    (* (- u0) (fma u0 (fma u0 -0.3333333333333333 -0.5) -1.0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.999999960041972e-12f) {
		tmp = (u0 * (alphax * (alphax * fmaf(u0, fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), 1.0f)))) / cos2phi;
	} else {
		tmp = ((alphay * alphay) / sin2phi) * (-u0 * fmaf(u0, fmaf(u0, -0.3333333333333333f, -0.5f), -1.0f));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.999999960041972e-12))
		tmp = Float32(Float32(u0 * Float32(alphax * Float32(alphax * fma(u0, fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0))))) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * Float32(Float32(-u0) * fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999996e-12

   1. Initial program 56.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 93.8

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

   5. Simplified 93.8%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f32 66.7

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

   8. Simplified 66.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      9. accelerator-lowering-fma.f32 66.8

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

   10. Applied egg-rr 66.8%

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

   if 9.99999996e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.1%

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

   3. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 92.0

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

   5. Simplified 92.0%

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

   6. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f32 85.8

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

   8. Simplified 85.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot \left(alphax \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 80.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.999999960041972e-12)
   (*
    (* u0 alphax)
    (*
     alphax
     (/ (fma u0 (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) 1.0) cos2phi)))
   (*
    (/ (* alphay alphay) sin2phi)
    (* (- u0) (fma u0 (fma u0 -0.3333333333333333 -0.5) -1.0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.999999960041972e-12f) {
		tmp = (u0 * alphax) * (alphax * (fmaf(u0, fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), 1.0f) / cos2phi));
	} else {
		tmp = ((alphay * alphay) / sin2phi) * (-u0 * fmaf(u0, fmaf(u0, -0.3333333333333333f, -0.5f), -1.0f));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.999999960041972e-12))
		tmp = Float32(Float32(u0 * alphax) * Float32(alphax * Float32(fma(u0, fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)) / cos2phi)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * Float32(Float32(-u0) * fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999996e-12

   1. Initial program 56.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 93.8

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

   5. Simplified 93.8%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f32 66.7

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

   8. Simplified 66.7%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. accelerator-lowering-fma.f32 66.7

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

   10. Applied egg-rr 66.7%

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

   if 9.99999996e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.1%

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

   3. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 92.0

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

   5. Simplified 92.0%

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

   6. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f32 85.8

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

   8. Simplified 85.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \left(alphax \cdot \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 91.1% 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 60.3%

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

3. Taylor expanded in u0 around 0

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

   1. *-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 91.7

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

5. Simplified 91.7%

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

Alternative 20: 79.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.999999960041972e-12)
   (/
    (* (fma u0 (fma u0 0.3333333333333333 0.5) 1.0) (* u0 (* alphax alphax)))
    cos2phi)
   (*
    (/ (* alphay alphay) sin2phi)
    (* (- u0) (fma u0 (fma u0 -0.3333333333333333 -0.5) -1.0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.999999960041972e-12f) {
		tmp = (fmaf(u0, fmaf(u0, 0.3333333333333333f, 0.5f), 1.0f) * (u0 * (alphax * alphax))) / cos2phi;
	} else {
		tmp = ((alphay * alphay) / sin2phi) * (-u0 * fmaf(u0, fmaf(u0, -0.3333333333333333f, -0.5f), -1.0f));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.999999960041972e-12))
		tmp = Float32(Float32(fma(u0, fma(u0, Float32(0.3333333333333333), Float32(0.5)), Float32(1.0)) * Float32(u0 * Float32(alphax * alphax))) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * Float32(Float32(-u0) * fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999996e-12

   1. Initial program 56.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 93.8

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

   5. Simplified 93.8%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f32 66.7

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

   8. Simplified 66.7%

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

   9. Taylor expanded in u0 around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f32 65.8

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

   11. Simplified 65.8%

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

   if 9.99999996e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.1%

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

   3. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 92.0

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

   5. Simplified 92.0%

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

   6. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f32 85.8

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

   8. Simplified 85.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 77.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.999999960041972e-12)
   (/
    (* (fma u0 (fma u0 0.3333333333333333 0.5) 1.0) (* u0 (* alphax alphax)))
    cos2phi)
   (* u0 (/ (fma alphay alphay (* 0.5 (* u0 (* alphay alphay)))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.999999960041972e-12f) {
		tmp = (fmaf(u0, fmaf(u0, 0.3333333333333333f, 0.5f), 1.0f) * (u0 * (alphax * alphax))) / cos2phi;
	} else {
		tmp = u0 * (fmaf(alphay, alphay, (0.5f * (u0 * (alphay * alphay)))) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.999999960041972e-12))
		tmp = Float32(Float32(fma(u0, fma(u0, Float32(0.3333333333333333), Float32(0.5)), Float32(1.0)) * Float32(u0 * Float32(alphax * alphax))) / cos2phi);
	else
		tmp = Float32(u0 * Float32(fma(alphay, alphay, Float32(Float32(0.5) * Float32(u0 * Float32(alphay * alphay)))) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999996e-12

   1. Initial program 56.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 93.8

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

   5. Simplified 93.8%

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

   6. Taylor expanded in cos2phi around inf

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

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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f32 66.7

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

   8. Simplified 66.7%

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

   9. Taylor expanded in u0 around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f32 65.8

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

   11. Simplified 65.8%

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

   if 9.99999996e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.1%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. remove-double-neg N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. remove-double-neg N/A

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

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

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

      8. sub-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      16. *-lowering-*.f32 98.2

          \[\leadsto \mathsf{log1p}\left(-u0\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-u0\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      6. unpow2 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      9. unpow2 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \color{blue}{\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      14. unpow2 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      15. *-lowering-*.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      16. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}}\right) \]

      17. unpow2 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}\right) \]

      18. *-lowering-*.f32 87.2

          \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}\right) \]

   7. Simplified 87.2%

      \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(0.5, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

   8. Taylor expanded in sin2phi around inf

      \[\leadsto u0 \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right) + {alphay}^{2}}{sin2phi}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right) + {alphay}^{2}}{sin2phi}} \]

      2. +-commutative N/A

         \[\leadsto u0 \cdot \frac{\color{blue}{{alphay}^{2} + \frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right)}}{sin2phi} \]

      3. unpow2 N/A

         \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay} + \frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right)}{sin2phi} \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto u0 \cdot \frac{\color{blue}{\mathsf{fma}\left(alphay, alphay, \frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right)\right)}}{sin2phi} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, \color{blue}{\frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right)}\right)}{sin2phi} \]

      6. *-commutative N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, \frac{1}{2} \cdot \color{blue}{\left(u0 \cdot {alphay}^{2}\right)}\right)}{sin2phi} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, \frac{1}{2} \cdot \color{blue}{\left(u0 \cdot {alphay}^{2}\right)}\right)}{sin2phi} \]

      8. unpow2 N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, \frac{1}{2} \cdot \left(u0 \cdot \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)}{sin2phi} \]

      9. *-lowering-*.f32 82.2

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(u0 \cdot \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)}{sin2phi} \]

   10. Simplified 82.2%

       \[\leadsto u0 \cdot \color{blue}{\frac{\mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right) \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 76.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphax, alphax, 0.5 \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.999999960041972e-12)
   (* u0 (/ (fma alphax alphax (* 0.5 (* u0 (* alphax alphax)))) cos2phi))
   (* u0 (/ (fma alphay alphay (* 0.5 (* u0 (* alphay alphay)))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.999999960041972e-12f) {
		tmp = u0 * (fmaf(alphax, alphax, (0.5f * (u0 * (alphax * alphax)))) / cos2phi);
	} else {
		tmp = u0 * (fmaf(alphay, alphay, (0.5f * (u0 * (alphay * alphay)))) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.999999960041972e-12))
		tmp = Float32(u0 * Float32(fma(alphax, alphax, Float32(Float32(0.5) * Float32(u0 * Float32(alphax * alphax)))) / cos2phi));
	else
		tmp = Float32(u0 * Float32(fma(alphay, alphay, Float32(Float32(0.5) * Float32(u0 * Float32(alphay * alphay)))) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999996e-12

   1. Initial program 56.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \left(1 - u0\right)\right)\right)\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \left(1 - u0\right)\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. remove-double-neg N/A

         \[\leadsto \color{blue}{\log \left(1 - u0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

      4. frac-2neg N/A

         \[\leadsto \log \left(1 - u0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \log \left(1 - u0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right)} \]

      6. remove-double-neg N/A

         \[\leadsto \log \left(1 - u0\right) \cdot \frac{-1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\log \left(1 - u0\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      8. sub-neg N/A

         \[\leadsto \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)} \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)} \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \color{blue}{\frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      12. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      16. *-lowering-*.f32 98.5

          \[\leadsto \mathsf{log1p}\left(-u0\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-u0\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      6. unpow2 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      9. unpow2 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \color{blue}{\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      14. unpow2 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      15. *-lowering-*.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      16. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}}\right) \]

      17. unpow2 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}\right) \]

      18. *-lowering-*.f32 87.7

          \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}\right) \]

   7. Simplified 87.7%

      \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(0.5, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

   8. Taylor expanded in cos2phi around inf

      \[\leadsto u0 \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) + {alphax}^{2}}{cos2phi}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) + {alphax}^{2}}{cos2phi}} \]

      2. +-commutative N/A

         \[\leadsto u0 \cdot \frac{\color{blue}{{alphax}^{2} + \frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right)}}{cos2phi} \]

      3. unpow2 N/A

         \[\leadsto u0 \cdot \frac{\color{blue}{alphax \cdot alphax} + \frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right)}{cos2phi} \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto u0 \cdot \frac{\color{blue}{\mathsf{fma}\left(alphax, alphax, \frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right)\right)}}{cos2phi} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphax, alphax, \color{blue}{\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right)}\right)}{cos2phi} \]

      6. *-commutative N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphax, alphax, \frac{1}{2} \cdot \color{blue}{\left(u0 \cdot {alphax}^{2}\right)}\right)}{cos2phi} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphax, alphax, \frac{1}{2} \cdot \color{blue}{\left(u0 \cdot {alphax}^{2}\right)}\right)}{cos2phi} \]

      8. unpow2 N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphax, alphax, \frac{1}{2} \cdot \left(u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}\right)\right)}{cos2phi} \]

      9. *-lowering-*.f32 64.1

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphax, alphax, 0.5 \cdot \left(u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}\right)\right)}{cos2phi} \]

   10. Simplified 64.1%

       \[\leadsto u0 \cdot \color{blue}{\frac{\mathsf{fma}\left(alphax, alphax, 0.5 \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)\right)}{cos2phi}} \]

   if 9.99999996e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \left(1 - u0\right)\right)\right)\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \left(1 - u0\right)\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. remove-double-neg N/A

         \[\leadsto \color{blue}{\log \left(1 - u0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

      4. frac-2neg N/A

         \[\leadsto \log \left(1 - u0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \log \left(1 - u0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right)} \]

      6. remove-double-neg N/A

         \[\leadsto \log \left(1 - u0\right) \cdot \frac{-1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\log \left(1 - u0\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      8. sub-neg N/A

         \[\leadsto \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)} \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)} \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \color{blue}{\frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      12. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      16. *-lowering-*.f32 98.2

          \[\leadsto \mathsf{log1p}\left(-u0\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-u0\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      6. unpow2 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      9. unpow2 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \color{blue}{\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      14. unpow2 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      15. *-lowering-*.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      16. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}}\right) \]

      17. unpow2 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}\right) \]

      18. *-lowering-*.f32 87.2

          \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}\right) \]

   7. Simplified 87.2%

      \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(0.5, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

   8. Taylor expanded in sin2phi around inf

      \[\leadsto u0 \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right) + {alphay}^{2}}{sin2phi}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right) + {alphay}^{2}}{sin2phi}} \]

      2. +-commutative N/A

         \[\leadsto u0 \cdot \frac{\color{blue}{{alphay}^{2} + \frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right)}}{sin2phi} \]

      3. unpow2 N/A

         \[\leadsto u0 \cdot \frac{\color{blue}{alphay \cdot alphay} + \frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right)}{sin2phi} \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto u0 \cdot \frac{\color{blue}{\mathsf{fma}\left(alphay, alphay, \frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right)\right)}}{sin2phi} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, \color{blue}{\frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right)}\right)}{sin2phi} \]

      6. *-commutative N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, \frac{1}{2} \cdot \color{blue}{\left(u0 \cdot {alphay}^{2}\right)}\right)}{sin2phi} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, \frac{1}{2} \cdot \color{blue}{\left(u0 \cdot {alphay}^{2}\right)}\right)}{sin2phi} \]

      8. unpow2 N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, \frac{1}{2} \cdot \left(u0 \cdot \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)}{sin2phi} \]

      9. *-lowering-*.f32 82.2

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(u0 \cdot \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)}{sin2phi} \]

   10. Simplified 82.2%

       \[\leadsto u0 \cdot \color{blue}{\frac{\mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}{sin2phi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 76.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphax, alphax, 0.5 \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.999999960041972e-12)
   (* u0 (/ (fma alphax alphax (* 0.5 (* u0 (* alphax alphax)))) cos2phi))
   (* (* alphay alphay) (/ (* u0 (fma u0 -0.5 -1.0)) (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.999999960041972e-12f) {
		tmp = u0 * (fmaf(alphax, alphax, (0.5f * (u0 * (alphax * alphax)))) / cos2phi);
	} else {
		tmp = (alphay * alphay) * ((u0 * fmaf(u0, -0.5f, -1.0f)) / -sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.999999960041972e-12))
		tmp = Float32(u0 * Float32(fma(alphax, alphax, Float32(Float32(0.5) * Float32(u0 * Float32(alphax * alphax)))) / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 * fma(u0, Float32(-0.5), Float32(-1.0))) / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999996e-12

   1. Initial program 56.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \left(1 - u0\right)\right)\right)\right)}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \left(1 - u0\right)\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. remove-double-neg N/A

         \[\leadsto \color{blue}{\log \left(1 - u0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

      4. frac-2neg N/A

         \[\leadsto \log \left(1 - u0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \log \left(1 - u0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right)} \]

      6. remove-double-neg N/A

         \[\leadsto \log \left(1 - u0\right) \cdot \frac{-1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\log \left(1 - u0\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      8. sub-neg N/A

         \[\leadsto \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)} \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      9. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)} \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \color{blue}{\frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      12. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      16. *-lowering-*.f32 98.5

          \[\leadsto \mathsf{log1p}\left(-u0\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-u0\right) \cdot \frac{-1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      6. unpow2 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      9. unpow2 N/A

         \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \color{blue}{\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}}\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      14. unpow2 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      15. *-lowering-*.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

      16. /-lowering-/.f32 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}}\right) \]

      17. unpow2 N/A

          \[\leadsto u0 \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}\right) \]

      18. *-lowering-*.f32 87.7

          \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}}\right) \]

   7. Simplified 87.7%

      \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(0.5, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}, \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]

   8. Taylor expanded in cos2phi around inf

      \[\leadsto u0 \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) + {alphax}^{2}}{cos2phi}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto u0 \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right) + {alphax}^{2}}{cos2phi}} \]

      2. +-commutative N/A

         \[\leadsto u0 \cdot \frac{\color{blue}{{alphax}^{2} + \frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right)}}{cos2phi} \]

      3. unpow2 N/A

         \[\leadsto u0 \cdot \frac{\color{blue}{alphax \cdot alphax} + \frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right)}{cos2phi} \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto u0 \cdot \frac{\color{blue}{\mathsf{fma}\left(alphax, alphax, \frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right)\right)}}{cos2phi} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphax, alphax, \color{blue}{\frac{1}{2} \cdot \left({alphax}^{2} \cdot u0\right)}\right)}{cos2phi} \]

      6. *-commutative N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphax, alphax, \frac{1}{2} \cdot \color{blue}{\left(u0 \cdot {alphax}^{2}\right)}\right)}{cos2phi} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphax, alphax, \frac{1}{2} \cdot \color{blue}{\left(u0 \cdot {alphax}^{2}\right)}\right)}{cos2phi} \]

      8. unpow2 N/A

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphax, alphax, \frac{1}{2} \cdot \left(u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}\right)\right)}{cos2phi} \]

      9. *-lowering-*.f32 64.1

         \[\leadsto u0 \cdot \frac{\mathsf{fma}\left(alphax, alphax, 0.5 \cdot \left(u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}\right)\right)}{cos2phi} \]

   10. Simplified 64.1%

       \[\leadsto u0 \cdot \color{blue}{\frac{\mathsf{fma}\left(alphax, alphax, 0.5 \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)\right)}{cos2phi}} \]

   if 9.99999996e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-add N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{\color{blue}{\frac{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi}{\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi} \cdot \left(\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

   4. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{log1p}\left(-u0\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{u0 \cdot \left(\color{blue}{u0 \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \left(u0 \cdot \frac{-1}{2} + \color{blue}{-1}\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      5. accelerator-lowering-fma.f32 87.6

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, -0.5, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

   7. Simplified 87.6%

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

   8. Taylor expanded in alphax around inf

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}{sin2phi}\right)} \cdot \left(alphay \cdot alphay\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}{sin2phi}\right)\right)} \cdot \left(alphay \cdot alphay\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}{\mathsf{neg}\left(sin2phi\right)}} \cdot \left(alphay \cdot alphay\right) \]

      3. neg-mul-1 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}{\color{blue}{-1 \cdot sin2phi}} \cdot \left(alphay \cdot alphay\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}{-1 \cdot sin2phi}} \cdot \left(alphay \cdot alphay\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}}{-1 \cdot sin2phi} \cdot \left(alphay \cdot alphay\right) \]

      6. sub-neg N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)}}{-1 \cdot sin2phi} \cdot \left(alphay \cdot alphay\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(\color{blue}{u0 \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)}{-1 \cdot sin2phi} \cdot \left(alphay \cdot alphay\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \frac{-1}{2} + \color{blue}{-1}\right)}{-1 \cdot sin2phi} \cdot \left(alphay \cdot alphay\right) \]

      9. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{2}, -1\right)}}{-1 \cdot sin2phi} \cdot \left(alphay \cdot alphay\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{2}, -1\right)}{\color{blue}{\mathsf{neg}\left(sin2phi\right)}} \cdot \left(alphay \cdot alphay\right) \]

      11. neg-lowering-neg.f32 81.9

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\color{blue}{-sin2phi}} \cdot \left(alphay \cdot alphay\right) \]

   10. Simplified 81.9%

       \[\leadsto \color{blue}{\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-sin2phi}} \cdot \left(alphay \cdot alphay\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(alphax, alphax, 0.5 \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 76.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(0 - u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.999999960041972e-12)
   (/ (* (fma u0 0.5 1.0) (* u0 (* alphax alphax))) cos2phi)
   (* (/ (* alphay alphay) sin2phi) (- 0.0 (* u0 (fma u0 -0.5 -1.0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.999999960041972e-12f) {
		tmp = (fmaf(u0, 0.5f, 1.0f) * (u0 * (alphax * alphax))) / cos2phi;
	} else {
		tmp = ((alphay * alphay) / sin2phi) * (0.0f - (u0 * fmaf(u0, -0.5f, -1.0f)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.999999960041972e-12))
		tmp = Float32(Float32(fma(u0, Float32(0.5), Float32(1.0)) * Float32(u0 * Float32(alphax * alphax))) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * Float32(Float32(0.0) - Float32(u0 * fma(u0, Float32(-0.5), Float32(-1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999996e-12

   1. Initial program 56.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   4. Step-by-step derivation

      1. *-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 93.8

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right)}, 0.5\right), 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Simplified 93.8%

      \[\leadsto \frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{cos2phi}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{cos2phi}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left({alphax}^{2} \cdot u0\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{cos2phi} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\left({alphax}^{2} \cdot u0\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{cos2phi} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(u0 \cdot {alphax}^{2}\right)} \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{cos2phi} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\left(u0 \cdot {alphax}^{2}\right)} \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{cos2phi} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{cos2phi} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \frac{\left(u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{cos2phi} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)}}{cos2phi} \]

      9. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), 1\right)}}{cos2phi} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}}, 1\right)}{cos2phi} \]

      11. accelerator-lowering-fma.f32 N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{1}{3} + \frac{1}{4} \cdot u0, \frac{1}{2}\right)}, 1\right)}{cos2phi} \]

      12. +-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, \frac{1}{2}\right), 1\right)}{cos2phi} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{1}{4}} + \frac{1}{3}, \frac{1}{2}\right), 1\right)}{cos2phi} \]

      14. accelerator-lowering-fma.f32 66.7

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right)}, 0.5\right), 1\right)}{cos2phi} \]

   8. Simplified 66.7%

      \[\leadsto \color{blue}{\frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}{cos2phi}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot u0\right)}}{cos2phi} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot u0 + 1\right)}}{cos2phi} \]

       2. *-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(\color{blue}{u0 \cdot \frac{1}{2}} + 1\right)}{cos2phi} \]

       3. accelerator-lowering-fma.f32 63.7

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u0, 0.5, 1\right)}}{cos2phi} \]

   11. Simplified 63.7%

       \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u0, 0.5, 1\right)}}{cos2phi} \]

   if 9.99999996e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\log \left(1 - u0\right) \cdot {alphay}^{2}}}{sin2phi}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\log \left(1 + \color{blue}{-1 \cdot u0}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      8. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot u0\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      10. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

      13. *-lowering-*.f32 92.0

          \[\leadsto -\mathsf{log1p}\left(-u0\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   5. Simplified 92.0%

      \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

   6. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)} \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)} \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(u0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\left(u0 \cdot \left(\color{blue}{u0 \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\left(u0 \cdot \left(u0 \cdot \frac{-1}{2} + \color{blue}{-1}\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      5. accelerator-lowering-fma.f32 81.9

         \[\leadsto -\left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, -0.5, -1\right)}\right) \cdot \frac{alphay \cdot alphay}{sin2phi} \]

   8. Simplified 81.9%

      \[\leadsto -\color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)\right)} \cdot \frac{alphay \cdot alphay}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(0 - u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 76.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.999999960041972e-12)
   (/ (* (fma u0 0.5 1.0) (* u0 (* alphax alphax))) cos2phi)
   (* (* alphay alphay) (/ (* u0 (fma u0 -0.5 -1.0)) (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.999999960041972e-12f) {
		tmp = (fmaf(u0, 0.5f, 1.0f) * (u0 * (alphax * alphax))) / cos2phi;
	} else {
		tmp = (alphay * alphay) * ((u0 * fmaf(u0, -0.5f, -1.0f)) / -sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.999999960041972e-12))
		tmp = Float32(Float32(fma(u0, Float32(0.5), Float32(1.0)) * Float32(u0 * Float32(alphax * alphax))) / cos2phi);
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 * fma(u0, Float32(-0.5), Float32(-1.0))) / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999996e-12

   1. Initial program 56.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   4. Step-by-step derivation

      1. *-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 93.8

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right)}, 0.5\right), 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Simplified 93.8%

      \[\leadsto \frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{cos2phi}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{cos2phi}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left({alphax}^{2} \cdot u0\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{cos2phi} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\left({alphax}^{2} \cdot u0\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{cos2phi} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(u0 \cdot {alphax}^{2}\right)} \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{cos2phi} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\left(u0 \cdot {alphax}^{2}\right)} \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{cos2phi} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{cos2phi} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \frac{\left(u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{cos2phi} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)}}{cos2phi} \]

      9. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), 1\right)}}{cos2phi} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}}, 1\right)}{cos2phi} \]

      11. accelerator-lowering-fma.f32 N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{1}{3} + \frac{1}{4} \cdot u0, \frac{1}{2}\right)}, 1\right)}{cos2phi} \]

      12. +-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, \frac{1}{2}\right), 1\right)}{cos2phi} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{1}{4}} + \frac{1}{3}, \frac{1}{2}\right), 1\right)}{cos2phi} \]

      14. accelerator-lowering-fma.f32 66.7

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right)}, 0.5\right), 1\right)}{cos2phi} \]

   8. Simplified 66.7%

      \[\leadsto \color{blue}{\frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}{cos2phi}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot u0\right)}}{cos2phi} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot u0 + 1\right)}}{cos2phi} \]

       2. *-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(\color{blue}{u0 \cdot \frac{1}{2}} + 1\right)}{cos2phi} \]

       3. accelerator-lowering-fma.f32 63.7

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u0, 0.5, 1\right)}}{cos2phi} \]

   11. Simplified 63.7%

       \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u0, 0.5, 1\right)}}{cos2phi} \]

   if 9.99999996e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-add N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{\color{blue}{\frac{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi}{\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi} \cdot \left(\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

   4. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{log1p}\left(-u0\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{u0 \cdot \left(\color{blue}{u0 \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{u0 \cdot \left(u0 \cdot \frac{-1}{2} + \color{blue}{-1}\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      5. accelerator-lowering-fma.f32 87.6

         \[\leadsto \left(\frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, -0.5, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

   7. Simplified 87.6%

      \[\leadsto \left(\frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

   8. Taylor expanded in alphax around inf

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}{sin2phi}\right)} \cdot \left(alphay \cdot alphay\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}{sin2phi}\right)\right)} \cdot \left(alphay \cdot alphay\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}{\mathsf{neg}\left(sin2phi\right)}} \cdot \left(alphay \cdot alphay\right) \]

      3. neg-mul-1 N/A

         \[\leadsto \frac{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}{\color{blue}{-1 \cdot sin2phi}} \cdot \left(alphay \cdot alphay\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}{-1 \cdot sin2phi}} \cdot \left(alphay \cdot alphay\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}}{-1 \cdot sin2phi} \cdot \left(alphay \cdot alphay\right) \]

      6. sub-neg N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)}}{-1 \cdot sin2phi} \cdot \left(alphay \cdot alphay\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{u0 \cdot \left(\color{blue}{u0 \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)}{-1 \cdot sin2phi} \cdot \left(alphay \cdot alphay\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{u0 \cdot \left(u0 \cdot \frac{-1}{2} + \color{blue}{-1}\right)}{-1 \cdot sin2phi} \cdot \left(alphay \cdot alphay\right) \]

      9. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{2}, -1\right)}}{-1 \cdot sin2phi} \cdot \left(alphay \cdot alphay\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{2}, -1\right)}{\color{blue}{\mathsf{neg}\left(sin2phi\right)}} \cdot \left(alphay \cdot alphay\right) \]

      11. neg-lowering-neg.f32 81.9

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\color{blue}{-sin2phi}} \cdot \left(alphay \cdot alphay\right) \]

   10. Simplified 81.9%

       \[\leadsto \color{blue}{\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-sin2phi}} \cdot \left(alphay \cdot alphay\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 69.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.999999960041972e-12)
   (/ (* (fma u0 0.5 1.0) (* u0 (* alphax alphax))) cos2phi)
   (* u0 (/ (* alphay alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.999999960041972e-12f) {
		tmp = (fmaf(u0, 0.5f, 1.0f) * (u0 * (alphax * alphax))) / cos2phi;
	} else {
		tmp = u0 * ((alphay * alphay) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.999999960041972e-12))
		tmp = Float32(Float32(fma(u0, Float32(0.5), Float32(1.0)) * Float32(u0 * Float32(alphax * alphax))) / cos2phi);
	else
		tmp = Float32(u0 * Float32(Float32(alphay * alphay) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999996e-12

   1. Initial program 56.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   4. Step-by-step derivation

      1. *-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 93.8

         \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right)}, 0.5\right), 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Simplified 93.8%

      \[\leadsto \frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{cos2phi}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}{cos2phi}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left({alphax}^{2} \cdot u0\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{cos2phi} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\left({alphax}^{2} \cdot u0\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{cos2phi} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(u0 \cdot {alphax}^{2}\right)} \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{cos2phi} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\left(u0 \cdot {alphax}^{2}\right)} \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{cos2phi} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{cos2phi} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \frac{\left(u0 \cdot \color{blue}{\left(alphax \cdot alphax\right)}\right) \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}{cos2phi} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)}}{cos2phi} \]

      9. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), 1\right)}}{cos2phi} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}}, 1\right)}{cos2phi} \]

      11. accelerator-lowering-fma.f32 N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{1}{3} + \frac{1}{4} \cdot u0, \frac{1}{2}\right)}, 1\right)}{cos2phi} \]

      12. +-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, \frac{1}{2}\right), 1\right)}{cos2phi} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{1}{4}} + \frac{1}{3}, \frac{1}{2}\right), 1\right)}{cos2phi} \]

      14. accelerator-lowering-fma.f32 66.7

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right)}, 0.5\right), 1\right)}{cos2phi} \]

   8. Simplified 66.7%

      \[\leadsto \color{blue}{\frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}{cos2phi}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot u0\right)}}{cos2phi} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot u0 + 1\right)}}{cos2phi} \]

       2. *-commutative N/A

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(\color{blue}{u0 \cdot \frac{1}{2}} + 1\right)}{cos2phi} \]

       3. accelerator-lowering-fma.f32 63.7

          \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u0, 0.5, 1\right)}}{cos2phi} \]

   11. Simplified 63.7%

       \[\leadsto \frac{\left(u0 \cdot \left(alphax \cdot alphax\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u0, 0.5, 1\right)}}{cos2phi} \]

   if 9.99999996e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\log \left(1 - u0\right) \cdot {alphay}^{2}}}{sin2phi}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\log \left(1 + \color{blue}{-1 \cdot u0}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      8. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot u0\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      10. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

      13. *-lowering-*.f32 92.0

          \[\leadsto -\mathsf{log1p}\left(-u0\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   5. Simplified 92.0%

      \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

   6. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot u0\right)} \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(u0\right)\right)} \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      2. neg-lowering-neg.f32 72.1

         \[\leadsto -\color{blue}{\left(-u0\right)} \cdot \frac{alphay \cdot alphay}{sin2phi} \]

   8. Simplified 72.1%

      \[\leadsto -\color{blue}{\left(-u0\right)} \cdot \frac{alphay \cdot alphay}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \left(u0 \cdot \left(alphax \cdot alphax\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 66.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.999999936531045e-21)
   (* (* 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 / (alphay * alphay)) <= 1.999999936531045e-21f) {
		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 / (alphay * alphay)) <= 1.999999936531045e-21) 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 (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.999999936531045e-21))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(u0 * Float32(Float32(alphay * alphay) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.999999936531045e-21))
		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 (/.f32 sin2phi (*.f32 alphay alphay)) < 1.9999999e-21

   1. Initial program 52.6%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

      9. *-lowering-*.f32 75.7

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   5. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   6. Taylor expanded in sin2phi around 0

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   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 65.0

         \[\leadsto \frac{\color{blue}{\left(alphax \cdot alphax\right)} \cdot u0}{cos2phi} \]

   8. Simplified 65.0%

      \[\leadsto \color{blue}{\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)} \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{cos2phi}} \cdot \left(alphax \cdot alphax\right) \]

      5. *-lowering-*.f32 65.4

         \[\leadsto \frac{u0}{cos2phi} \cdot \color{blue}{\left(alphax \cdot alphax\right)} \]

   10. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)} \]

   if 1.9999999e-21 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\log \left(1 - u0\right) \cdot {alphay}^{2}}}{sin2phi}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\log \left(1 + \color{blue}{-1 \cdot u0}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      8. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot u0\right)} \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      10. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left(u0\right)}\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi}\right) \]

      13. *-lowering-*.f32 86.5

          \[\leadsto -\mathsf{log1p}\left(-u0\right) \cdot \frac{\color{blue}{alphay \cdot alphay}}{sin2phi} \]

   5. Simplified 86.5%

      \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

   6. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot u0\right)} \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(u0\right)\right)} \cdot \frac{alphay \cdot alphay}{sin2phi}\right) \]

      2. neg-lowering-neg.f32 68.0

         \[\leadsto -\color{blue}{\left(-u0\right)} \cdot \frac{alphay \cdot alphay}{sin2phi} \]

   8. Simplified 68.0%

      \[\leadsto -\color{blue}{\left(-u0\right)} \cdot \frac{alphay \cdot alphay}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 66.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.999999936531045e-21)
   (* (* 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 / (alphay * alphay)) <= 1.999999936531045e-21f) {
		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 / (alphay * alphay)) <= 1.999999936531045e-21) 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 (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.999999936531045e-21))
		tmp = Float32(Float32(alphax * 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 / (alphay * alphay)) <= single(1.999999936531045e-21))
		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 (/.f32 sin2phi (*.f32 alphay alphay)) < 1.9999999e-21

   1. Initial program 52.6%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

      9. *-lowering-*.f32 75.7

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   5. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   6. Taylor expanded in sin2phi around 0

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   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 65.0

         \[\leadsto \frac{\color{blue}{\left(alphax \cdot alphax\right)} \cdot u0}{cos2phi} \]

   8. Simplified 65.0%

      \[\leadsto \color{blue}{\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)} \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{cos2phi}} \cdot \left(alphax \cdot alphax\right) \]

      5. *-lowering-*.f32 65.4

         \[\leadsto \frac{u0}{cos2phi} \cdot \color{blue}{\left(alphax \cdot alphax\right)} \]

   10. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)} \]

   if 1.9999999e-21 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

      9. *-lowering-*.f32 75.8

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   5. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   6. Taylor expanded in sin2phi around inf

      \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}} \]
   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 68.0

         \[\leadsto \frac{\color{blue}{\left(alphay \cdot alphay\right)} \cdot u0}{sin2phi} \]

   8. Simplified 68.0%

      \[\leadsto \color{blue}{\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 66.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.999999936531045e-21)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.999999936531045e-21f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.999999936531045e-21) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.999999936531045e-21))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.999999936531045e-21))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.9999999e-21

   1. Initial program 52.6%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

      9. *-lowering-*.f32 75.7

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   5. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   6. Taylor expanded in sin2phi around 0

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   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 65.0

         \[\leadsto \frac{\color{blue}{\left(alphax \cdot alphax\right)} \cdot u0}{cos2phi} \]

   8. Simplified 65.0%

      \[\leadsto \color{blue}{\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)} \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{cos2phi}} \cdot \left(alphax \cdot alphax\right) \]

      5. *-lowering-*.f32 65.4

         \[\leadsto \frac{u0}{cos2phi} \cdot \color{blue}{\left(alphax \cdot alphax\right)} \]

   10. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)} \]

   if 1.9999999e-21 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-add N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{\color{blue}{\frac{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi}{\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi} \cdot \left(\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{neg}\left(\log \left(1 - u0\right)\right)}{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

   4. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{log1p}\left(-u0\right)}{0 - \mathsf{fma}\left(alphax \cdot alphax, sin2phi, cos2phi \cdot \left(alphay \cdot alphay\right)\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right)} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\color{blue}{\frac{u0}{{alphax}^{2} \cdot sin2phi + {alphay}^{2} \cdot cos2phi}} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \left(\color{blue}{\frac{u0}{{alphax}^{2} \cdot sin2phi + {alphay}^{2} \cdot cos2phi}} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{u0}{\color{blue}{{alphay}^{2} \cdot cos2phi + {alphax}^{2} \cdot sin2phi}} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      3. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{u0}{\color{blue}{\mathsf{fma}\left({alphay}^{2}, cos2phi, {alphax}^{2} \cdot sin2phi\right)}} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{u0}{\mathsf{fma}\left(\color{blue}{alphay \cdot alphay}, cos2phi, {alphax}^{2} \cdot sin2phi\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{u0}{\mathsf{fma}\left(\color{blue}{alphay \cdot alphay}, cos2phi, {alphax}^{2} \cdot sin2phi\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \left(\frac{u0}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, \color{blue}{{alphax}^{2} \cdot sin2phi}\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{u0}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, \color{blue}{\left(alphax \cdot alphax\right)} \cdot sin2phi\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

      8. *-lowering-*.f32 76.1

         \[\leadsto \left(\frac{u0}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, \color{blue}{\left(alphax \cdot alphax\right)} \cdot sin2phi\right)} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

   7. Simplified 76.1%

      \[\leadsto \left(\color{blue}{\frac{u0}{\mathsf{fma}\left(alphay \cdot alphay, cos2phi, \left(alphax \cdot alphax\right) \cdot sin2phi\right)}} \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(alphay \cdot alphay\right) \]

   8. Taylor expanded in alphay around 0

      \[\leadsto \color{blue}{\frac{u0}{sin2phi}} \cdot \left(alphay \cdot alphay\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 67.9

         \[\leadsto \color{blue}{\frac{u0}{sin2phi}} \cdot \left(alphay \cdot alphay\right) \]

   10. Simplified 67.9%

       \[\leadsto \color{blue}{\frac{u0}{sin2phi}} \cdot \left(alphay \cdot alphay\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 23.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (* (* alphax alphax) (/ u0 cos2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (alphax * alphax) * (u0 / cos2phi);
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (alphax * alphax) * (u0 / cos2phi)
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (alphax * alphax) * (u0 / cos2phi);
end

Derivation

1. Initial program 60.3%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   9. *-lowering-*.f32 75.8

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

5. Simplified 75.8%

   \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

6. Taylor expanded in sin2phi around 0

   \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
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 23.9

      \[\leadsto \frac{\color{blue}{\left(alphax \cdot alphax\right)} \cdot u0}{cos2phi} \]

8. Simplified 23.9%

   \[\leadsto \color{blue}{\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}} \]
9. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{cos2phi}} \cdot \left(alphax \cdot alphax\right) \]

   5. *-lowering-*.f32 24.0

      \[\leadsto \frac{u0}{cos2phi} \cdot \color{blue}{\left(alphax \cdot alphax\right)} \]

10. Applied egg-rr 24.0%

    \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)} \]

11. Final simplification 24.0%

    \[\leadsto \left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi} \]
12. Add Preprocessing

Alternative 31: 23.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ alphax \cdot \frac{u0 \cdot alphax}{cos2phi} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (* alphax (/ (* u0 alphax) cos2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return alphax * ((u0 * alphax) / cos2phi);
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = alphax * ((u0 * alphax) / cos2phi)
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(alphax * Float32(Float32(u0 * alphax) / cos2phi))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = alphax * ((u0 * alphax) / cos2phi);
end

Derivation

1. Initial program 60.3%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   9. *-lowering-*.f32 75.8

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

5. Simplified 75.8%

   \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

6. Taylor expanded in sin2phi around 0

   \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
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 23.9

      \[\leadsto \frac{\color{blue}{\left(alphax \cdot alphax\right)} \cdot u0}{cos2phi} \]

8. Simplified 23.9%

   \[\leadsto \color{blue}{\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}} \]
9. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{alphax \cdot \left(alphax \cdot u0\right)}}{cos2phi} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{alphax \cdot \frac{alphax \cdot u0}{cos2phi}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{alphax \cdot \frac{alphax \cdot u0}{cos2phi}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto alphax \cdot \color{blue}{\frac{alphax \cdot u0}{cos2phi}} \]

   5. *-commutative N/A

      \[\leadsto alphax \cdot \frac{\color{blue}{u0 \cdot alphax}}{cos2phi} \]

   6. *-lowering-*.f32 23.9

      \[\leadsto alphax \cdot \frac{\color{blue}{u0 \cdot alphax}}{cos2phi} \]

10. Applied egg-rr 23.9%

    \[\leadsto \color{blue}{alphax \cdot \frac{u0 \cdot alphax}{cos2phi}} \]
11. Add Preprocessing

Alternative 32: 23.1% 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 60.3%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   9. *-lowering-*.f32 75.8

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

5. Simplified 75.8%

   \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

6. Taylor expanded in sin2phi around 0

   \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
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 23.9

      \[\leadsto \frac{\color{blue}{\left(alphax \cdot alphax\right)} \cdot u0}{cos2phi} \]

8. Simplified 23.9%

   \[\leadsto \color{blue}{\frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi}} \]
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 23.9

      \[\leadsto alphax \cdot \left(alphax \cdot \color{blue}{\frac{u0}{cos2phi}}\right) \]

10. Applied egg-rr 23.9%

    \[\leadsto \color{blue}{alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(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)))))