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

Percentage Accurate: 60.2% → 97.6%

Time: 14.0s

Alternatives: 19

Speedup: 3.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ t_1 := \mathsf{fma}\left(alphay \cdot alphay, t\_0, sin2phi\right)\\ t_2 := t\_0 + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;-\frac{\log \left(1 - u0\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \mathsf{fma}\left(u0, \frac{u0}{t\_2} \cdot \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), u0 \cdot \mathsf{fma}\left(\frac{alphay \cdot alphay}{t\_1}, 0.5, \frac{alphay \cdot alphay}{u0 \cdot t\_1}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ cos2phi (* alphax alphax)))
        (t_1 (fma (* alphay alphay) t_0 sin2phi))
        (t_2 (+ t_0 (/ sin2phi (* alphay alphay)))))
   (if (<= (- 1.0 u0) 0.9599999785423279)
     (- (/ (log (- 1.0 u0)) t_2))
     (*
      u0
      (fma
       u0
       (* (/ u0 t_2) (fma u0 0.25 0.3333333333333333))
       (*
        u0
        (fma
         (/ (* alphay alphay) t_1)
         0.5
         (/ (* alphay alphay) (* u0 t_1)))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float t_1 = fmaf((alphay * alphay), t_0, sin2phi);
	float t_2 = t_0 + (sin2phi / (alphay * alphay));
	float tmp;
	if ((1.0f - u0) <= 0.9599999785423279f) {
		tmp = -(logf((1.0f - u0)) / t_2);
	} else {
		tmp = u0 * fmaf(u0, ((u0 / t_2) * fmaf(u0, 0.25f, 0.3333333333333333f)), (u0 * fmaf(((alphay * alphay) / t_1), 0.5f, ((alphay * alphay) / (u0 * t_1)))));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	t_1 = fma(Float32(alphay * alphay), t_0, sin2phi)
	t_2 = Float32(t_0 + Float32(sin2phi / Float32(alphay * alphay)))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9599999785423279))
		tmp = Float32(-Float32(log(Float32(Float32(1.0) - u0)) / t_2));
	else
		tmp = Float32(u0 * fma(u0, Float32(Float32(u0 / t_2) * fma(u0, Float32(0.25), Float32(0.3333333333333333))), Float32(u0 * fma(Float32(Float32(alphay * alphay) / t_1), Float32(0.5), Float32(Float32(alphay * alphay) / Float32(u0 * t_1))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.959999979

   1. Initial program 94.9%

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

   if 0.959999979 < (-.f32 #s(literal 1 binary32) u0)

   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

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

      1. lower-*.f32 N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

   5. Simplified 98.1%

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

   6. Taylor expanded in alphay around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 98.1

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

   8. Simplified 98.1%

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

   9. Taylor expanded in u0 around inf

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

       1. lower-*.f32 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f32 N/A

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

   11. Simplified 98.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;-\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \mathsf{fma}\left(u0, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \cdot \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), u0 \cdot \mathsf{fma}\left(\frac{alphay \cdot alphay}{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}, 0.5, \frac{alphay \cdot alphay}{u0 \cdot \mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;-\frac{\log \left(1 - u0\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \mathsf{fma}\left(u0, \frac{u0}{t\_0} \cdot \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), \mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{1}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))
   (if (<= (- 1.0 u0) 0.9599999785423279)
     (- (/ (log (- 1.0 u0)) t_0))
     (*
      u0
      (fma
       u0
       (* (/ u0 t_0) (fma u0 0.25 0.3333333333333333))
       (* (fma u0 0.5 1.0) (/ 1.0 t_0)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay));
	float tmp;
	if ((1.0f - u0) <= 0.9599999785423279f) {
		tmp = -(logf((1.0f - u0)) / t_0);
	} else {
		tmp = u0 * fmaf(u0, ((u0 / t_0) * fmaf(u0, 0.25f, 0.3333333333333333f)), (fmaf(u0, 0.5f, 1.0f) * (1.0f / t_0)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9599999785423279))
		tmp = Float32(-Float32(log(Float32(Float32(1.0) - u0)) / t_0));
	else
		tmp = Float32(u0 * fma(u0, Float32(Float32(u0 / t_0) * fma(u0, Float32(0.25), Float32(0.3333333333333333))), Float32(fma(u0, Float32(0.5), Float32(1.0)) * Float32(Float32(1.0) / t_0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.959999979

   1. Initial program 94.9%

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

   if 0.959999979 < (-.f32 #s(literal 1 binary32) u0)

   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

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

      1. lower-*.f32 N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

   5. Simplified 98.1%

      \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(u0, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \cdot \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), \mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;-\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \mathsf{fma}\left(u0, \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \cdot \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), \mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 500:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\mathsf{log1p}\left(-u0\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 500.0)
     (/
      (* (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0) (- u0))
      (+ (/ cos2phi (* alphax alphax)) t_0))
     (* (* alphay alphay) (/ (log1p (- u0)) (- sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 500.0f) {
		tmp = (fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f) * -u0) / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (alphay * alphay) * (log1pf(-u0) / -sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(500.0))
		tmp = Float32(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)) + t_0));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(log1p(Float32(-u0)) / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 59.3%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f32 92.6

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

   5. Simplified 92.6%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.6%

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

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f32 N/A

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

      11. sub-neg N/A

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

      12. lower-log1p.f32 N/A

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

      13. lower-neg.f32 N/A

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

      14. lower-neg.f32 98.6

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

   5. Simplified 98.6%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\mathsf{log1p}\left(-u0\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))
	return Float32(u0 * fma(u0, Float32(Float32(u0 / t_0) * fma(u0, Float32(0.25), Float32(0.3333333333333333))), Float32(fma(u0, Float32(0.5), Float32(1.0)) * Float32(Float32(1.0) / t_0))))
end

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+l+ N/A

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

5. Simplified 92.2%

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

Alternative 5: 90.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 500:\\ \;\;\;\;u0 \cdot \left(\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(u0 \cdot \left(alphay \cdot \left(-alphay\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 500.0)
     (* u0 (* (fma u0 0.5 1.0) (/ 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 <= 500.0f) {
		tmp = u0 * (fmaf(u0, 0.5f, 1.0f) * (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(500.0))
		tmp = Float32(u0 * Float32(fma(u0, Float32(0.5), Float32(1.0)) * Float32(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 * Float32(-alphay)))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 59.3%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f32 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft1-in N/A

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

   5. Simplified 84.7%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.6%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f32 91.5

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

   5. Simplified 91.5%

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

   6. Taylor expanded in cos2phi around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

   8. Simplified 92.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;u0 \cdot \left(\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(u0 \cdot \left(alphay \cdot \left(-alphay\right)\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 500:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-\left(\frac{cos2phi}{alphax \cdot alphax} + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(u0 \cdot \left(alphay \cdot \left(-alphay\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 500.0)
     (/ (* 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 <= 500.0f) {
		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(500.0))
		tmp = Float32(Float32(u0 * fma(u0, Float32(-0.5), Float32(-1.0))) / Float32(-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 * Float32(-alphay)))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 59.3%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 84.5

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

   5. Simplified 84.5%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.6%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f32 91.5

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

   5. Simplified 91.5%

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

   6. Taylor expanded in cos2phi around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

   8. Simplified 92.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(u0 \cdot \left(alphay \cdot \left(-alphay\right)\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 2.1× speedup

Math

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

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(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 60.4%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-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. Final simplification 92.1%

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

Alternative 8: 80.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.000000067449534 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(-u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.000000067449534e-16)
   (/
    (*
     (* alphax alphax)
     (* (- u0) (fma u0 (fma u0 -0.3333333333333333 -0.5) -1.0)))
    cos2phi)
   (*
    (* alphay alphay)
    (/
     (* (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0) (- u0))
     sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.000000067449534e-16f) {
		tmp = ((alphax * alphax) * (-u0 * fmaf(u0, fmaf(u0, -0.3333333333333333f, -0.5f), -1.0f))) / cos2phi;
	} else {
		tmp = (alphay * alphay) * ((fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f) * -u0) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.000000067449534e-16))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(Float32(-u0) * fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0)))) / cos2phi);
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)) * Float32(-u0)) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.00000007e-16

   1. Initial program 61.9%

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

   3. Taylor expanded in cos2phi around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-neg.f32 69.1

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

   5. Simplified 69.1%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f32 61.6

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

   8. Simplified 61.6%

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

   if 4.00000007e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

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

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f32 N/A

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

      11. sub-neg N/A

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

      12. lower-log1p.f32 N/A

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

      13. lower-neg.f32 N/A

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

      14. lower-neg.f32 87.2

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

   5. Simplified 87.2%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f32 82.2

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

   8. Simplified 82.2%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.000000067449534 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(-u0\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.2% accurate, 2.3× speedup

Math

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

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 89.7

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

5. Simplified 89.7%

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

6. Final simplification 89.7%

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

Alternative 10: 79.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.000000067449534 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot t\_0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{t\_0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* (- u0) (fma u0 (fma u0 -0.3333333333333333 -0.5) -1.0))))
   (if (<= (/ sin2phi (* alphay alphay)) 4.000000067449534e-16)
     (/ (* (* alphax alphax) t_0) cos2phi)
     (* (* alphay alphay) (/ t_0 sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = -u0 * fmaf(u0, fmaf(u0, -0.3333333333333333f, -0.5f), -1.0f);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.000000067449534e-16f) {
		tmp = ((alphax * alphax) * t_0) / cos2phi;
	} else {
		tmp = (alphay * alphay) * (t_0 / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(-u0) * fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0)))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.000000067449534e-16))
		tmp = Float32(Float32(Float32(alphax * alphax) * t_0) / cos2phi);
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(t_0 / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.00000007e-16

   1. Initial program 61.9%

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

   3. Taylor expanded in cos2phi around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-neg.f32 69.1

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

   5. Simplified 69.1%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f32 61.6

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

   8. Simplified 61.6%

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

   if 4.00000007e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

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

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f32 N/A

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

      11. sub-neg N/A

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

      12. lower-log1p.f32 N/A

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

      13. lower-neg.f32 N/A

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

      14. lower-neg.f32 87.2

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

   5. Simplified 87.2%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f32 80.1

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

   8. Simplified 80.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.000000067449534 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.000000067449534 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.000000067449534e-16)
   (/ (* (* alphax alphax) (* u0 (fma u0 -0.5 -1.0))) (- cos2phi))
   (*
    (* alphay alphay)
    (/ (* (- u0) (fma u0 (fma u0 -0.3333333333333333 -0.5) -1.0)) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.000000067449534e-16f) {
		tmp = ((alphax * alphax) * (u0 * fmaf(u0, -0.5f, -1.0f))) / -cos2phi;
	} else {
		tmp = (alphay * alphay) * ((-u0 * fmaf(u0, fmaf(u0, -0.3333333333333333f, -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(4.000000067449534e-16))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(u0 * fma(u0, Float32(-0.5), Float32(-1.0)))) / Float32(-cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(Float32(-u0) * fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.00000007e-16

   1. Initial program 61.9%

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

   3. Taylor expanded in cos2phi around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-neg.f32 69.1

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

   5. Simplified 69.1%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 58.0

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

   8. Simplified 58.0%

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

   if 4.00000007e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

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

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f32 N/A

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

      11. sub-neg N/A

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

      12. lower-log1p.f32 N/A

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

      13. lower-neg.f32 N/A

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

      14. lower-neg.f32 87.2

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

   5. Simplified 87.2%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f32 80.1

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

   8. Simplified 80.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.000000067449534 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)\right)}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.000000067449534 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot t\_0}{-cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{t\_0}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* u0 (fma u0 -0.5 -1.0))))
   (if (<= (/ sin2phi (* alphay alphay)) 4.000000067449534e-16)
     (/ (* (* alphax alphax) t_0) (- cos2phi))
     (* (* alphay alphay) (/ t_0 (- sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = u0 * fmaf(u0, -0.5f, -1.0f);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.000000067449534e-16f) {
		tmp = ((alphax * alphax) * t_0) / -cos2phi;
	} else {
		tmp = (alphay * alphay) * (t_0 / -sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(u0 * fma(u0, Float32(-0.5), Float32(-1.0)))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.000000067449534e-16))
		tmp = Float32(Float32(Float32(alphax * alphax) * t_0) / Float32(-cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(t_0 / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.00000007e-16

   1. Initial program 61.9%

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

   3. Taylor expanded in cos2phi around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-neg.f32 69.1

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

   5. Simplified 69.1%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 58.0

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

   8. Simplified 58.0%

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

   if 4.00000007e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

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

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f32 N/A

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

      11. sub-neg N/A

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

      12. lower-log1p.f32 N/A

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

      13. lower-neg.f32 N/A

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

      14. lower-neg.f32 87.2

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

   5. Simplified 87.2%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 76.5

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

   8. Simplified 76.5%

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

Alternative 13: 73.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.000000031374395e-22)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (* (* 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)) <= 1.000000031374395e-22f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} 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(1.000000031374395e-22))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	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)) < 1.00000003e-22

   1. Initial program 63.7%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 69.1

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

   5. Simplified 69.1%

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

   6. Taylor expanded in cos2phi around inf

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

      1. lower-/.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 57.8

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

   8. Simplified 57.8%

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

   if 1.00000003e-22 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.9%

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

   3. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f32 N/A

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

      11. sub-neg N/A

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

      12. lower-log1p.f32 N/A

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

      13. lower-neg.f32 N/A

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

      14. lower-neg.f32 83.4

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

   5. Simplified 83.4%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 73.3

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

   8. Simplified 73.3%

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

Alternative 14: 84.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.000000026176508 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(u0 \cdot \left(alphay \cdot \left(-alphay\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.000000026176508e-9)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay 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 <= 3.000000026176508e-9f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * 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 (sin2phi <= Float32(3.000000026176508e-9))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * 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 * Float32(-alphay)))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.00000003e-9

   1. Initial program 57.8%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 72.4

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

   5. Simplified 72.4%

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

   if 3.00000003e-9 < sin2phi

   1. Initial program 61.9%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f32 91.7

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

   5. Simplified 91.7%

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

   6. Taylor expanded in cos2phi around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

   8. Simplified 88.6%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.000000026176508 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(u0 \cdot \left(alphay \cdot \left(-alphay\right)\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 84.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.000000026176508 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(-u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.000000026176508e-9)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (*
    (* alphay alphay)
    (/
     (* (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0) (- u0))
     sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.000000026176508e-9f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (alphay * alphay) * ((fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f) * -u0) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.000000026176508e-9))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)) * Float32(-u0)) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.00000003e-9

   1. Initial program 57.8%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 72.4

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

   5. Simplified 72.4%

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

   if 3.00000003e-9 < sin2phi

   1. Initial program 61.9%

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

   3. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f32 N/A

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

      11. sub-neg N/A

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

      12. lower-log1p.f32 N/A

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

      13. lower-neg.f32 N/A

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

      14. lower-neg.f32 94.3

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

   5. Simplified 94.3%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f32 88.4

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

   8. Simplified 88.4%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.000000026176508 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(-u0\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.000000067449534 \cdot 10^{-16}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(u0 \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.000000067449534e-16)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (/ (* alphay (* u0 alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.000000067449534e-16f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} else {
		tmp = (alphay * (u0 * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.000000067449534e-16) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    else
        tmp = (alphay * (u0 * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.000000067449534e-16))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(alphay * Float32(u0 * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.000000067449534e-16))
		tmp = u0 / (cos2phi / (alphax * alphax));
	else
		tmp = (alphay * (u0 * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.00000007e-16

   1. Initial program 61.9%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 68.9

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

   5. Simplified 68.9%

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

   6. Taylor expanded in cos2phi around inf

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

      1. lower-/.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 51.3

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

   8. Simplified 51.3%

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

   if 4.00000007e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 75.6

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

   5. Simplified 75.6%

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

   6. Taylor expanded in cos2phi around 0

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

      1. lower-/.f32 N/A

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

      2. lower-*.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. lower-*.f32 68.6

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

   8. Simplified 68.6%

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

   9. Taylor expanded in alphay around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f32 N/A

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

       4. lower-*.f32 68.7

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

   11. Simplified 68.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.000000067449534 \cdot 10^{-16}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(u0 \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.000000067449534 \cdot 10^{-16}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(u0 \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.000000067449534e-16)
   (/ (* u0 (* alphax alphax)) cos2phi)
   (/ (* alphay (* u0 alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.000000067449534e-16f) {
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	} else {
		tmp = (alphay * (u0 * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.000000067449534e-16) then
        tmp = (u0 * (alphax * alphax)) / cos2phi
    else
        tmp = (alphay * (u0 * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.000000067449534e-16))
		tmp = Float32(Float32(u0 * Float32(alphax * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(alphay * Float32(u0 * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.000000067449534e-16))
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	else
		tmp = (alphay * (u0 * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.00000007e-16

   1. Initial program 61.9%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 68.9

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

   5. Simplified 68.9%

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

   6. Taylor expanded in cos2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.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. lower-*.f32 51.2

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

   8. Simplified 51.2%

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

   if 4.00000007e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 75.6

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

   5. Simplified 75.6%

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

   6. Taylor expanded in cos2phi around 0

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

      1. lower-/.f32 N/A

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

      2. lower-*.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. lower-*.f32 68.6

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

   8. Simplified 68.6%

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

   9. Taylor expanded in alphay around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f32 N/A

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

       4. lower-*.f32 68.7

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

   11. Simplified 68.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.000000067449534 \cdot 10^{-16}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(u0 \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.8% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \frac{alphay \cdot \left(u0 \cdot alphay\right)}{sin2phi} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 74.2

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

5. Simplified 74.2%

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

6. Taylor expanded in cos2phi around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.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. lower-*.f32 59.3

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

8. Simplified 59.3%

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

9. Taylor expanded in alphay around 0

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

    1. unpow2 N/A

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

    2. associate-*l* N/A

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

    3. lower-*.f32 N/A

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

    4. lower-*.f32 59.4

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

11. Simplified 59.4%

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

12. Final simplification 59.4%

    \[\leadsto \frac{alphay \cdot \left(u0 \cdot alphay\right)}{sin2phi} \]
13. Add Preprocessing

Alternative 19: 58.8% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ alphay \cdot \left(alphay \cdot \frac{u0}{sin2phi}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 74.2

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

5. Simplified 74.2%

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

6. Taylor expanded in cos2phi around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.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. lower-*.f32 59.3

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

8. Simplified 59.3%

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

9. Taylor expanded in alphay around 0

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

    1. associate-/l* N/A

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

    2. unpow2 N/A

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

    3. associate-*l* N/A

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

    4. lower-*.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. lower-/.f32 59.2

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

11. Simplified 59.2%

    \[\leadsto \color{blue}{alphay \cdot \left(alphay \cdot \frac{u0}{sin2phi}\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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)))))