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

Percentage Accurate: 61.0% → 96.1%

Time: 11.3s

Alternatives: 10

Speedup: 3.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 61.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u0\right)\\ \mathbf{if}\;-t\_0 \leq 0.0031999999191612005:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\frac{-1}{alphay} \cdot \left(\frac{\sqrt{sin2phi}}{alphay} \cdot \sqrt{sin2phi}\right) - \frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0))))
   (if (<= (- t_0) 0.0031999999191612005)
     (/
      (- (* (+ 1.0 (* -0.5 u0)) u0) (* (- u0) u0))
      (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
     (/
      t_0
      (-
       (* (/ -1.0 alphay) (* (/ (sqrt sin2phi) alphay) (sqrt sin2phi)))
       (/ 1.0 (/ (* alphax alphax) cos2phi)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = logf((1.0f - u0));
	float tmp;
	if (-t_0 <= 0.0031999999191612005f) {
		tmp = (((1.0f + (-0.5f * u0)) * u0) - (-u0 * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = t_0 / (((-1.0f / alphay) * ((sqrtf(sin2phi) / alphay) * sqrtf(sin2phi))) - (1.0f / ((alphax * alphax) / cos2phi)));
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = log((1.0e0 - u0))
    if (-t_0 <= 0.0031999999191612005e0) then
        tmp = (((1.0e0 + ((-0.5e0) * u0)) * u0) - (-u0 * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    else
        tmp = t_0 / ((((-1.0e0) / alphay) * ((sqrt(sin2phi) / alphay) * sqrt(sin2phi))) - (1.0e0 / ((alphax * alphax) / cos2phi)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (Float32(-t_0) <= Float32(0.0031999999191612005))
		tmp = Float32(Float32(Float32(Float32(Float32(1.0) + Float32(Float32(-0.5) * u0)) * u0) - Float32(Float32(-u0) * u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(t_0 / Float32(Float32(Float32(Float32(-1.0) / alphay) * Float32(Float32(sqrt(sin2phi) / alphay) * sqrt(sin2phi))) - Float32(Float32(1.0) / Float32(Float32(alphax * alphax) / cos2phi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log((single(1.0) - u0));
	tmp = single(0.0);
	if (-t_0 <= single(0.0031999999191612005))
		tmp = (((single(1.0) + (single(-0.5) * u0)) * u0) - (-u0 * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	else
		tmp = t_0 / (((single(-1.0) / alphay) * ((sqrt(sin2phi) / alphay) * sqrt(sin2phi))) - (single(1.0) / ((alphax * alphax) / cos2phi)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0))) < 0.00319999992

   1. Initial program 47.2%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

         \[\leadsto \frac{-\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0 \cdot u0\right)\right)} - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

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

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u0\right)\right) \cdot u0}\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u0\right)\right) \cdot u0}\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lower-neg.f32 N/A

          \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u0\right)} \cdot u0\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. lower-log1p.f32 87.1

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

   4. Applied rewrites 87.1%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(\color{blue}{-1 \cdot {u0}^{2}} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{-\left(\color{blue}{\left(\mathsf{neg}\left({u0}^{2}\right)\right)} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. unpow2 N/A

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

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

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

      4. mul-1-neg N/A

         \[\leadsto \frac{-\left(\color{blue}{\left(-1 \cdot u0\right)} \cdot u0 - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{-\left(\color{blue}{\left(-1 \cdot u0\right) \cdot u0} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. mul-1-neg N/A

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

      7. lower-neg.f32 87.1

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 87.1

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

   10. Applied rewrites 86.6%

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

   12. Applied rewrites 97.8%

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

   if 0.00319999992 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0)))

   1. Initial program 92.2%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lower-/.f32 92.3

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

   4. Applied rewrites 92.3%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f32 92.1

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

   6. Applied rewrites 92.1%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f32 N/A

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

      6. div-inv N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. sqrt-unprod N/A

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

      10. lift-sqrt.f32 N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. lift-*.f32 N/A

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

      14. sqrt-prod N/A

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

      15. rem-square-sqrt N/A

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

      16. lower-*.f32 N/A

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

   8. Applied rewrites 92.4%

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\frac{1}{\frac{alphax \cdot alphax}{cos2phi}} + \color{blue}{\left(\frac{\sqrt{sin2phi}}{alphay} \cdot \sqrt{sin2phi}\right) \cdot \frac{1}{alphay}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u0\right) \leq 0.0031999999191612005:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{-1}{alphay} \cdot \left(\frac{\sqrt{sin2phi}}{alphay} \cdot \sqrt{sin2phi}\right) - \frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ t_1 := -\log \left(1 - u0\right)\\ \mathbf{if}\;t\_1 \leq 0.0031999999191612005:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{t\_0 + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_0 + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ cos2phi (* alphax alphax))) (t_1 (- (log (- 1.0 u0)))))
   (if (<= t_1 0.0031999999191612005)
     (/
      (- (* (+ 1.0 (* -0.5 u0)) u0) (* (- u0) u0))
      (+ t_0 (/ sin2phi (* alphay alphay))))
     (/ t_1 (+ t_0 (/ (/ sin2phi alphay) alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float t_1 = -logf((1.0f - u0));
	float tmp;
	if (t_1 <= 0.0031999999191612005f) {
		tmp = (((1.0f + (-0.5f * u0)) * u0) - (-u0 * u0)) / (t_0 + (sin2phi / (alphay * alphay)));
	} else {
		tmp = t_1 / (t_0 + ((sin2phi / alphay) / alphay));
	}
	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) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = cos2phi / (alphax * alphax)
    t_1 = -log((1.0e0 - u0))
    if (t_1 <= 0.0031999999191612005e0) then
        tmp = (((1.0e0 + ((-0.5e0) * u0)) * u0) - (-u0 * u0)) / (t_0 + (sin2phi / (alphay * alphay)))
    else
        tmp = t_1 / (t_0 + ((sin2phi / alphay) / alphay))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	t_1 = Float32(-log(Float32(Float32(1.0) - u0)))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.0031999999191612005))
		tmp = Float32(Float32(Float32(Float32(Float32(1.0) + Float32(Float32(-0.5) * u0)) * u0) - Float32(Float32(-u0) * u0)) / Float32(t_0 + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(t_1 / Float32(t_0 + Float32(Float32(sin2phi / alphay) / alphay)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = cos2phi / (alphax * alphax);
	t_1 = -log((single(1.0) - u0));
	tmp = single(0.0);
	if (t_1 <= single(0.0031999999191612005))
		tmp = (((single(1.0) + (single(-0.5) * u0)) * u0) - (-u0 * u0)) / (t_0 + (sin2phi / (alphay * alphay)));
	else
		tmp = t_1 / (t_0 + ((sin2phi / alphay) / alphay));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0))) < 0.00319999992

   1. Initial program 47.2%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

         \[\leadsto \frac{-\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0 \cdot u0\right)\right)} - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

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

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u0\right)\right) \cdot u0}\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u0\right)\right) \cdot u0}\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lower-neg.f32 N/A

          \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u0\right)} \cdot u0\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. lower-log1p.f32 87.1

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

   4. Applied rewrites 87.1%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(\color{blue}{-1 \cdot {u0}^{2}} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{-\left(\color{blue}{\left(\mathsf{neg}\left({u0}^{2}\right)\right)} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. unpow2 N/A

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

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

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

      4. mul-1-neg N/A

         \[\leadsto \frac{-\left(\color{blue}{\left(-1 \cdot u0\right)} \cdot u0 - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{-\left(\color{blue}{\left(-1 \cdot u0\right) \cdot u0} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. mul-1-neg N/A

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

      7. lower-neg.f32 87.1

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 87.1

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

   10. Applied rewrites 86.6%

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

   12. Applied rewrites 97.8%

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

   if 0.00319999992 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0)))

   1. Initial program 92.2%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 92.4

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

   4. Applied rewrites 92.4%

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u0\right) \leq 0.0031999999191612005:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ t_1 := -\log \left(1 - u0\right)\\ \mathbf{if}\;t\_1 \leq 0.0031999999191612005:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
        (t_1 (- (log (- 1.0 u0)))))
   (if (<= t_1 0.0031999999191612005)
     (/ (- (* (+ 1.0 (* -0.5 u0)) u0) (* (- u0) u0)) t_0)
     (/ t_1 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 t_1 = -logf((1.0f - u0));
	float tmp;
	if (t_1 <= 0.0031999999191612005f) {
		tmp = (((1.0f + (-0.5f * u0)) * u0) - (-u0 * u0)) / t_0;
	} else {
		tmp = t_1 / t_0;
	}
	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) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay))
    t_1 = -log((1.0e0 - u0))
    if (t_1 <= 0.0031999999191612005e0) then
        tmp = (((1.0e0 + ((-0.5e0) * u0)) * u0) - (-u0 * u0)) / t_0
    else
        tmp = t_1 / t_0
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))
	t_1 = Float32(-log(Float32(Float32(1.0) - u0)))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.0031999999191612005))
		tmp = Float32(Float32(Float32(Float32(Float32(1.0) + Float32(Float32(-0.5) * u0)) * u0) - Float32(Float32(-u0) * u0)) / t_0);
	else
		tmp = Float32(t_1 / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay));
	t_1 = -log((single(1.0) - u0));
	tmp = single(0.0);
	if (t_1 <= single(0.0031999999191612005))
		tmp = (((single(1.0) + (single(-0.5) * u0)) * u0) - (-u0 * u0)) / t_0;
	else
		tmp = t_1 / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0))) < 0.00319999992

   1. Initial program 47.2%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-log1p.f32 N/A

         \[\leadsto \frac{-\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0 \cdot u0\right)\right)} - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

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

         \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u0\right)\right) \cdot u0}\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u0\right)\right) \cdot u0}\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      11. lower-neg.f32 N/A

          \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u0\right)} \cdot u0\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. lower-log1p.f32 87.1

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

   4. Applied rewrites 87.1%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(\color{blue}{-1 \cdot {u0}^{2}} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{-\left(\color{blue}{\left(\mathsf{neg}\left({u0}^{2}\right)\right)} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. unpow2 N/A

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

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

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

      4. mul-1-neg N/A

         \[\leadsto \frac{-\left(\color{blue}{\left(-1 \cdot u0\right)} \cdot u0 - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{-\left(\color{blue}{\left(-1 \cdot u0\right) \cdot u0} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. mul-1-neg N/A

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

      7. lower-neg.f32 87.1

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 87.1

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

   10. Applied rewrites 86.6%

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

   12. Applied rewrites 97.8%

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

   if 0.00319999992 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0)))

   1. Initial program 92.2%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u0\right) \leq 0.0031999999191612005:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot u0\right) \cdot u0 - \left(-u0\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (((1.0f + (-0.5f * u0)) * u0) - (-u0 * 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 = (((1.0e0 + ((-0.5e0) * u0)) * u0) - (-u0 * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

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

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (((single(1.0) + (single(-0.5) * u0)) * u0) - (-u0 * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Derivation

1. Initial program 59.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f32 N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower-log1p.f32 N/A

      \[\leadsto \frac{-\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0 \cdot u0\right)\right)} - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

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

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u0\right)\right) \cdot u0}\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u0\right)\right) \cdot u0}\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   11. lower-neg.f32 N/A

       \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u0\right)} \cdot u0\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   12. lower-log1p.f32 76.0

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

4. Applied rewrites 76.0%

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

5. Taylor expanded in u0 around 0

   \[\leadsto \frac{-\left(\color{blue}{-1 \cdot {u0}^{2}} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{-\left(\color{blue}{\left(\mathsf{neg}\left({u0}^{2}\right)\right)} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. unpow2 N/A

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

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

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

   4. mul-1-neg N/A

      \[\leadsto \frac{-\left(\color{blue}{\left(-1 \cdot u0\right)} \cdot u0 - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{-\left(\color{blue}{\left(-1 \cdot u0\right) \cdot u0} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. mul-1-neg N/A

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

   7. lower-neg.f32 76.0

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

7. Applied rewrites 76.0%

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

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 76.0

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

10. Applied rewrites 75.9%

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

12. Applied rewrites 87.5%

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

13. Final simplification 87.5%

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

Alternative 5: 75.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return ((1.0f * u0) - (-u0 * 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 = ((1.0e0 * u0) - (-u0 * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f32 N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower-log1p.f32 N/A

      \[\leadsto \frac{-\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0 \cdot u0\right)\right)} - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

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

      \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u0\right)\right) \cdot u0}\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u0\right)\right) \cdot u0}\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   11. lower-neg.f32 N/A

       \[\leadsto \frac{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u0\right)} \cdot u0\right) - \log \left(1 + u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   12. lower-log1p.f32 76.0

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

4. Applied rewrites 76.0%

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

5. Taylor expanded in u0 around 0

   \[\leadsto \frac{-\left(\color{blue}{-1 \cdot {u0}^{2}} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{-\left(\color{blue}{\left(\mathsf{neg}\left({u0}^{2}\right)\right)} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. unpow2 N/A

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

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

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

   4. mul-1-neg N/A

      \[\leadsto \frac{-\left(\color{blue}{\left(-1 \cdot u0\right)} \cdot u0 - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{-\left(\color{blue}{\left(-1 \cdot u0\right) \cdot u0} - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. mul-1-neg N/A

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

   7. lower-neg.f32 76.0

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

7. Applied rewrites 76.0%

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

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 76.0

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

10. Applied rewrites 75.9%

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

11. Taylor expanded in u0 around 0

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

13. Applied rewrites 76.0%

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

14. Final simplification 76.0%

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

Alternative 6: 75.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.5%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 75.4

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

5. Applied rewrites 75.4%

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

6. Final simplification 75.4%

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

Alternative 7: 66.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{alphax \cdot u0}{cos2phi} \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.99999996490334e-14)
   (* (/ (* alphax u0) cos2phi) alphax)
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14f) {
		tmp = ((alphax * u0) / cos2phi) * alphax;
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14) then
        tmp = ((alphax * u0) / cos2phi) * alphax
    else
        tmp = ((alphay * alphay) * u0) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.99999996490334e-14))
		tmp = Float32(Float32(Float32(alphax * u0) / cos2phi) * alphax);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.99999996490334e-14))
		tmp = ((alphax * u0) / cos2phi) * alphax;
	else
		tmp = ((alphay * alphay) * u0) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999996e-14

   1. Initial program 49.9%

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

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 61.3%

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

   10. Applied rewrites 61.4%

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

   if 1.99999996e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.9%

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

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 74.2

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

   5. Applied rewrites 74.2%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 69.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{alphax \cdot u0}{cos2phi} \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{alphax \cdot u0}{cos2phi} \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.99999996490334e-14)
   (* (/ (* alphax u0) cos2phi) alphax)
   (* (/ u0 sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14f) {
		tmp = ((alphax * u0) / cos2phi) * alphax;
	} else {
		tmp = (u0 / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14) then
        tmp = ((alphax * u0) / cos2phi) * alphax
    else
        tmp = (u0 / sin2phi) * (alphay * alphay)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.99999996490334e-14))
		tmp = Float32(Float32(Float32(alphax * u0) / cos2phi) * alphax);
	else
		tmp = Float32(Float32(u0 / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.99999996490334e-14))
		tmp = ((alphax * u0) / cos2phi) * alphax;
	else
		tmp = (u0 / sin2phi) * (alphay * alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999996e-14

   1. Initial program 49.9%

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

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 61.3%

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

   10. Applied rewrites 61.4%

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

   if 1.99999996e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.9%

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

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 74.2

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

   5. Applied rewrites 74.2%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 11.4%

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

   9. Taylor expanded in alphay around 0

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

   11. Applied rewrites 68.8%

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

   12. Taylor expanded in alphax around inf

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

   14. Applied rewrites 69.4%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{alphax \cdot u0}{cos2phi} \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 23.3% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.5%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 75.4

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

5. Applied rewrites 75.4%

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

6. Taylor expanded in alphax around 0

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

8. Applied rewrites 27.0%

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

10. Applied rewrites 27.0%

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

11. Final simplification 27.0%

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

Alternative 10: 23.3% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.5%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 75.4

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

5. Applied rewrites 75.4%

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

6. Taylor expanded in alphax around 0

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

8. Applied rewrites 27.0%

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

10. Applied rewrites 27.0%

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

11. Final simplification 27.0%

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

Reproduce

herbie shell --seed 2024304 
(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)))))