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

Percentage Accurate: 61.0% → 96.3%

Time: 16.3s

Alternatives: 17

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: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{alphax \cdot alphax}{cos2phi}\right)}^{-0.5}\\ \mathbf{if}\;1 - u0 \leq 0.9965000152587891:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{sin2phi}{-alphay \cdot alphay} - t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (pow (/ (* alphax alphax) cos2phi) -0.5)))
   (if (<= (- 1.0 u0) 0.9965000152587891)
     (/ (log (- 1.0 u0)) (- (/ sin2phi (- (* alphay alphay))) (* t_0 t_0)))
     (/
      (- u0 (* u0 (* u0 -0.5)))
      (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = powf(((alphax * alphax) / cos2phi), -0.5f);
	float tmp;
	if ((1.0f - u0) <= 0.9965000152587891f) {
		tmp = logf((1.0f - u0)) / ((sin2phi / -(alphay * alphay)) - (t_0 * t_0));
	} else {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	}
	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 = ((alphax * alphax) / cos2phi) ** (-0.5e0)
    if ((1.0e0 - u0) <= 0.9965000152587891e0) then
        tmp = log((1.0e0 - u0)) / ((sin2phi / -(alphay * alphay)) - (t_0 * t_0))
    else
        tmp = (u0 - (u0 * (u0 * (-0.5e0)))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(alphax * alphax) / cos2phi) ^ Float32(-0.5)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9965000152587891))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) / Float32(Float32(sin2phi / Float32(-Float32(alphay * alphay))) - Float32(t_0 * t_0)));
	else
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = ((alphax * alphax) / cos2phi) ^ single(-0.5);
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9965000152587891))
		tmp = log((single(1.0) - u0)) / ((sin2phi / -(alphay * alphay)) - (t_0 * t_0));
	else
		tmp = (u0 - (u0 * (u0 * single(-0.5)))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99650002

   1. Initial program 93.4%

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

      1. lift-*.f32 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.f32 N/A

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

      10. lower-/.f32 N/A

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

      11. metadata-eval 93.5

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

   4. Applied rewrites 93.5%

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

   if 0.99650002 < (-.f32 #s(literal 1 binary32) u0)

   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

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 17.6

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

   5. Applied rewrites 17.5%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 98.1

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

   7. Applied rewrites 98.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9965000152587891:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{sin2phi}{-alphay \cdot alphay} - {\left(\frac{alphax \cdot alphax}{cos2phi}\right)}^{-0.5} \cdot {\left(\frac{alphax \cdot alphax}{cos2phi}\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;-\log \left(1 - u0\right) \leq 0.027000000700354576:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{1 - u0}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= (- (log (- 1.0 u0))) 0.027000000700354576)
     (/ (- u0 (* u0 (* u0 -0.5))) (+ t_0 (/ cos2phi (* alphax alphax))))
     (/ (log (/ 1.0 (- 1.0 u0))) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (-logf((1.0f - u0)) <= 0.027000000700354576f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = logf((1.0f / (1.0f - u0))) / 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) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (-log((1.0e0 - u0)) <= 0.027000000700354576e0) then
        tmp = (u0 - (u0 * (u0 * (-0.5e0)))) / (t_0 + (cos2phi / (alphax * alphax)))
    else
        tmp = log((1.0e0 / (1.0e0 - u0))) / t_0
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (Float32(-log(Float32(Float32(1.0) - u0))) <= Float32(0.027000000700354576))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u0))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (-log((single(1.0) - u0)) <= single(0.027000000700354576))
		tmp = (u0 - (u0 * (u0 * single(-0.5)))) / (t_0 + (cos2phi / (alphax * alphax)));
	else
		tmp = log((single(1.0) / (single(1.0) - u0))) / 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))) < 0.0270000007

   1. Initial program 54.8%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 16.4

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

   5. Applied rewrites 16.3%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 95.0

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

   7. Applied rewrites 95.0%

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

   if 0.0270000007 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0)))

   1. Initial program 96.3%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. remove-double-neg N/A

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

      4. lift-log.f32 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. clear-num N/A

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

      8. log-rec N/A

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

      9. clear-num N/A

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

      10. lower-log.f32 N/A

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

      11. lower-/.f32 95.2

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in cos2phi around 0

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

      1. lower-/.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 74.1

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

   7. Applied rewrites 74.1%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u0\right) \leq 0.027000000700354576:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{1 - u0}\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;1 - u0 \leq 0.9965000152587891:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{1}{\frac{-1}{t\_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))
   (if (<= (- 1.0 u0) 0.9965000152587891)
     (/ (log (- 1.0 u0)) (/ 1.0 (/ -1.0 t_0)))
     (/ (- u0 (* u0 (* u0 -0.5))) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax));
	float tmp;
	if ((1.0f - u0) <= 0.9965000152587891f) {
		tmp = logf((1.0f - u0)) / (1.0f / (-1.0f / t_0));
	} else {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / 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) :: tmp
    t_0 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax))
    if ((1.0e0 - u0) <= 0.9965000152587891e0) then
        tmp = log((1.0e0 - u0)) / (1.0e0 / ((-1.0e0) / t_0))
    else
        tmp = (u0 - (u0 * (u0 * (-0.5e0)))) / t_0
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax)))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9965000152587891))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) / Float32(Float32(1.0) / Float32(Float32(-1.0) / t_0)));
	else
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax));
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9965000152587891))
		tmp = log((single(1.0) - u0)) / (single(1.0) / (single(-1.0) / t_0));
	else
		tmp = (u0 - (u0 * (u0 * single(-0.5)))) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99650002

   1. Initial program 93.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. remove-double-neg N/A

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

      4. lift-log.f32 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. clear-num N/A

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

      8. log-rec N/A

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

      9. clear-num N/A

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

      10. lower-log.f32 N/A

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

      11. lower-/.f32 91.3

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

   4. Applied rewrites 91.3%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. lower-/.f32 N/A

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

   6. Applied rewrites 63.2%

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

   8. Applied rewrites 91.3%

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

      1. lift--.f32 N/A

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

      2. /-rgt-identity N/A

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

      3. clear-num N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. log-rec N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-log.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-neg.f32 93.4

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

   10. Applied rewrites 93.4%

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

   if 0.99650002 < (-.f32 #s(literal 1 binary32) u0)

   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

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 17.6

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

   5. Applied rewrites 17.5%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 98.1

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

   7. Applied rewrites 98.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9965000152587891:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{1}{\frac{-1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9965000152587891:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{cos2phi \cdot \frac{-1}{alphax \cdot alphax} - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= (- 1.0 u0) 0.9965000152587891)
     (/ (log (- 1.0 u0)) (- (* cos2phi (/ -1.0 (* alphax alphax))) t_0))
     (/ (- u0 (* u0 (* u0 -0.5))) (+ t_0 (/ cos2phi (* alphax alphax)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if ((1.0f - u0) <= 0.9965000152587891f) {
		tmp = logf((1.0f - u0)) / ((cos2phi * (-1.0f / (alphax * alphax))) - t_0);
	} else {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (t_0 + (cos2phi / (alphax * alphax)));
	}
	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 = sin2phi / (alphay * alphay)
    if ((1.0e0 - u0) <= 0.9965000152587891e0) then
        tmp = log((1.0e0 - u0)) / ((cos2phi * ((-1.0e0) / (alphax * alphax))) - t_0)
    else
        tmp = (u0 - (u0 * (u0 * (-0.5e0)))) / (t_0 + (cos2phi / (alphax * alphax)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9965000152587891))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) / Float32(Float32(cos2phi * Float32(Float32(-1.0) / Float32(alphax * alphax))) - t_0));
	else
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9965000152587891))
		tmp = log((single(1.0) - u0)) / ((cos2phi * (single(-1.0) / (alphax * alphax))) - t_0);
	else
		tmp = (u0 - (u0 * (u0 * single(-0.5)))) / (t_0 + (cos2phi / (alphax * alphax)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99650002

   1. Initial program 93.4%

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

      1. lift-*.f32 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 93.4

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

   4. Applied rewrites 93.4%

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

   if 0.99650002 < (-.f32 #s(literal 1 binary32) u0)

   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

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 17.6

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

   5. Applied rewrites 17.5%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 98.1

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

   7. Applied rewrites 98.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9965000152587891:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{cos2phi \cdot \frac{-1}{alphax \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9965000152587891:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot \left(-alphax\right)} - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= (- 1.0 u0) 0.9965000152587891)
     (/ (log (- 1.0 u0)) (- (/ cos2phi (* alphax (- alphax))) t_0))
     (/ (- u0 (* u0 (* u0 -0.5))) (+ t_0 (/ cos2phi (* alphax alphax)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if ((1.0f - u0) <= 0.9965000152587891f) {
		tmp = logf((1.0f - u0)) / ((cos2phi / (alphax * -alphax)) - t_0);
	} else {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (t_0 + (cos2phi / (alphax * alphax)));
	}
	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 = sin2phi / (alphay * alphay)
    if ((1.0e0 - u0) <= 0.9965000152587891e0) then
        tmp = log((1.0e0 - u0)) / ((cos2phi / (alphax * -alphax)) - t_0)
    else
        tmp = (u0 - (u0 * (u0 * (-0.5e0)))) / (t_0 + (cos2phi / (alphax * alphax)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9965000152587891))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) / Float32(Float32(cos2phi / Float32(alphax * Float32(-alphax))) - t_0));
	else
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9965000152587891))
		tmp = log((single(1.0) - u0)) / ((cos2phi / (alphax * -alphax)) - t_0);
	else
		tmp = (u0 - (u0 * (u0 * single(-0.5)))) / (t_0 + (cos2phi / (alphax * alphax)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99650002

   1. Initial program 93.4%

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

   if 0.99650002 < (-.f32 #s(literal 1 binary32) u0)

   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

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 17.6

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

   5. Applied rewrites 17.5%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 98.1

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

   7. Applied rewrites 98.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9965000152587891:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot \left(-alphax\right)} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.800000011920929:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 - -0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.800000011920929)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
   (* (* alphay alphay) (/ (- u0 (* -0.5 (* u0 u0))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.800000011920929f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = (alphay * alphay) * ((u0 - (-0.5f * (u0 * 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)) <= 0.800000011920929e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
    else
        tmp = (alphay * alphay) * ((u0 - ((-0.5e0) * (u0 * 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(0.800000011920929))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 - Float32(Float32(-0.5) * Float32(u0 * 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(0.800000011920929))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	else
		tmp = (alphay * alphay) * ((u0 - (single(-0.5) * (u0 * u0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.800000012

   1. Initial program 54.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.8

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

   5. Applied rewrites 75.8%

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

      1. associate-/r* N/A

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

      2. lower-/.f32 N/A

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

      3. lower-/.f32 75.9

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

   7. Applied rewrites 75.9%

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

   if 0.800000012 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.1%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 23.8

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

   5. Applied rewrites 23.7%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 87.4

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

   7. Applied rewrites 87.4%

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

      1. associate-/r* N/A

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

      2. lower-/.f32 N/A

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

      3. lower-/.f32 87.4

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

   9. Applied rewrites 87.4%

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

   10. Taylor expanded in cos2phi around 0

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

       1. associate-/l* N/A

          \[\leadsto \color{blue}{{alphay}^{2} \cdot \frac{u0 - \frac{-1}{2} \cdot {u0}^{2}}{sin2phi}} \]

       2. div-sub N/A

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

       3. associate-*r/ N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. associate-*r/ N/A

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

       8. div-sub N/A

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

       9. lower-/.f32 N/A

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

       10. lower--.f32 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 86.6

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

   12. Applied rewrites 86.6%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.800000011920929:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 - -0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 0.800000011920929:\\ \;\;\;\;\frac{u0}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 - -0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.800000011920929)
     (/ u0 (+ t_0 (/ cos2phi (* alphax alphax))))
     (* (* alphay alphay) (/ (- u0 (* -0.5 (* u0 u0))) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.800000011920929f) {
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = (alphay * alphay) * ((u0 - (-0.5f * (u0 * 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) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.800000011920929e0) then
        tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)))
    else
        tmp = (alphay * alphay) * ((u0 - ((-0.5e0) * (u0 * u0))) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.800000011920929))
		tmp = Float32(u0 / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 - Float32(Float32(-0.5) * Float32(u0 * u0))) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.800000011920929))
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	else
		tmp = (alphay * alphay) * ((u0 - (single(-0.5) * (u0 * u0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.800000012

   1. Initial program 54.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.8

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

   5. Applied rewrites 75.8%

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

   if 0.800000012 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.1%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 23.8

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

   5. Applied rewrites 23.7%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 87.4

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

   7. Applied rewrites 87.4%

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

      1. associate-/r* N/A

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

      2. lower-/.f32 N/A

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

      3. lower-/.f32 87.4

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

   9. Applied rewrites 87.4%

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

   10. Taylor expanded in cos2phi around 0

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

       1. associate-/l* N/A

          \[\leadsto \color{blue}{{alphay}^{2} \cdot \frac{u0 - \frac{-1}{2} \cdot {u0}^{2}}{sin2phi}} \]

       2. div-sub N/A

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

       3. associate-*r/ N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. associate-*r/ N/A

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

       8. div-sub N/A

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

       9. lower-/.f32 N/A

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

       10. lower--.f32 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 86.6

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

   12. Applied rewrites 86.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.800000011920929:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 - -0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 6.999999885293207 \cdot 10^{-11}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 - -0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 6.999999885293207e-11)
   (/ (- u0 (* u0 (* u0 -0.5))) (/ cos2phi (* alphax alphax)))
   (* (* alphay alphay) (/ (- u0 (* -0.5 (* u0 u0))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 6.999999885293207e-11f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (cos2phi / (alphax * alphax));
	} else {
		tmp = (alphay * alphay) * ((u0 - (-0.5f * (u0 * 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)) <= 6.999999885293207e-11) then
        tmp = (u0 - (u0 * (u0 * (-0.5e0)))) / (cos2phi / (alphax * alphax))
    else
        tmp = (alphay * alphay) * ((u0 - ((-0.5e0) * (u0 * 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(6.999999885293207e-11))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 - Float32(Float32(-0.5) * Float32(u0 * 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(6.999999885293207e-11))
		tmp = (u0 - (u0 * (u0 * single(-0.5)))) / (cos2phi / (alphax * alphax));
	else
		tmp = (alphay * alphay) * ((u0 - (single(-0.5) * (u0 * u0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 6.99999989e-11

   1. Initial program 57.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 2.8

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

   5. Applied rewrites 2.9%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 86.1

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in cos2phi around inf

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

      1. lower-/.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 60.8

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

   10. Applied rewrites 60.8%

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

   if 6.99999989e-11 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 19.5

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

   5. Applied rewrites 19.5%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 88.6

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

   7. Applied rewrites 88.6%

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

      1. associate-/r* N/A

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

      2. lower-/.f32 N/A

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

      3. lower-/.f32 88.6

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

   9. Applied rewrites 88.6%

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

   10. Taylor expanded in cos2phi around 0

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

       1. associate-/l* N/A

          \[\leadsto \color{blue}{{alphay}^{2} \cdot \frac{u0 - \frac{-1}{2} \cdot {u0}^{2}}{sin2phi}} \]

       2. div-sub N/A

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

       3. associate-*r/ N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. associate-*r/ N/A

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

       8. div-sub N/A

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

       9. lower-/.f32 N/A

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

       10. lower--.f32 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 81.5

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

   12. Applied rewrites 81.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 6.999999885293207 \cdot 10^{-11}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 - -0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 14.8

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

5. Applied rewrites 14.7%

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

   1. distribute-rgt-in N/A

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

   2. neg-mul-1 N/A

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

   3. unsub-neg N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 87.9

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

7. Applied rewrites 87.9%

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

8. Final simplification 87.9%

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

Alternative 10: 87.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 14.8

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

5. Applied rewrites 14.7%

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 87.7

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

7. Applied rewrites 87.7%

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

8. Final simplification 87.7%

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

Alternative 11: 75.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 6.999999885293207 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot 0.5\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 - -0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 6.999999885293207e-11)
   (/ (* (* alphax alphax) (+ u0 (* (* u0 u0) 0.5))) cos2phi)
   (* (* alphay alphay) (/ (- u0 (* -0.5 (* u0 u0))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 6.999999885293207e-11f) {
		tmp = ((alphax * alphax) * (u0 + ((u0 * u0) * 0.5f))) / cos2phi;
	} else {
		tmp = (alphay * alphay) * ((u0 - (-0.5f * (u0 * 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)) <= 6.999999885293207e-11) then
        tmp = ((alphax * alphax) * (u0 + ((u0 * u0) * 0.5e0))) / cos2phi
    else
        tmp = (alphay * alphay) * ((u0 - ((-0.5e0) * (u0 * 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(6.999999885293207e-11))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(u0 + Float32(Float32(u0 * u0) * Float32(0.5)))) / cos2phi);
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 - Float32(Float32(-0.5) * Float32(u0 * 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(6.999999885293207e-11))
		tmp = ((alphax * alphax) * (u0 + ((u0 * u0) * single(0.5)))) / cos2phi;
	else
		tmp = (alphay * alphay) * ((u0 - (single(-0.5) * (u0 * u0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 6.99999989e-11

   1. Initial program 57.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 2.8

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

   5. Applied rewrites 2.9%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 86.1

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in cos2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 60.8

          \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot \left(u0 + 0.5 \cdot \color{blue}{\left(u0 \cdot u0\right)}\right)}{cos2phi} \]

   10. Applied rewrites 60.8%

       \[\leadsto \color{blue}{\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 + 0.5 \cdot \left(u0 \cdot u0\right)\right)}{cos2phi}} \]

   if 6.99999989e-11 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 19.5

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

   5. Applied rewrites 19.5%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 88.6

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

   7. Applied rewrites 88.6%

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

      1. associate-/r* N/A

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

      2. lower-/.f32 N/A

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

      3. lower-/.f32 88.6

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

   9. Applied rewrites 88.6%

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

   10. Taylor expanded in cos2phi around 0

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

       1. associate-/l* N/A

          \[\leadsto \color{blue}{{alphay}^{2} \cdot \frac{u0 - \frac{-1}{2} \cdot {u0}^{2}}{sin2phi}} \]

       2. div-sub N/A

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

       3. associate-*r/ N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. associate-*r/ N/A

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

       8. div-sub N/A

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

       9. lower-/.f32 N/A

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

       10. lower--.f32 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 81.5

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

   12. Applied rewrites 81.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 6.999999885293207 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot 0.5\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 - -0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u0 - -0.5 \cdot \left(u0 \cdot u0\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 6.999999885293207 \cdot 10^{-11}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{t\_0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{t\_0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (- u0 (* -0.5 (* u0 u0)))))
   (if (<= (/ sin2phi (* alphay alphay)) 6.999999885293207e-11)
     (* (* 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 - (-0.5f * (u0 * u0));
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 6.999999885293207e-11f) {
		tmp = (alphax * alphax) * (t_0 / cos2phi);
	} else {
		tmp = (alphay * alphay) * (t_0 / 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) :: t_0
    real(4) :: tmp
    t_0 = u0 - ((-0.5e0) * (u0 * u0))
    if ((sin2phi / (alphay * alphay)) <= 6.999999885293207e-11) then
        tmp = (alphax * alphax) * (t_0 / cos2phi)
    else
        tmp = (alphay * alphay) * (t_0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(u0 - Float32(Float32(-0.5) * Float32(u0 * u0)))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(6.999999885293207e-11))
		tmp = Float32(Float32(alphax * alphax) * Float32(t_0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(t_0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = u0 - (single(-0.5) * (u0 * u0));
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(6.999999885293207e-11))
		tmp = (alphax * alphax) * (t_0 / cos2phi);
	else
		tmp = (alphay * alphay) * (t_0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 6.99999989e-11

   1. Initial program 57.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 2.8

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

   5. Applied rewrites 2.9%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 86.1

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

   7. Applied rewrites 86.1%

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

      1. associate-/r* N/A

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

      2. lower-/.f32 N/A

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

      3. lower-/.f32 86.0

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

   9. Applied rewrites 86.0%

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

   10. Taylor expanded in cos2phi around inf

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

       1. associate-/l* N/A

          \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{u0 - \frac{-1}{2} \cdot {u0}^{2}}{cos2phi}} \]

       2. div-sub N/A

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

       3. associate-*r/ N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. associate-*r/ N/A

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

       8. div-sub N/A

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

       9. lower-/.f32 N/A

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

       10. lower--.f32 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 60.8

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

   12. Applied rewrites 60.8%

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

   if 6.99999989e-11 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 19.5

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

   5. Applied rewrites 19.5%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 88.6

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

   7. Applied rewrites 88.6%

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

      1. associate-/r* N/A

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

      2. lower-/.f32 N/A

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

      3. lower-/.f32 88.6

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

   9. Applied rewrites 88.6%

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

   10. Taylor expanded in cos2phi around 0

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

       1. associate-/l* N/A

          \[\leadsto \color{blue}{{alphay}^{2} \cdot \frac{u0 - \frac{-1}{2} \cdot {u0}^{2}}{sin2phi}} \]

       2. div-sub N/A

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

       3. associate-*r/ N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. associate-*r/ N/A

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

       8. div-sub N/A

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

       9. lower-/.f32 N/A

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

       10. lower--.f32 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 81.5

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

   12. Applied rewrites 81.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 6.999999885293207 \cdot 10^{-11}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0 - -0.5 \cdot \left(u0 \cdot u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 - -0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0 - -0.5 \cdot \left(u0 \cdot u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.999999858590343e-10)
   (* (* alphax alphax) (/ (- u0 (* -0.5 (* u0 u0))) cos2phi))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.999999858590343e-10f) {
		tmp = (alphax * alphax) * ((u0 - (-0.5f * (u0 * u0))) / cos2phi);
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.999999858590343e-10) then
        tmp = (alphax * alphax) * ((u0 - ((-0.5e0) * (u0 * u0))) / cos2phi)
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.999999858590343e-10))
		tmp = Float32(Float32(alphax * alphax) * Float32(Float32(u0 - Float32(Float32(-0.5) * Float32(u0 * u0))) / cos2phi));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.999999858590343e-10))
		tmp = (alphax * alphax) * ((u0 - (single(-0.5) * (u0 * u0))) / cos2phi);
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999986e-10

   1. Initial program 56.8%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 2.9

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

   5. Applied rewrites 2.9%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 86.5

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

   7. Applied rewrites 86.5%

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

      1. associate-/r* N/A

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

      2. lower-/.f32 N/A

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

      3. lower-/.f32 86.5

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

   9. Applied rewrites 86.5%

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

   10. Taylor expanded in cos2phi around inf

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

       1. associate-/l* N/A

          \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{u0 - \frac{-1}{2} \cdot {u0}^{2}}{cos2phi}} \]

       2. div-sub N/A

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

       3. associate-*r/ N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. associate-*r/ N/A

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

       8. div-sub N/A

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

       9. lower-/.f32 N/A

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

       10. lower--.f32 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 60.1

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

   12. Applied rewrites 60.1%

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

   if 4.99999986e-10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 77.7

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

   5. Applied rewrites 77.7%

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

   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. *-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 72.8

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

   8. Applied rewrites 72.8%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0 - -0.5 \cdot \left(u0 \cdot u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.999999858590343e-10)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.999999858590343e-10f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.999999858590343e-10) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.999999858590343e-10))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.999999858590343e-10))
		tmp = u0 / (cos2phi / (alphax * alphax));
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999986e-10

   1. Initial program 56.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 73.9

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

   5. Applied rewrites 73.9%

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

   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 51.9

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

   8. Applied rewrites 51.9%

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

   if 4.99999986e-10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 77.7

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

   5. Applied rewrites 77.7%

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

   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. *-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 72.8

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

   8. Applied rewrites 72.8%

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

Alternative 15: 66.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;u0 \cdot \left(\left(alphax \cdot alphax\right) \cdot \frac{1}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.999999858590343e-10)
   (* u0 (* (* alphax alphax) (/ 1.0 cos2phi)))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.999999858590343e-10f) {
		tmp = u0 * ((alphax * alphax) * (1.0f / cos2phi));
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.999999858590343e-10) then
        tmp = u0 * ((alphax * alphax) * (1.0e0 / cos2phi))
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.999999858590343e-10))
		tmp = Float32(u0 * Float32(Float32(alphax * alphax) * Float32(Float32(1.0) / cos2phi)));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.999999858590343e-10))
		tmp = u0 * ((alphax * alphax) * (single(1.0) / cos2phi));
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999986e-10

   1. Initial program 56.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 73.9

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

   5. Applied rewrites 73.9%

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

   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. *-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 51.9

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

   8. Applied rewrites 51.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 51.9

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

   10. Applied rewrites 51.9%

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

       1. lift-*.f32 N/A

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

       2. lift-/.f32 N/A

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

       3. *-commutative N/A

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

       4. lift-/.f32 N/A

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

       5. div-inv N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f32 N/A

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

       8. lower-*.f32 N/A

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

       9. lower-/.f32 51.9

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

   12. Applied rewrites 51.9%

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

   if 4.99999986e-10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 77.7

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

   5. Applied rewrites 77.7%

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

   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. *-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 72.8

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

   8. Applied rewrites 72.8%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;u0 \cdot \left(\left(alphax \cdot alphax\right) \cdot \frac{1}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.999999858590343e-10)
   (* (* alphax alphax) (/ u0 cos2phi))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.999999858590343e-10f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.999999858590343e-10) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.999999858590343e-10))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.999999858590343e-10))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999986e-10

   1. Initial program 56.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 73.9

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

   5. Applied rewrites 73.9%

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

   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. *-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 51.9

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

   8. Applied rewrites 51.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 51.9

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

   10. Applied rewrites 51.9%

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

   if 4.99999986e-10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 77.7

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

   5. Applied rewrites 77.7%

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

   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. *-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 72.8

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

   8. Applied rewrites 72.8%

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

Alternative 17: 23.7% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
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 76.6

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

5. Applied rewrites 76.6%

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

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. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 23.4

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

8. Applied rewrites 23.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 23.4

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

10. Applied rewrites 23.4%

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

Reproduce

herbie shell --seed 2024216 
(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)))))