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

Percentage Accurate: 60.7% → 98.3%

Time: 18.6s

Alternatives: 13

Speedup: 12.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.3%

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

   1. sub-neg 61.3%

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

   2. log1p-def 98.4%

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

3. Simplified 98.4%

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

   1. clear-num 98.3%

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

   2. associate-/r/ 98.3%

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

   3. pow2 98.3%

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

   4. pow-flip 98.4%

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

   5. metadata-eval 98.4%

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

5. Applied egg-rr 98.4%

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

6. Taylor expanded in alphax around 0 98.4%

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

   1. +-commutative 98.4%

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

   2. unpow2 98.4%

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

   3. unpow2 98.4%

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

   4. associate-/r* 98.5%

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

8. Simplified 98.5%

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

9. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.3%

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

   1. sub-neg 61.3%

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

   2. log1p-def 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 3: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.999999848427251 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.999999848427251e-5)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ (/ sin2phi alphay) alphay) (/ cos2phi (* alphax alphax))))
   (/ (- (log1p (- u0))) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.999999848427251e-5f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (((sin2phi / alphay) / alphay) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = -log1pf(-u0) / (sin2phi / (alphay * alphay));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.999999848427251e-5))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(Float32(sin2phi / alphay) / alphay) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / Float32(sin2phi / Float32(alphay * alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.99999985e-5

   1. Initial program 52.7%

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

      1. sub-neg 52.7%

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

      2. log1p-def 98.7%

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

   3. Simplified 98.7%

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

      1. clear-num 98.5%

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

      2. associate-/r/ 98.5%

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

      3. pow2 98.5%

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

      4. pow-flip 98.7%

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

      5. metadata-eval 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in alphax around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. unpow2 98.7%

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

      3. unpow2 98.7%

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

      4. associate-/r* 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in u0 around 0 86.7%

      \[\leadsto \frac{-\color{blue}{\left(-1 \cdot u0 + -0.5 \cdot {u0}^{2}\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
   10. Step-by-step derivation

       1. +-commutative 86.7%

          \[\leadsto \frac{-\color{blue}{\left(-0.5 \cdot {u0}^{2} + -1 \cdot u0\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       2. neg-mul-1 86.7%

          \[\leadsto \frac{-\left(-0.5 \cdot {u0}^{2} + \color{blue}{\left(-u0\right)}\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       3. unsub-neg 86.7%

          \[\leadsto \frac{-\color{blue}{\left(-0.5 \cdot {u0}^{2} - u0\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       4. *-commutative 86.7%

          \[\leadsto \frac{-\left(\color{blue}{{u0}^{2} \cdot -0.5} - u0\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       5. unpow2 86.7%

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

       6. associate-*l* 86.7%

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

   11. Simplified 86.7%

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

   if 5.99999985e-5 < sin2phi

   1. Initial program 68.2%

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

      1. sub-neg 68.2%

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

      2. log1p-def 98.2%

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

   3. Simplified 98.2%

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

      1. clear-num 98.2%

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

      2. associate-/r/ 98.2%

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

      3. pow2 98.2%

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

      4. pow-flip 98.2%

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

      5. metadata-eval 98.2%

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

   5. Applied egg-rr 98.2%

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

   6. Taylor expanded in alphax around inf 97.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 97.2%

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

   8. Simplified 97.2%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.999999848427251 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

Alternative 4: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{sin2phi}{alphay}}{alphay}\\ \mathbf{if}\;sin2phi \leq 5.999999848427251 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ (/ sin2phi alphay) alphay)))
   (if (<= sin2phi 5.999999848427251e-5)
     (/ (- u0 (* u0 (* u0 -0.5))) (+ t_0 (/ cos2phi (* alphax alphax))))
     (/ (- (log1p (- 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 (sin2phi <= 5.999999848427251e-5f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = -log1pf(-u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(sin2phi / alphay) / alphay)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.999999848427251e-5))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.99999985e-5

   1. Initial program 52.7%

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

      1. sub-neg 52.7%

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

      2. log1p-def 98.7%

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

   3. Simplified 98.7%

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

      1. clear-num 98.5%

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

      2. associate-/r/ 98.5%

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

      3. pow2 98.5%

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

      4. pow-flip 98.7%

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

      5. metadata-eval 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in alphax around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. unpow2 98.7%

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

      3. unpow2 98.7%

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

      4. associate-/r* 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in u0 around 0 86.7%

      \[\leadsto \frac{-\color{blue}{\left(-1 \cdot u0 + -0.5 \cdot {u0}^{2}\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
   10. Step-by-step derivation

       1. +-commutative 86.7%

          \[\leadsto \frac{-\color{blue}{\left(-0.5 \cdot {u0}^{2} + -1 \cdot u0\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       2. neg-mul-1 86.7%

          \[\leadsto \frac{-\left(-0.5 \cdot {u0}^{2} + \color{blue}{\left(-u0\right)}\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       3. unsub-neg 86.7%

          \[\leadsto \frac{-\color{blue}{\left(-0.5 \cdot {u0}^{2} - u0\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       4. *-commutative 86.7%

          \[\leadsto \frac{-\left(\color{blue}{{u0}^{2} \cdot -0.5} - u0\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       5. unpow2 86.7%

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

       6. associate-*l* 86.7%

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

   11. Simplified 86.7%

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

   if 5.99999985e-5 < sin2phi

   1. Initial program 68.2%

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

      1. sub-neg 68.2%

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

      2. log1p-def 98.2%

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

   3. Simplified 98.2%

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

      1. clear-num 98.2%

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

      2. associate-/r/ 98.2%

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

      3. pow2 98.2%

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

      4. pow-flip 98.2%

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

      5. metadata-eval 98.2%

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

   5. Applied egg-rr 98.2%

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

   6. Taylor expanded in alphax around inf 97.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 97.2%

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

      2. associate-/r* 97.3%

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

   8. Simplified 97.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.999999848427251 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]

Alternative 5: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.999999848427251 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.999999848427251e-5)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ (/ sin2phi alphay) alphay) (/ cos2phi (* alphax alphax))))
   (/ (* alphay (* (log1p (- u0)) (- alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.999999848427251e-5f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (((sin2phi / alphay) / alphay) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = (alphay * (log1pf(-u0) * -alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.999999848427251e-5))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(Float32(sin2phi / alphay) / alphay) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(alphay * Float32(log1p(Float32(-u0)) * Float32(-alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.99999985e-5

   1. Initial program 52.7%

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

      1. sub-neg 52.7%

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

      2. log1p-def 98.7%

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

   3. Simplified 98.7%

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

      1. clear-num 98.5%

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

      2. associate-/r/ 98.5%

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

      3. pow2 98.5%

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

      4. pow-flip 98.7%

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

      5. metadata-eval 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in alphax around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. unpow2 98.7%

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

      3. unpow2 98.7%

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

      4. associate-/r* 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in u0 around 0 86.7%

      \[\leadsto \frac{-\color{blue}{\left(-1 \cdot u0 + -0.5 \cdot {u0}^{2}\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
   10. Step-by-step derivation

       1. +-commutative 86.7%

          \[\leadsto \frac{-\color{blue}{\left(-0.5 \cdot {u0}^{2} + -1 \cdot u0\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       2. neg-mul-1 86.7%

          \[\leadsto \frac{-\left(-0.5 \cdot {u0}^{2} + \color{blue}{\left(-u0\right)}\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       3. unsub-neg 86.7%

          \[\leadsto \frac{-\color{blue}{\left(-0.5 \cdot {u0}^{2} - u0\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       4. *-commutative 86.7%

          \[\leadsto \frac{-\left(\color{blue}{{u0}^{2} \cdot -0.5} - u0\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       5. unpow2 86.7%

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

       6. associate-*l* 86.7%

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

   11. Simplified 86.7%

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

   if 5.99999985e-5 < sin2phi

   1. Initial program 68.2%

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

      1. sub-neg 68.2%

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

      2. log1p-def 98.2%

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

   3. Simplified 98.2%

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

      1. clear-num 98.2%

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

      2. associate-/r/ 98.2%

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

      3. pow2 98.2%

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

      4. pow-flip 98.2%

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

      5. metadata-eval 98.2%

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

   5. Applied egg-rr 98.2%

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

   6. Taylor expanded in alphax around inf 68.8%

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

      1. associate-*r/ 68.8%

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

      2. mul-1-neg 68.8%

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

      3. unpow2 68.8%

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

      4. associate-*l* 68.8%

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

      5. sub-neg 68.8%

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

      6. log1p-def 98.1%

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

   8. Simplified 98.1%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.999999848427251 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 6: 89.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.999999848427251 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - sin2phi \cdot \left(u0 \cdot -0.08333333333333333\right)\right) - \left(u0 \cdot u0\right) \cdot \left(sin2phi \cdot -0.08333333333333333 + sin2phi \cdot 0.041666666666666664\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.999999848427251e-5)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ (/ sin2phi alphay) alphay) (/ cos2phi (* alphax alphax))))
   (/
    (* alphay (- alphay))
    (-
     (-
      (- (* sin2phi 0.5) (/ sin2phi u0))
      (* sin2phi (* u0 -0.08333333333333333)))
     (*
      (* u0 u0)
      (+
       (* sin2phi -0.08333333333333333)
       (* sin2phi 0.041666666666666664)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.999999848427251e-5f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (((sin2phi / alphay) / alphay) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = (alphay * -alphay) / ((((sin2phi * 0.5f) - (sin2phi / u0)) - (sin2phi * (u0 * -0.08333333333333333f))) - ((u0 * u0) * ((sin2phi * -0.08333333333333333f) + (sin2phi * 0.041666666666666664f))));
	}
	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 <= 5.999999848427251e-5) then
        tmp = (u0 - (u0 * (u0 * (-0.5e0)))) / (((sin2phi / alphay) / alphay) + (cos2phi / (alphax * alphax)))
    else
        tmp = (alphay * -alphay) / ((((sin2phi * 0.5e0) - (sin2phi / u0)) - (sin2phi * (u0 * (-0.08333333333333333e0)))) - ((u0 * u0) * ((sin2phi * (-0.08333333333333333e0)) + (sin2phi * 0.041666666666666664e0))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.999999848427251e-5))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(Float32(sin2phi / alphay) / alphay) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)) - Float32(sin2phi * Float32(u0 * Float32(-0.08333333333333333)))) - Float32(Float32(u0 * u0) * Float32(Float32(sin2phi * Float32(-0.08333333333333333)) + Float32(sin2phi * Float32(0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(5.999999848427251e-5))
		tmp = (u0 - (u0 * (u0 * single(-0.5)))) / (((sin2phi / alphay) / alphay) + (cos2phi / (alphax * alphax)));
	else
		tmp = (alphay * -alphay) / ((((sin2phi * single(0.5)) - (sin2phi / u0)) - (sin2phi * (u0 * single(-0.08333333333333333)))) - ((u0 * u0) * ((sin2phi * single(-0.08333333333333333)) + (sin2phi * single(0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.99999985e-5

   1. Initial program 52.7%

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

      1. sub-neg 52.7%

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

      2. log1p-def 98.7%

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

   3. Simplified 98.7%

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

      1. clear-num 98.5%

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

      2. associate-/r/ 98.5%

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

      3. pow2 98.5%

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

      4. pow-flip 98.7%

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

      5. metadata-eval 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in alphax around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. unpow2 98.7%

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

      3. unpow2 98.7%

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

      4. associate-/r* 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in u0 around 0 86.7%

      \[\leadsto \frac{-\color{blue}{\left(-1 \cdot u0 + -0.5 \cdot {u0}^{2}\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
   10. Step-by-step derivation

       1. +-commutative 86.7%

          \[\leadsto \frac{-\color{blue}{\left(-0.5 \cdot {u0}^{2} + -1 \cdot u0\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       2. neg-mul-1 86.7%

          \[\leadsto \frac{-\left(-0.5 \cdot {u0}^{2} + \color{blue}{\left(-u0\right)}\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       3. unsub-neg 86.7%

          \[\leadsto \frac{-\color{blue}{\left(-0.5 \cdot {u0}^{2} - u0\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       4. *-commutative 86.7%

          \[\leadsto \frac{-\left(\color{blue}{{u0}^{2} \cdot -0.5} - u0\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

       5. unpow2 86.7%

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

       6. associate-*l* 86.7%

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

   11. Simplified 86.7%

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

   if 5.99999985e-5 < sin2phi

   1. Initial program 68.2%

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

      1. sub-neg 68.2%

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

      2. log1p-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in cos2phi around 0 68.8%

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

      1. mul-1-neg 68.8%

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

      2. unpow2 68.8%

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

      3. associate-/l* 68.0%

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

      4. distribute-neg-frac 68.0%

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

      5. distribute-rgt-neg-out 68.0%

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

      6. sub-neg 68.0%

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

      7. log1p-def 96.7%

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

   6. Simplified 96.7%

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

   7. Taylor expanded in u0 around 0 94.0%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{-1 \cdot \left({u0}^{2} \cdot \left(-0.25 \cdot sin2phi + \left(0.16666666666666666 \cdot sin2phi + -0.5 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right)\right)\right) + \left(-1 \cdot \left(u0 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 94.0%

         \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(-1 \cdot \left(u0 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right) + -1 \cdot \left({u0}^{2} \cdot \left(-0.25 \cdot sin2phi + \left(0.16666666666666666 \cdot sin2phi + -0.5 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right)\right)\right)}} \]

      2. mul-1-neg 94.0%

         \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\left(-1 \cdot \left(u0 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right) + \color{blue}{\left(-{u0}^{2} \cdot \left(-0.25 \cdot sin2phi + \left(0.16666666666666666 \cdot sin2phi + -0.5 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right)\right)\right)}} \]

      3. unsub-neg 94.0%

         \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(-1 \cdot \left(u0 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right) - {u0}^{2} \cdot \left(-0.25 \cdot sin2phi + \left(0.16666666666666666 \cdot sin2phi + -0.5 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right)\right)}} \]

   9. Simplified 94.0%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - sin2phi \cdot \left(-0.08333333333333333 \cdot u0\right)\right) - \left(u0 \cdot u0\right) \cdot \left(sin2phi \cdot -0.08333333333333333 + sin2phi \cdot 0.041666666666666664\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.999999848427251 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - sin2phi \cdot \left(u0 \cdot -0.08333333333333333\right)\right) - \left(u0 \cdot u0\right) \cdot \left(sin2phi \cdot -0.08333333333333333 + sin2phi \cdot 0.041666666666666664\right)}\\ \end{array} \]

Alternative 7: 83.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - sin2phi \cdot \left(u0 \cdot -0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 4.999999873689376e-5)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/
      (* alphay (- alphay))
      (-
       (- (* sin2phi 0.5) (/ sin2phi u0))
       (* sin2phi (* u0 -0.08333333333333333)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 4.999999873689376e-5f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (alphay * -alphay) / (((sin2phi * 0.5f) - (sin2phi / u0)) - (sin2phi * (u0 * -0.08333333333333333f)));
	}
	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 <= 4.999999873689376e-5) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0)
    else
        tmp = (alphay * -alphay) / (((sin2phi * 0.5e0) - (sin2phi / u0)) - (sin2phi * (u0 * (-0.08333333333333333e0))))
    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(4.999999873689376e-5))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)) - Float32(sin2phi * Float32(u0 * Float32(-0.08333333333333333)))));
	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(4.999999873689376e-5))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	else
		tmp = (alphay * -alphay) / (((sin2phi * single(0.5)) - (sin2phi / u0)) - (sin2phi * (u0 * single(-0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999987e-5

   1. Initial program 53.2%

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

      1. sub-neg 53.2%

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

      2. log1p-def 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in u0 around 0 75.4%

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

      1. unpow2 75.4%

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

      2. unpow2 75.4%

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

   6. Simplified 75.4%

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

   if 4.99999987e-5 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.3%

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

      1. sub-neg 66.3%

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

      2. log1p-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in cos2phi around 0 66.2%

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

      1. mul-1-neg 66.2%

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

      2. unpow2 66.2%

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

      3. associate-/l* 65.5%

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

      4. distribute-neg-frac 65.5%

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

      5. distribute-rgt-neg-out 65.5%

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

      6. sub-neg 65.5%

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

      7. log1p-def 95.1%

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

   6. Simplified 95.1%

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

   7. Taylor expanded in u0 around 0 90.9%

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

      1. +-commutative 90.9%

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

      2. mul-1-neg 90.9%

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

      3. unsub-neg 90.9%

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

      4. +-commutative 90.9%

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

      5. mul-1-neg 90.9%

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

      6. unsub-neg 90.9%

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

      7. *-commutative 90.9%

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

      8. *-commutative 90.9%

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

      9. distribute-rgt-out 90.9%

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

      10. associate-*l* 90.9%

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

      11. metadata-eval 90.9%

          \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - sin2phi \cdot \left(\color{blue}{-0.08333333333333333} \cdot u0\right)} \]

   9. Simplified 90.9%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - sin2phi \cdot \left(-0.08333333333333333 \cdot u0\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - sin2phi \cdot \left(u0 \cdot -0.08333333333333333\right)}\\ \end{array} \]

Alternative 8: 87.5% accurate, 6.1× speedup

Math

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

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(Float32(sin2phi / 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 61.3%

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

   1. sub-neg 61.3%

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

   2. log1p-def 98.4%

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

3. Simplified 98.4%

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

   1. clear-num 98.3%

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

   2. associate-/r/ 98.3%

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

   3. pow2 98.3%

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

   4. pow-flip 98.4%

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

   5. metadata-eval 98.4%

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

5. Applied egg-rr 98.4%

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

6. Taylor expanded in alphax around 0 98.4%

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

   1. +-commutative 98.4%

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

   2. unpow2 98.4%

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

   3. unpow2 98.4%

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

   4. associate-/r* 98.5%

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

8. Simplified 98.5%

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

9. Taylor expanded in u0 around 0 87.3%

   \[\leadsto \frac{-\color{blue}{\left(-1 \cdot u0 + -0.5 \cdot {u0}^{2}\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
10. Step-by-step derivation

    1. +-commutative 87.3%

       \[\leadsto \frac{-\color{blue}{\left(-0.5 \cdot {u0}^{2} + -1 \cdot u0\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

    2. neg-mul-1 87.3%

       \[\leadsto \frac{-\left(-0.5 \cdot {u0}^{2} + \color{blue}{\left(-u0\right)}\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

    3. unsub-neg 87.3%

       \[\leadsto \frac{-\color{blue}{\left(-0.5 \cdot {u0}^{2} - u0\right)}}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

    4. *-commutative 87.3%

       \[\leadsto \frac{-\left(\color{blue}{{u0}^{2} \cdot -0.5} - u0\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

    5. unpow2 87.3%

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

    6. associate-*l* 87.3%

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

11. Simplified 87.3%

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

12. Final simplification 87.3%

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

Alternative 9: 81.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.9999999494757503e-5)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (/ (* alphay (- alphay)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.9999999494757503e-5f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (alphay * -alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.9999999494757503e-5) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    else
        tmp = (alphay * -alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.9999999494757503e-5))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.9999999494757503e-5))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	else
		tmp = (alphay * -alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999995e-5

   1. Initial program 53.5%

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

      1. sub-neg 53.5%

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

      2. log1p-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in u0 around 0 75.4%

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

      1. unpow2 75.4%

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

      2. unpow2 75.4%

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

   6. Simplified 75.4%

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

   if 1.99999995e-5 < sin2phi

   1. Initial program 67.1%

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

      1. sub-neg 67.1%

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

      2. log1p-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in cos2phi around 0 67.4%

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

      1. mul-1-neg 67.4%

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

      2. unpow2 67.4%

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

      3. associate-/l* 66.7%

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

      4. distribute-neg-frac 66.7%

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

      5. distribute-rgt-neg-out 66.7%

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

      6. sub-neg 66.7%

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

      7. log1p-def 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in u0 around 0 88.5%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi}} \]
   8. Step-by-step derivation

      1. +-commutative 88.5%

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

      2. mul-1-neg 88.5%

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

      3. unsub-neg 88.5%

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

      4. *-commutative 88.5%

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

   9. Simplified 88.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 10: 74.3% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax}}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.999999967550318e-17)
   (/ 1.0 (/ (/ cos2phi (* alphax alphax)) u0))
   (/ (* alphay (- alphay)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.999999967550318e-17f) {
		tmp = 1.0f / ((cos2phi / (alphax * alphax)) / u0);
	} else {
		tmp = (alphay * -alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.999999967550318e-17) then
        tmp = 1.0e0 / ((cos2phi / (alphax * alphax)) / u0)
    else
        tmp = (alphay * -alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.999999967550318e-17))
		tmp = Float32(Float32(1.0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) / u0));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.999999967550318e-17))
		tmp = single(1.0) / ((cos2phi / (alphax * alphax)) / u0);
	else
		tmp = (alphay * -alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999997e-17

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      2. log1p-def 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in u0 around 0 77.2%

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

      1. unpow2 77.2%

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

      2. unpow2 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in cos2phi around inf 62.0%

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

      1. associate-/l* 62.1%

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

      2. unpow2 62.1%

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

   9. Simplified 62.1%

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

       1. associate-/r/ 62.0%

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

   11. Applied egg-rr 62.0%

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

       1. associate-/r/ 62.1%

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

       2. clear-num 62.2%

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

   13. Applied egg-rr 62.2%

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

   if 1.99999997e-17 < sin2phi

   1. Initial program 64.9%

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

      1. sub-neg 64.9%

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

      2. log1p-def 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in cos2phi around 0 62.1%

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

      1. mul-1-neg 62.1%

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

      2. unpow2 62.1%

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

      3. associate-/l* 61.5%

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

      4. distribute-neg-frac 61.5%

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

      5. distribute-rgt-neg-out 61.5%

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

      6. sub-neg 61.5%

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

      7. log1p-def 90.2%

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

   6. Simplified 90.2%

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

   7. Taylor expanded in u0 around 0 82.5%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi}} \]
   8. Step-by-step derivation

      1. +-commutative 82.5%

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

      2. mul-1-neg 82.5%

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

      3. unsub-neg 82.5%

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

      4. *-commutative 82.5%

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

   9. Simplified 82.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax}}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 11: 66.8% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax}}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.999999967550318e-17)
   (/ 1.0 (/ (/ cos2phi (* alphax alphax)) u0))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.999999967550318e-17f) {
		tmp = 1.0f / ((cos2phi / (alphax * alphax)) / u0);
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.999999967550318e-17) then
        tmp = 1.0e0 / ((cos2phi / (alphax * alphax)) / u0)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.999999967550318e-17))
		tmp = Float32(Float32(1.0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) / u0));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.999999967550318e-17))
		tmp = single(1.0) / ((cos2phi / (alphax * alphax)) / u0);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999997e-17

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      2. log1p-def 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in u0 around 0 77.2%

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

      1. unpow2 77.2%

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

      2. unpow2 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in cos2phi around inf 62.0%

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

      1. associate-/l* 62.1%

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

      2. unpow2 62.1%

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

   9. Simplified 62.1%

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

       1. associate-/r/ 62.0%

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

   11. Applied egg-rr 62.0%

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

       1. associate-/r/ 62.1%

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

       2. clear-num 62.2%

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

   13. Applied egg-rr 62.2%

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

   if 1.99999997e-17 < sin2phi

   1. Initial program 64.9%

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

      1. sub-neg 64.9%

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

      2. log1p-def 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in cos2phi around 0 62.1%

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

      1. mul-1-neg 62.1%

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

      2. unpow2 62.1%

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

      3. associate-/l* 61.5%

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

      4. distribute-neg-frac 61.5%

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

      5. distribute-rgt-neg-out 61.5%

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

      6. sub-neg 61.5%

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

      7. log1p-def 90.2%

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

   6. Simplified 90.2%

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

   7. Taylor expanded in u0 around 0 70.7%

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

      1. associate-/l* 70.4%

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

      2. associate-/r/ 70.7%

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

      3. unpow2 70.7%

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

   9. Simplified 70.7%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax}}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 12: 66.9% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.999999967550318e-17)
   (* alphax (* alphax (/ u0 cos2phi)))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.999999967550318e-17f) {
		tmp = alphax * (alphax * (u0 / cos2phi));
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.999999967550318e-17) then
        tmp = alphax * (alphax * (u0 / cos2phi))
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.999999967550318e-17))
		tmp = Float32(alphax * Float32(alphax * Float32(u0 / cos2phi)));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.999999967550318e-17))
		tmp = alphax * (alphax * (u0 / cos2phi));
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999997e-17

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      2. log1p-def 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in u0 around 0 77.2%

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

      1. unpow2 77.2%

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

      2. unpow2 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in cos2phi around inf 62.0%

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

      1. associate-/l* 62.1%

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

      2. unpow2 62.1%

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

   9. Simplified 62.1%

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

       1. associate-/r/ 62.0%

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

   11. Applied egg-rr 62.0%

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

   12. Taylor expanded in u0 around 0 62.0%

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

       1. unpow2 62.0%

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

       2. associate-*l/ 62.0%

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

       3. *-commutative 62.0%

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

       4. associate-*l* 62.2%

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

   14. Simplified 62.2%

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

   if 1.99999997e-17 < sin2phi

   1. Initial program 64.9%

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

      1. sub-neg 64.9%

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

      2. log1p-def 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in cos2phi around 0 62.1%

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

      1. mul-1-neg 62.1%

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

      2. unpow2 62.1%

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

      3. associate-/l* 61.5%

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

      4. distribute-neg-frac 61.5%

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

      5. distribute-rgt-neg-out 61.5%

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

      6. sub-neg 61.5%

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

      7. log1p-def 90.2%

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

   6. Simplified 90.2%

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

   7. Taylor expanded in u0 around 0 70.7%

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

      1. associate-/l* 70.4%

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

      2. associate-/r/ 70.7%

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

      3. unpow2 70.7%

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

   9. Simplified 70.7%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 13: 23.6% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.3%

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

   1. sub-neg 61.3%

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

   2. log1p-def 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in u0 around 0 76.0%

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

   1. unpow2 76.0%

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

   2. unpow2 76.0%

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

6. Simplified 76.0%

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

7. Taylor expanded in cos2phi around inf 25.1%

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

   1. associate-/l* 25.1%

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

   2. unpow2 25.1%

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

9. Simplified 25.1%

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

    1. associate-/r/ 25.1%

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

11. Applied egg-rr 25.1%

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

12. Taylor expanded in u0 around 0 25.1%

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

    1. unpow2 25.1%

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

    2. associate-*l/ 25.1%

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

    3. *-commutative 25.1%

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

    4. associate-*l* 25.1%

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

14. Simplified 25.1%

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

15. Final simplification 25.1%

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

Reproduce

herbie shell --seed 2023274 
(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)))))