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

Percentage Accurate: 59.8% → 98.4%

Time: 15.9s

Alternatives: 12

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: 59.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{log1p}\left(-u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (+ (* cos2phi (/ 1.0 (* alphax alphax))) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -log1pf(-u0) / ((cos2phi * (1.0f / (alphax * alphax))) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(cos2phi * Float32(Float32(1.0) / Float32(alphax * alphax))) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 62.2%

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

   1. sub-neg 62.2%

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

   2. log1p-def 98.5%

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

3. Simplified 98.5%

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

   1. frac-2neg 98.5%

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

   2. div-inv 98.6%

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

   3. distribute-rgt-neg-in 98.6%

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

5. Applied egg-rr 98.6%

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

6. Taylor expanded in alphax around 0 98.6%

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

   1. unpow2 98.6%

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

8. Simplified 98.6%

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

   1. associate-*r/ 98.5%

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

10. Applied egg-rr 98.5%

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

    1. unpow2 98.5%

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

    2. associate-*r/ 98.6%

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

    3. unpow2 98.6%

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

    4. distribute-lft-neg-in 98.6%

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

    5. distribute-rgt-neg-in 98.6%

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

    6. unpow2 98.6%

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

    7. distribute-neg-frac 98.6%

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

    8. metadata-eval 98.6%

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

    9. unpow2 98.6%

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

12. Simplified 98.6%

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

13. Final simplification 98.6%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 62.2%

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

   1. sub-neg 62.2%

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

   2. log1p-def 98.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 62.2%

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

   1. sub-neg 62.2%

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

   2. log1p-def 98.5%

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

3. Simplified 98.5%

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

   1. frac-2neg 98.5%

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

   2. div-inv 98.6%

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

   3. distribute-rgt-neg-in 98.6%

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

5. Applied egg-rr 98.6%

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

   1. un-div-inv 98.5%

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

   2. distribute-rgt-neg-out 98.5%

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

   3. frac-2neg 98.5%

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

   4. associate-/r* 98.6%

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

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 4: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.10000000149011612:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot \left(-alphay\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.10000000149011612)
   (/
    (- 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 <= 0.10000000149011612f) {
		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(0.10000000149011612))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) / sin2phi) * Float32(alphay * Float32(-alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.100000001

   1. Initial program 59.3%

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

   2. Taylor expanded in u0 around 0 84.3%

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

      1. +-commutative 44.3%

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

      2. mul-1-neg 44.3%

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

      3. unsub-neg 44.3%

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

      4. *-commutative 44.3%

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

      5. unpow2 44.3%

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

      6. associate-*l* 44.3%

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

   4. Simplified 84.3%

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

   if 0.100000001 < sin2phi

   1. Initial program 66.2%

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

      1. sub-neg 66.2%

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

      2. log1p-def 98.5%

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

   3. Simplified 98.5%

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

      1. frac-2neg 98.5%

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

      2. div-inv 98.5%

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

      3. distribute-rgt-neg-in 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in alphax around 0 98.5%

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

      1. unpow2 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in cos2phi around 0 65.9%

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

       1. mul-1-neg 65.9%

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

       2. *-commutative 65.9%

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

       3. associate-*l/ 65.9%

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

       4. sub-neg 65.9%

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

       5. log1p-def 98.0%

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

       6. unpow2 98.0%

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

   11. Simplified 98.0%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.10000000149011612:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot \left(-alphay\right)\right)\\ \end{array} \]

Alternative 5: 81.7% accurate, 5.5× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 10.0)
   (/
    u0
    (+ (/ cos2phi (* alphax alphax)) (* (/ sin2phi alphay) (/ 1.0 alphay))))
   (/ (* (* alphay alphay) (- u0 (* u0 (* u0 -0.5)))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 10.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0f / alphay)));
	} else {
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * -0.5f)))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 10.0e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0e0 / alphay)))
    else
        tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * (-0.5e0))))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(10.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) * Float32(Float32(1.0) / alphay))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5))))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(10.0))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (single(1.0) / alphay)));
	else
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * single(-0.5))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 60.0%

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

   2. Taylor expanded in u0 around 0 70.3%

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

      1. +-commutative 70.3%

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

      2. unpow2 70.3%

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

      3. unpow2 70.3%

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

   4. Simplified 70.3%

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

      1. associate-/r* 70.3%

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

      2. div-inv 70.4%

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

   6. Applied egg-rr 70.4%

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

   if 10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.9%

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

   2. Taylor expanded in cos2phi around 0 64.2%

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

      1. mul-1-neg 64.2%

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

      2. unpow2 64.2%

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

      3. *-commutative 64.2%

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

   4. Simplified 64.2%

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

   5. Taylor expanded in u0 around 0 86.7%

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

      1. +-commutative 86.7%

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

      2. mul-1-neg 86.7%

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

      3. unsub-neg 86.7%

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

      4. *-commutative 86.7%

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

      5. unpow2 86.7%

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

      6. associate-*l* 86.7%

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

   7. Simplified 86.7%

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

4. Final simplification 77.8%

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

Alternative 6: 81.7% accurate, 5.8× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 10.0)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
   (* alphay (/ (* alphay (* u0 (- (* u0 (- -0.5)) -1.0))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 10.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = alphay * ((alphay * (u0 * ((u0 * -(-0.5f)) - -1.0f))) / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 10.0e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
    else
        tmp = alphay * ((alphay * (u0 * ((u0 * -(-0.5e0)) - (-1.0e0)))) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(10.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(alphay * Float32(Float32(alphay * Float32(u0 * Float32(Float32(u0 * Float32(-Float32(-0.5))) - Float32(-1.0)))) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(10.0))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	else
		tmp = alphay * ((alphay * (u0 * ((u0 * -single(-0.5)) - single(-1.0)))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 60.0%

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

   2. Taylor expanded in u0 around 0 70.3%

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

      1. +-commutative 70.3%

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

      2. unpow2 70.3%

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

      3. unpow2 70.3%

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

   4. Simplified 70.3%

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

      1. frac-2neg 70.3%

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

      2. div-inv 70.3%

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

      3. distribute-rgt-neg-in 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. un-div-inv 70.3%

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

      2. distribute-rgt-neg-out 70.3%

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

      3. frac-2neg 70.3%

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

      4. associate-/r* 70.3%

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

   8. Applied egg-rr 70.3%

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

   if 10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.9%

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

   2. Taylor expanded in cos2phi around 0 64.2%

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

      1. mul-1-neg 64.2%

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

      2. unpow2 64.2%

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

      3. *-commutative 64.2%

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

   4. Simplified 64.2%

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

   5. Taylor expanded in u0 around 0 86.7%

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

      1. +-commutative 86.7%

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

      2. mul-1-neg 86.7%

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

      3. unsub-neg 86.7%

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

      4. *-commutative 86.7%

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

      5. unpow2 86.7%

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

      6. associate-*l* 86.7%

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

   7. Simplified 86.7%

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

   8. Taylor expanded in alphay around 0 86.7%

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

      1. sub-neg 86.7%

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

      2. +-commutative 86.7%

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

      3. +-commutative 86.7%

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

      4. *-commutative 86.7%

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

      5. unpow2 86.7%

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

      6. associate-*r* 86.7%

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

      7. sub-neg 86.7%

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

      8. unpow2 86.7%

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

      9. associate-*l* 86.6%

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

      10. *-lft-identity 86.6%

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

      11. times-frac 86.8%

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

      12. /-rgt-identity 86.8%

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

      13. sub-neg 86.8%

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

      14. +-commutative 86.8%

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

      15. neg-mul-1 86.8%

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

      16. *-commutative 86.8%

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

      17. distribute-rgt-out 86.7%

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

   10. Simplified 86.7%

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

4. Final simplification 77.8%

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

Alternative 7: 81.7% accurate, 6.1× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 10.0)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
   (/ (* (* alphay alphay) (- u0 (* u0 (* u0 -0.5)))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 10.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * -0.5f)))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 10.0e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
    else
        tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * (-0.5e0))))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(10.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5))))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(10.0))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	else
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * single(-0.5))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 60.0%

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

   2. Taylor expanded in u0 around 0 70.3%

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

      1. +-commutative 70.3%

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

      2. unpow2 70.3%

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

      3. unpow2 70.3%

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

   4. Simplified 70.3%

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

      1. frac-2neg 70.3%

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

      2. div-inv 70.3%

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

      3. distribute-rgt-neg-in 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. un-div-inv 70.3%

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

      2. distribute-rgt-neg-out 70.3%

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

      3. frac-2neg 70.3%

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

      4. associate-/r* 70.3%

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

   8. Applied egg-rr 70.3%

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

   if 10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.9%

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

   2. Taylor expanded in cos2phi around 0 64.2%

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

      1. mul-1-neg 64.2%

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

      2. unpow2 64.2%

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

      3. *-commutative 64.2%

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

   4. Simplified 64.2%

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

   5. Taylor expanded in u0 around 0 86.7%

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

      1. +-commutative 86.7%

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

      2. mul-1-neg 86.7%

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

      3. unsub-neg 86.7%

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

      4. *-commutative 86.7%

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

      5. unpow2 86.7%

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

      6. associate-*l* 86.7%

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

   7. Simplified 86.7%

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

4. Final simplification 77.8%

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

Alternative 8: 87.7% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (u0 - (u0 * (u0 * (-0.5e0)))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 62.2%

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

2. Taylor expanded in u0 around 0 85.3%

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

   1. +-commutative 62.1%

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

   2. mul-1-neg 62.1%

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

   3. unsub-neg 62.1%

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

   4. *-commutative 62.1%

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

   5. unpow2 62.1%

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

   6. associate-*l* 62.1%

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

4. Simplified 85.3%

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

5. Final simplification 85.3%

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

Alternative 9: 76.1% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.2%

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

2. Taylor expanded in u0 around 0 72.8%

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

   1. +-commutative 72.8%

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

   2. unpow2 72.8%

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

   3. unpow2 72.8%

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

4. Simplified 72.8%

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

5. Final simplification 72.8%

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

Alternative 10: 67.1% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.9999999996399175e-23)
   (* u0 (* alphax (/ alphax cos2phi)))
   (* u0 (* alphay (/ alphay sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.9999999996399175e-23f) {
		tmp = u0 * (alphax * (alphax / cos2phi));
	} else {
		tmp = u0 * (alphay * (alphay / sin2phi));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.9999999996399175e-23) then
        tmp = u0 * (alphax * (alphax / cos2phi))
    else
        tmp = u0 * (alphay * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.9999999996399175e-23))
		tmp = Float32(u0 * Float32(alphax * Float32(alphax / cos2phi)));
	else
		tmp = Float32(u0 * Float32(alphay * Float32(alphay / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.9999999996399175e-23))
		tmp = u0 * (alphax * (alphax / cos2phi));
	else
		tmp = u0 * (alphay * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2e-23

   1. Initial program 64.6%

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

   2. Taylor expanded in u0 around 0 65.8%

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

      1. +-commutative 65.8%

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

      2. unpow2 65.8%

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

      3. unpow2 65.8%

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

   4. Simplified 65.8%

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

   5. Taylor expanded in sin2phi around 0 56.6%

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

      1. *-commutative 56.6%

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

      2. associate-*r/ 56.6%

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

      3. unpow2 56.6%

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

      4. associate-/l* 56.5%

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

   7. Simplified 56.5%

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

      1. associate-/r/ 56.6%

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

   9. Applied egg-rr 56.6%

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

   if 2e-23 < sin2phi

   1. Initial program 61.6%

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

   2. Taylor expanded in u0 around 0 74.7%

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

      1. +-commutative 74.7%

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

      2. unpow2 74.7%

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

      3. unpow2 74.7%

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

   4. Simplified 74.7%

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

   5. Taylor expanded in sin2phi around inf 64.8%

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

      1. *-commutative 64.8%

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

      2. associate-*r/ 64.8%

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

      3. unpow2 64.8%

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

      4. associate-*r/ 64.8%

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

   7. Simplified 64.8%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 11: 67.1% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;u0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.9999999996399175e-23)
   (* u0 (/ (* alphax alphax) cos2phi))
   (* u0 (* alphay (/ alphay sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.9999999996399175e-23f) {
		tmp = u0 * ((alphax * alphax) / cos2phi);
	} else {
		tmp = u0 * (alphay * (alphay / sin2phi));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.9999999996399175e-23) then
        tmp = u0 * ((alphax * alphax) / cos2phi)
    else
        tmp = u0 * (alphay * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.9999999996399175e-23))
		tmp = Float32(u0 * Float32(Float32(alphax * alphax) / cos2phi));
	else
		tmp = Float32(u0 * Float32(alphay * Float32(alphay / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.9999999996399175e-23))
		tmp = u0 * ((alphax * alphax) / cos2phi);
	else
		tmp = u0 * (alphay * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2e-23

   1. Initial program 64.6%

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

   2. Taylor expanded in u0 around 0 65.8%

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

      1. +-commutative 65.8%

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

      2. unpow2 65.8%

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

      3. unpow2 65.8%

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

   4. Simplified 65.8%

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

      1. frac-2neg 65.8%

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

      2. div-inv 65.8%

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

      3. distribute-rgt-neg-in 65.8%

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

   6. Applied egg-rr 65.8%

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

   7. Taylor expanded in sin2phi around 0 56.6%

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

      1. *-commutative 56.6%

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

      2. associate-*r/ 56.6%

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

      3. unpow2 56.6%

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

   9. Simplified 56.6%

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

   if 2e-23 < sin2phi

   1. Initial program 61.6%

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

   2. Taylor expanded in u0 around 0 74.7%

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

      1. +-commutative 74.7%

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

      2. unpow2 74.7%

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

      3. unpow2 74.7%

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

   4. Simplified 74.7%

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

   5. Taylor expanded in sin2phi around inf 64.8%

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

      1. *-commutative 64.8%

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

      2. associate-*r/ 64.8%

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

      3. unpow2 64.8%

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

      4. associate-*r/ 64.8%

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

   7. Simplified 64.8%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;u0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 12: 59.5% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.2%

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

2. Taylor expanded in u0 around 0 72.8%

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

   1. +-commutative 72.8%

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

   2. unpow2 72.8%

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

   3. unpow2 72.8%

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

4. Simplified 72.8%

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

5. Taylor expanded in sin2phi around inf 54.8%

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

   1. *-commutative 54.8%

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

   2. associate-*r/ 54.8%

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

   3. unpow2 54.8%

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

   4. associate-*r/ 54.8%

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

7. Simplified 54.8%

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

8. Final simplification 54.8%

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

Reproduce

herbie shell --seed 2023272 
(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)))))