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

Percentage Accurate: 61.1% → 98.4%

Time: 15.2s

Alternatives: 9

Speedup: 9.7×

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: 61.1% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{log1p}\left(-u0\right)}{{alphax}^{-2} \cdot cos2phi + \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (+ (* (pow alphax -2.0) cos2phi) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -log1pf(-u0) / ((powf(alphax, -2.0f) * cos2phi) + ((sin2phi / alphay) / alphay));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32((alphax ^ Float32(-2.0)) * cos2phi) + Float32(Float32(sin2phi / alphay) / alphay)))
end

Derivation

1. Initial program 58.9%

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

   1. sqr-neg 58.9%

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

   2. sub-neg 58.9%

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

   3. log1p-def 98.1%

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

   4. sqr-neg 98.1%

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

   5. associate-/r* 98.0%

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

3. Simplified 98.0%

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

   1. clear-num 98.0%

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

   2. associate-/r/ 98.0%

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

   3. pow2 98.0%

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

   4. pow-flip 98.1%

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

   5. metadata-eval 98.1%

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

5. Applied egg-rr 98.1%

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

6. Final simplification 98.1%

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

Alternative 2: 98.4% 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 58.9%

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

   1. sub-neg 58.9%

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

   2. log1p-def 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 58.9%

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

   1. sqr-neg 58.9%

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

   2. sub-neg 58.9%

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

   3. log1p-def 98.1%

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

   4. sqr-neg 98.1%

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

   5. associate-/r* 98.0%

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

3. Simplified 98.0%

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

   1. clear-num 98.0%

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

   2. associate-/r/ 98.0%

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

   3. pow2 98.0%

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

   4. pow-flip 98.1%

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

   5. metadata-eval 98.1%

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

5. Applied egg-rr 98.1%

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

   1. *-commutative 98.1%

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

   2. metadata-eval 98.1%

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

   3. pow-flip 98.0%

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

   4. pow2 98.0%

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

   5. div-inv 98.0%

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

   6. associate-/r* 98.1%

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

7. Applied egg-rr 98.1%

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

8. Final simplification 98.1%

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

Alternative 4: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{{alphay}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot \left(-alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.000000013351432e-10)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (pow alphay 2.0))))
   (* (log1p (- u0)) (/ (* alphay (- alphay)) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.000000013351432e-10f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / powf(alphay, 2.0f)));
	} else {
		tmp = log1pf(-u0) * ((alphay * -alphay) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.000000013351432e-10))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / (alphay ^ Float32(2.0)))));
	else
		tmp = Float32(log1p(Float32(-u0)) * Float32(Float32(alphay * Float32(-alphay)) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.00000001e-10

   1. Initial program 51.6%

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

      1. associate-/r* 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in u0 around 0 77.0%

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

      1. mul-1-neg 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in sin2phi around 0 77.0%

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

   if 1.00000001e-10 < sin2phi

   1. Initial program 63.2%

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

      1. associate-/r* 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in cos2phi around 0 61.9%

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

      1. mul-1-neg 61.9%

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

      2. associate-/l* 61.7%

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

      3. distribute-neg-frac 61.7%

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

      4. sub-neg 61.7%

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

      5. mul-1-neg 61.7%

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

      6. log1p-def 93.5%

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

      7. mul-1-neg 93.5%

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

   6. Simplified 93.5%

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

      1. associate-/r/ 94.1%

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

   8. Applied egg-rr 94.1%

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

      1. unpow2 83.2%

         \[\leadsto \frac{-\color{blue}{alphay \cdot alphay}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}} \]

   10. Applied egg-rr 94.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{{alphay}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot \left(-alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 5: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot \left(-alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.000000013351432e-10)
   (/
    u0
    (+ (/ cos2phi (* alphax alphax)) (* (/ sin2phi alphay) (/ 1.0 alphay))))
   (* (log1p (- u0)) (/ (* alphay (- alphay)) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.000000013351432e-10f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0f / alphay)));
	} else {
		tmp = log1pf(-u0) * ((alphay * -alphay) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.000000013351432e-10))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) * Float32(Float32(1.0) / alphay))));
	else
		tmp = Float32(log1p(Float32(-u0)) * Float32(Float32(alphay * Float32(-alphay)) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.00000001e-10

   1. Initial program 51.6%

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

      1. associate-/r* 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in u0 around 0 77.0%

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

      1. mul-1-neg 77.0%

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

   6. Simplified 77.0%

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

      1. div-inv 77.0%

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

   8. Applied egg-rr 77.0%

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

   if 1.00000001e-10 < sin2phi

   1. Initial program 63.2%

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

      1. associate-/r* 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in cos2phi around 0 61.9%

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

      1. mul-1-neg 61.9%

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

      2. associate-/l* 61.7%

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

      3. distribute-neg-frac 61.7%

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

      4. sub-neg 61.7%

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

      5. mul-1-neg 61.7%

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

      6. log1p-def 93.5%

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

      7. mul-1-neg 93.5%

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

   6. Simplified 93.5%

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

      1. associate-/r/ 94.1%

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

   8. Applied egg-rr 94.1%

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

      1. unpow2 83.2%

         \[\leadsto \frac{-\color{blue}{alphay \cdot alphay}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}} \]

   10. Applied egg-rr 94.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot \left(-alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 6: 81.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot \left(0.5 + \frac{-1}{u0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.999999987845058e-8)
   (/ u0 (+ (/ (/ sin2phi alphay) alphay) (/ cos2phi (* alphax alphax))))
   (/ (* alphay (- alphay)) (* sin2phi (+ 0.5 (/ -1.0 u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.999999987845058e-8f) {
		tmp = u0 / (((sin2phi / alphay) / alphay) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = (alphay * -alphay) / (sin2phi * (0.5f + (-1.0f / 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 <= 1.999999987845058e-8) then
        tmp = u0 / (((sin2phi / alphay) / alphay) + (cos2phi / (alphax * alphax)))
    else
        tmp = (alphay * -alphay) / (sin2phi * (0.5e0 + ((-1.0e0) / u0)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.999999987845058e-8))
		tmp = Float32(u0 / Float32(Float32(Float32(sin2phi / alphay) / alphay) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(sin2phi * Float32(Float32(0.5) + Float32(Float32(-1.0) / u0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.999999987845058e-8))
		tmp = u0 / (((sin2phi / alphay) / alphay) + (cos2phi / (alphax * alphax)));
	else
		tmp = (alphay * -alphay) / (sin2phi * (single(0.5) + (single(-1.0) / u0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999999e-8

   1. Initial program 51.0%

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

      1. associate-/r* 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in u0 around 0 77.4%

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

      1. mul-1-neg 77.4%

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

   6. Simplified 77.4%

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

   if 1.99999999e-8 < sin2phi

   1. Initial program 63.7%

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

      1. associate-/r* 63.7%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in cos2phi around 0 62.4%

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

      1. mul-1-neg 62.4%

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

      2. associate-/l* 62.2%

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

      3. distribute-neg-frac 62.2%

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

      4. sub-neg 62.2%

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

      5. mul-1-neg 62.2%

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

      6. log1p-def 93.5%

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

      7. mul-1-neg 93.5%

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

   6. Simplified 93.5%

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

   7. Taylor expanded in u0 around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. mul-1-neg 83.0%

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

      3. unsub-neg 83.0%

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

      4. *-commutative 83.0%

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

   9. Simplified 83.0%

      \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}} \]
   10. Step-by-step derivation

       1. unpow2 83.0%

          \[\leadsto \frac{-\color{blue}{alphay \cdot alphay}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}} \]

   11. Applied egg-rr 83.0%

       \[\leadsto \frac{-\color{blue}{alphay \cdot alphay}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}} \]

   12. Taylor expanded in sin2phi around 0 83.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot \left(0.5 + \frac{-1}{u0}\right)}\\ \end{array} \]

Alternative 7: 67.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (alphay * -alphay) / (sin2phi * (0.5f + (-1.0f / u0)));
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.9%

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

   1. associate-/r* 58.9%

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

3. Simplified 58.9%

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

4. Taylor expanded in cos2phi around 0 48.2%

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

   1. mul-1-neg 48.2%

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

   2. associate-/l* 48.1%

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

   3. distribute-neg-frac 48.1%

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

   4. sub-neg 48.1%

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

   5. mul-1-neg 48.1%

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

   6. log1p-def 73.0%

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

   7. mul-1-neg 73.0%

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

6. Simplified 73.0%

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

7. Taylor expanded in u0 around 0 65.6%

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

   1. +-commutative 65.6%

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

   2. mul-1-neg 65.6%

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

   3. unsub-neg 65.6%

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

   4. *-commutative 65.6%

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

9. Simplified 65.6%

   \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}} \]
10. Step-by-step derivation

    1. unpow2 65.6%

       \[\leadsto \frac{-\color{blue}{alphay \cdot alphay}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}} \]

11. Applied egg-rr 65.6%

    \[\leadsto \frac{-\color{blue}{alphay \cdot alphay}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}} \]

12. Taylor expanded in sin2phi around 0 65.6%

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

13. Final simplification 65.6%

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

Alternative 8: 58.5% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.9%

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

   1. associate-/r* 58.9%

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

3. Simplified 58.9%

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

4. Taylor expanded in cos2phi around 0 48.2%

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

   1. mul-1-neg 48.2%

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

   2. associate-/l* 48.1%

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

   3. distribute-neg-frac 48.1%

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

   4. sub-neg 48.1%

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

   5. mul-1-neg 48.1%

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

   6. log1p-def 73.0%

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

   7. mul-1-neg 73.0%

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

6. Simplified 73.0%

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

7. Taylor expanded in u0 around 0 65.6%

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

   1. +-commutative 65.6%

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

   2. mul-1-neg 65.6%

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

   3. unsub-neg 65.6%

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

   4. *-commutative 65.6%

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

9. Simplified 65.6%

   \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}} \]
10. Step-by-step derivation

    1. unpow2 65.6%

       \[\leadsto \frac{-\color{blue}{alphay \cdot alphay}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}} \]

11. Applied egg-rr 65.6%

    \[\leadsto \frac{-\color{blue}{alphay \cdot alphay}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}} \]

12. Taylor expanded in u0 around 0 56.9%

    \[\leadsto \frac{-alphay \cdot alphay}{\color{blue}{-1 \cdot \frac{sin2phi}{u0}}} \]
13. Step-by-step derivation

    1. neg-mul-1 56.9%

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

    2. distribute-neg-frac 56.9%

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

14. Simplified 56.9%

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

15. Final simplification 56.9%

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

Alternative 9: 8.4% accurate, 14.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.9%

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

   1. associate-/r* 58.9%

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

3. Simplified 58.9%

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

4. Taylor expanded in cos2phi around 0 48.2%

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

   1. mul-1-neg 48.2%

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

   2. associate-/l* 48.1%

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

   3. distribute-neg-frac 48.1%

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

   4. sub-neg 48.1%

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

   5. mul-1-neg 48.1%

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

   6. log1p-def 73.0%

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

   7. mul-1-neg 73.0%

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

6. Simplified 73.0%

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

7. Taylor expanded in u0 around 0 65.6%

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

   1. +-commutative 65.6%

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

   2. mul-1-neg 65.6%

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

   3. unsub-neg 65.6%

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

   4. *-commutative 65.6%

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

9. Simplified 65.6%

   \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}} \]
10. Step-by-step derivation

    1. unpow2 65.6%

       \[\leadsto \frac{-\color{blue}{alphay \cdot alphay}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}} \]

11. Applied egg-rr 65.6%

    \[\leadsto \frac{-\color{blue}{alphay \cdot alphay}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}} \]

12. Taylor expanded in u0 around inf 6.8%

    \[\leadsto \frac{-alphay \cdot alphay}{\color{blue}{0.5 \cdot sin2phi}} \]
13. Step-by-step derivation

    1. *-commutative 6.8%

       \[\leadsto \frac{-alphay \cdot alphay}{\color{blue}{sin2phi \cdot 0.5}} \]

14. Simplified 6.8%

    \[\leadsto \frac{-alphay \cdot alphay}{\color{blue}{sin2phi \cdot 0.5}} \]

15. Final simplification 6.8%

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

Reproduce

herbie shell --seed 2023336 
(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)))))