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

Percentage Accurate: 61.2% → 98.4%

Time: 14.7s

Alternatives: 8

Speedup: 1.0×

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: 61.2% 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

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

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(log1p(Float32(-u0)) / Float32(Float32(Float32(cos2phi / alphax) / Float32(-alphax)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 59.1%

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

   1. distribute-frac-neg 59.1%

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

   2. distribute-neg-frac2 59.1%

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

   3. sub-neg 59.1%

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

   4. log1p-define 97.9%

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

   5. neg-sub0 97.9%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 97.9%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 97.9%

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

   8. associate-/r* 98.0%

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

   9. distribute-neg-frac2 98.0%

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

3. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\frac{u0}{\frac{1}{alphax \cdot \frac{alphax}{cos2phi}} + \frac{sin2phi}{{alphay}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.05000000074505806)
     (/
      u0
      (+ (/ 1.0 (* alphax (/ alphax cos2phi))) (/ sin2phi (pow alphay 2.0))))
     (/ (log1p (- u0)) (- (/ (/ cos2phi alphax) alphax) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.05000000074505806f) {
		tmp = u0 / ((1.0f / (alphax * (alphax / cos2phi))) + (sin2phi / powf(alphay, 2.0f)));
	} else {
		tmp = log1pf(-u0) / (((cos2phi / alphax) / alphax) - t_0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.05000000074505806))
		tmp = Float32(u0 / Float32(Float32(Float32(1.0) / Float32(alphax * Float32(alphax / cos2phi))) + Float32(sin2phi / (alphay ^ Float32(2.0)))));
	else
		tmp = Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(cos2phi / alphax) / alphax) - t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.0500000007

   1. Initial program 50.6%

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

      1. distribute-frac-neg 50.6%

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

      2. distribute-neg-frac2 50.6%

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

      3. neg-mul-1 50.6%

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

      4. associate-/r* 50.6%

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

      5. remove-double-neg 50.6%

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

      6. distribute-frac-neg 50.6%

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

      7. distribute-neg-frac2 50.6%

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

      8. metadata-eval 50.6%

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

      9. /-rgt-identity 50.6%

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

      10. sub-neg 50.6%

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

      11. log1p-define 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in u0 around 0 78.6%

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

      1. pow2 78.6%

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

      2. associate-/r* 78.6%

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

      3. clear-num 78.6%

         \[\leadsto \frac{u0}{\color{blue}{\frac{1}{\frac{alphax}{\frac{cos2phi}{alphax}}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      4. inv-pow 78.6%

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

   7. Applied egg-rr 78.6%

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

      1. unpow-1 78.6%

         \[\leadsto \frac{u0}{\color{blue}{\frac{1}{\frac{alphax}{\frac{cos2phi}{alphax}}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      2. associate-/r/ 78.7%

         \[\leadsto \frac{u0}{\frac{1}{\color{blue}{\frac{alphax}{cos2phi} \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

   9. Simplified 78.7%

      \[\leadsto \frac{u0}{\color{blue}{\frac{1}{\frac{alphax}{cos2phi} \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

   if 0.0500000007 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.0%

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

      1. distribute-frac-neg 65.0%

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

      2. distribute-neg-frac2 65.0%

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

      3. sub-neg 65.0%

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

      4. log1p-define 97.4%

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

      5. neg-sub0 97.4%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 97.4%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 97.4%

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

      8. associate-/r* 97.4%

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

      9. distribute-neg-frac2 97.4%

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

   3. Simplified 97.4%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 96.8%

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

      3. sqr-neg 96.8%

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

      4. sqrt-prod 96.8%

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

      5. add-sqr-sqrt 96.8%

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

      6. div-inv 96.8%

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

   6. Applied egg-rr 96.8%

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

      1. associate-*r/ 96.8%

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

      2. *-rgt-identity 96.8%

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

   8. Simplified 96.8%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.05000000074505806:\\ \;\;\;\;\frac{u0}{\frac{1}{alphax \cdot \frac{alphax}{cos2phi}} + \frac{sin2phi}{{alphay}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{1}{alphax \cdot \frac{alphax}{cos2phi}} + \frac{sin2phi}{{alphay}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;{alphay}^{2} \cdot \left(u0 \cdot \left(0.5 \cdot \frac{u0}{sin2phi} + \frac{1}{sin2phi}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.99999993922529e-9)
   (/
    u0
    (+ (/ 1.0 (* alphax (/ alphax cos2phi))) (/ sin2phi (pow alphay 2.0))))
   (* (pow alphay 2.0) (* u0 (+ (* 0.5 (/ u0 sin2phi)) (/ 1.0 sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.99999993922529e-9f) {
		tmp = u0 / ((1.0f / (alphax * (alphax / cos2phi))) + (sin2phi / powf(alphay, 2.0f)));
	} else {
		tmp = powf(alphay, 2.0f) * (u0 * ((0.5f * (u0 / sin2phi)) + (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 <= 9.99999993922529e-9) then
        tmp = u0 / ((1.0e0 / (alphax * (alphax / cos2phi))) + (sin2phi / (alphay ** 2.0e0)))
    else
        tmp = (alphay ** 2.0e0) * (u0 * ((0.5e0 * (u0 / sin2phi)) + (1.0e0 / sin2phi)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.99999993922529e-9))
		tmp = Float32(u0 / Float32(Float32(Float32(1.0) / Float32(alphax * Float32(alphax / cos2phi))) + Float32(sin2phi / (alphay ^ Float32(2.0)))));
	else
		tmp = Float32((alphay ^ Float32(2.0)) * Float32(u0 * Float32(Float32(Float32(0.5) * Float32(u0 / sin2phi)) + Float32(Float32(1.0) / sin2phi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(9.99999993922529e-9))
		tmp = u0 / ((single(1.0) / (alphax * (alphax / cos2phi))) + (sin2phi / (alphay ^ single(2.0))));
	else
		tmp = (alphay ^ single(2.0)) * (u0 * ((single(0.5) * (u0 / sin2phi)) + (single(1.0) / sin2phi)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.99999994e-9

   1. Initial program 51.3%

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

      1. distribute-frac-neg 51.3%

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

      2. distribute-neg-frac2 51.3%

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

      3. neg-mul-1 51.3%

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

      4. associate-/r* 51.3%

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

      5. remove-double-neg 51.3%

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

      6. distribute-frac-neg 51.3%

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

      7. distribute-neg-frac2 51.3%

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

      8. metadata-eval 51.3%

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

      9. /-rgt-identity 51.3%

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

      10. sub-neg 51.3%

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

      11. log1p-define 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in u0 around 0 78.2%

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

      1. pow2 78.2%

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

      2. associate-/r* 78.2%

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

      3. clear-num 78.2%

         \[\leadsto \frac{u0}{\color{blue}{\frac{1}{\frac{alphax}{\frac{cos2phi}{alphax}}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      4. inv-pow 78.2%

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

   7. Applied egg-rr 78.2%

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

      1. unpow-1 78.2%

         \[\leadsto \frac{u0}{\color{blue}{\frac{1}{\frac{alphax}{\frac{cos2phi}{alphax}}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      2. associate-/r/ 78.2%

         \[\leadsto \frac{u0}{\frac{1}{\color{blue}{\frac{alphax}{cos2phi} \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

   9. Simplified 78.2%

      \[\leadsto \frac{u0}{\color{blue}{\frac{1}{\frac{alphax}{cos2phi} \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

   if 9.99999994e-9 < sin2phi

   1. Initial program 63.8%

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

      1. distribute-frac-neg 63.8%

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

      2. distribute-neg-frac2 63.8%

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

      3. sub-neg 63.8%

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

      4. log1p-define 97.6%

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

      5. neg-sub0 97.6%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 97.6%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 97.6%

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

      8. associate-/r* 97.6%

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

      9. distribute-neg-frac2 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in cos2phi around 0 65.0%

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

      1. mul-1-neg 65.0%

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

      2. associate-/l* 65.1%

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

      3. distribute-rgt-neg-in 65.1%

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

      4. distribute-neg-frac2 65.1%

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

      5. sub-neg 65.1%

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

      6. log1p-define 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in u0 around 0 89.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{1}{alphax \cdot \frac{alphax}{cos2phi}} + \frac{sin2phi}{{alphay}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;{alphay}^{2} \cdot \left(u0 \cdot \left(0.5 \cdot \frac{u0}{sin2phi} + \frac{1}{sin2phi}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{alphax} \cdot \frac{1}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;{alphay}^{2} \cdot \left(u0 \cdot \left(0.5 \cdot \frac{u0}{sin2phi} + \frac{1}{sin2phi}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.99999993922529e-9)
   (/
    u0
    (+ (/ sin2phi (pow alphay 2.0)) (* (/ cos2phi alphax) (/ 1.0 alphax))))
   (* (pow alphay 2.0) (* u0 (+ (* 0.5 (/ u0 sin2phi)) (/ 1.0 sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.99999993922529e-9f) {
		tmp = u0 / ((sin2phi / powf(alphay, 2.0f)) + ((cos2phi / alphax) * (1.0f / alphax)));
	} else {
		tmp = powf(alphay, 2.0f) * (u0 * ((0.5f * (u0 / sin2phi)) + (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 <= 9.99999993922529e-9) then
        tmp = u0 / ((sin2phi / (alphay ** 2.0e0)) + ((cos2phi / alphax) * (1.0e0 / alphax)))
    else
        tmp = (alphay ** 2.0e0) * (u0 * ((0.5e0 * (u0 / sin2phi)) + (1.0e0 / sin2phi)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.99999993922529e-9))
		tmp = Float32(u0 / Float32(Float32(sin2phi / (alphay ^ Float32(2.0))) + Float32(Float32(cos2phi / alphax) * Float32(Float32(1.0) / alphax))));
	else
		tmp = Float32((alphay ^ Float32(2.0)) * Float32(u0 * Float32(Float32(Float32(0.5) * Float32(u0 / sin2phi)) + Float32(Float32(1.0) / sin2phi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(9.99999993922529e-9))
		tmp = u0 / ((sin2phi / (alphay ^ single(2.0))) + ((cos2phi / alphax) * (single(1.0) / alphax)));
	else
		tmp = (alphay ^ single(2.0)) * (u0 * ((single(0.5) * (u0 / sin2phi)) + (single(1.0) / sin2phi)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.99999994e-9

   1. Initial program 51.3%

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

      1. distribute-frac-neg 51.3%

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

      2. distribute-neg-frac2 51.3%

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

      3. neg-mul-1 51.3%

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

      4. associate-/r* 51.3%

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

      5. remove-double-neg 51.3%

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

      6. distribute-frac-neg 51.3%

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

      7. distribute-neg-frac2 51.3%

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

      8. metadata-eval 51.3%

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

      9. /-rgt-identity 51.3%

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

      10. sub-neg 51.3%

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

      11. log1p-define 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in u0 around 0 78.2%

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

      1. pow2 78.2%

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

      2. associate-/r* 78.2%

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

      3. div-inv 78.2%

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

   7. Applied egg-rr 78.2%

      \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

   if 9.99999994e-9 < sin2phi

   1. Initial program 63.8%

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

      1. distribute-frac-neg 63.8%

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

      2. distribute-neg-frac2 63.8%

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

      3. sub-neg 63.8%

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

      4. log1p-define 97.6%

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

      5. neg-sub0 97.6%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 97.6%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 97.6%

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

      8. associate-/r* 97.6%

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

      9. distribute-neg-frac2 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in cos2phi around 0 65.0%

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

      1. mul-1-neg 65.0%

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

      2. associate-/l* 65.1%

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

      3. distribute-rgt-neg-in 65.1%

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

      4. distribute-neg-frac2 65.1%

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

      5. sub-neg 65.1%

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

      6. log1p-define 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in u0 around 0 89.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{alphax} \cdot \frac{1}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;{alphay}^{2} \cdot \left(u0 \cdot \left(0.5 \cdot \frac{u0}{sin2phi} + \frac{1}{sin2phi}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.0000000781659255 \cdot 10^{-25}:\\ \;\;\;\;\frac{u0 \cdot {alphax}^{2}}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;{alphay}^{2} \cdot \left(u0 \cdot \left(0.5 \cdot \frac{u0}{sin2phi} + \frac{1}{sin2phi}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.0000000781659255e-25)
   (/ (* u0 (pow alphax 2.0)) cos2phi)
   (* (pow alphay 2.0) (* u0 (+ (* 0.5 (/ u0 sin2phi)) (/ 1.0 sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.0000000781659255e-25f) {
		tmp = (u0 * powf(alphax, 2.0f)) / cos2phi;
	} else {
		tmp = powf(alphay, 2.0f) * (u0 * ((0.5f * (u0 / sin2phi)) + (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 <= 4.0000000781659255e-25) then
        tmp = (u0 * (alphax ** 2.0e0)) / cos2phi
    else
        tmp = (alphay ** 2.0e0) * (u0 * ((0.5e0 * (u0 / sin2phi)) + (1.0e0 / sin2phi)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.0000000781659255e-25))
		tmp = Float32(Float32(u0 * (alphax ^ Float32(2.0))) / cos2phi);
	else
		tmp = Float32((alphay ^ Float32(2.0)) * Float32(u0 * Float32(Float32(Float32(0.5) * Float32(u0 / sin2phi)) + Float32(Float32(1.0) / sin2phi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.0000000781659255e-25))
		tmp = (u0 * (alphax ^ single(2.0))) / cos2phi;
	else
		tmp = (alphay ^ single(2.0)) * (u0 * ((single(0.5) * (u0 / sin2phi)) + (single(1.0) / sin2phi)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.00000008e-25

   1. Initial program 51.5%

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

      1. distribute-frac-neg 51.5%

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

      2. distribute-neg-frac2 51.5%

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

      3. neg-mul-1 51.5%

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

      4. associate-/r* 51.5%

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

      5. remove-double-neg 51.5%

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

      6. distribute-frac-neg 51.5%

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

      7. distribute-neg-frac2 51.5%

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

      8. metadata-eval 51.5%

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

      9. /-rgt-identity 51.5%

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

      10. sub-neg 51.5%

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

      11. log1p-define 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in u0 around 0 77.7%

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

   6. Taylor expanded in cos2phi around inf 62.9%

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

   if 4.00000008e-25 < sin2phi

   1. Initial program 60.7%

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

      1. distribute-frac-neg 60.7%

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

      2. distribute-neg-frac2 60.7%

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

      3. sub-neg 60.7%

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

      4. log1p-define 97.8%

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

      5. neg-sub0 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 97.8%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 97.8%

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

      8. associate-/r* 97.9%

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

      9. distribute-neg-frac2 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in cos2phi around 0 58.6%

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

      1. mul-1-neg 58.6%

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

      2. associate-/l* 58.7%

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

      3. distribute-rgt-neg-in 58.7%

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

      4. distribute-neg-frac2 58.7%

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

      5. sub-neg 58.7%

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

      6. log1p-define 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in u0 around 0 83.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.0000000781659255 \cdot 10^{-25}:\\ \;\;\;\;\frac{u0 \cdot {alphax}^{2}}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;{alphay}^{2} \cdot \left(u0 \cdot \left(0.5 \cdot \frac{u0}{sin2phi} + \frac{1}{sin2phi}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.0000000781659255 \cdot 10^{-25}:\\ \;\;\;\;\frac{u0 \cdot {alphax}^{2}}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;{alphay}^{2} \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.0000000781659255e-25)
   (/ (* u0 (pow alphax 2.0)) cos2phi)
   (* (pow alphay 2.0) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.0000000781659255e-25f) {
		tmp = (u0 * powf(alphax, 2.0f)) / cos2phi;
	} else {
		tmp = powf(alphay, 2.0f) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 4.0000000781659255e-25) then
        tmp = (u0 * (alphax ** 2.0e0)) / cos2phi
    else
        tmp = (alphay ** 2.0e0) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.0000000781659255e-25))
		tmp = Float32(Float32(u0 * (alphax ^ Float32(2.0))) / cos2phi);
	else
		tmp = Float32((alphay ^ Float32(2.0)) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.0000000781659255e-25))
		tmp = (u0 * (alphax ^ single(2.0))) / cos2phi;
	else
		tmp = (alphay ^ single(2.0)) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.00000008e-25

   1. Initial program 51.5%

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

      1. distribute-frac-neg 51.5%

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

      2. distribute-neg-frac2 51.5%

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

      3. neg-mul-1 51.5%

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

      4. associate-/r* 51.5%

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

      5. remove-double-neg 51.5%

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

      6. distribute-frac-neg 51.5%

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

      7. distribute-neg-frac2 51.5%

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

      8. metadata-eval 51.5%

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

      9. /-rgt-identity 51.5%

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

      10. sub-neg 51.5%

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

      11. log1p-define 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in u0 around 0 77.7%

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

   6. Taylor expanded in cos2phi around inf 62.9%

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

   if 4.00000008e-25 < sin2phi

   1. Initial program 60.7%

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

      1. distribute-frac-neg 60.7%

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

      2. distribute-neg-frac2 60.7%

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

      3. neg-mul-1 60.7%

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

      4. associate-/r* 60.7%

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

      5. remove-double-neg 60.7%

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

      6. distribute-frac-neg 60.7%

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

      7. distribute-neg-frac2 60.7%

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

      8. metadata-eval 60.7%

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

      9. /-rgt-identity 60.7%

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

      10. sub-neg 60.7%

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

      11. log1p-define 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in u0 around 0 77.4%

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

   6. Taylor expanded in cos2phi around 0 72.5%

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

      1. associate-/l* 72.5%

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

   8. Simplified 72.5%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.0000000781659255 \cdot 10^{-25}:\\ \;\;\;\;\frac{u0 \cdot {alphax}^{2}}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;{alphay}^{2} \cdot \frac{u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.0000000781659255 \cdot 10^{-25}:\\ \;\;\;\;{alphax}^{2} \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;{alphay}^{2} \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.0000000781659255e-25)
   (* (pow alphax 2.0) (/ u0 cos2phi))
   (* (pow alphay 2.0) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.0000000781659255e-25f) {
		tmp = powf(alphax, 2.0f) * (u0 / cos2phi);
	} else {
		tmp = powf(alphay, 2.0f) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 4.0000000781659255e-25) then
        tmp = (alphax ** 2.0e0) * (u0 / cos2phi)
    else
        tmp = (alphay ** 2.0e0) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.0000000781659255e-25))
		tmp = Float32((alphax ^ Float32(2.0)) * Float32(u0 / cos2phi));
	else
		tmp = Float32((alphay ^ Float32(2.0)) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.0000000781659255e-25))
		tmp = (alphax ^ single(2.0)) * (u0 / cos2phi);
	else
		tmp = (alphay ^ single(2.0)) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.00000008e-25

   1. Initial program 51.5%

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

      1. distribute-frac-neg 51.5%

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

      2. distribute-neg-frac2 51.5%

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

      3. neg-mul-1 51.5%

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

      4. associate-/r* 51.5%

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

      5. remove-double-neg 51.5%

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

      6. distribute-frac-neg 51.5%

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

      7. distribute-neg-frac2 51.5%

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

      8. metadata-eval 51.5%

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

      9. /-rgt-identity 51.5%

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

      10. sub-neg 51.5%

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

      11. log1p-define 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in u0 around 0 77.7%

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

   6. Taylor expanded in cos2phi around inf 62.9%

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

      1. associate-/l* 62.8%

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

   8. Simplified 62.8%

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

   if 4.00000008e-25 < sin2phi

   1. Initial program 60.7%

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

      1. distribute-frac-neg 60.7%

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

      2. distribute-neg-frac2 60.7%

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

      3. neg-mul-1 60.7%

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

      4. associate-/r* 60.7%

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

      5. remove-double-neg 60.7%

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

      6. distribute-frac-neg 60.7%

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

      7. distribute-neg-frac2 60.7%

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

      8. metadata-eval 60.7%

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

      9. /-rgt-identity 60.7%

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

      10. sub-neg 60.7%

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

      11. log1p-define 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in u0 around 0 77.4%

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

   6. Taylor expanded in cos2phi around 0 72.5%

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

      1. associate-/l* 72.5%

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

   8. Simplified 72.5%

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

Alternative 8: 23.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {alphax}^{2} \cdot \frac{u0}{cos2phi} \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (alphax ** 2.0e0) * (u0 / cos2phi)
end function

Julia

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

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (alphax ^ single(2.0)) * (u0 / cos2phi);
end

Derivation

1. Initial program 59.1%

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

   1. distribute-frac-neg 59.1%

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

   2. distribute-neg-frac2 59.1%

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

   3. neg-mul-1 59.1%

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

   4. associate-/r* 59.1%

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

   5. remove-double-neg 59.1%

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

   6. distribute-frac-neg 59.1%

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

   7. distribute-neg-frac2 59.1%

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

   8. metadata-eval 59.1%

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

   9. /-rgt-identity 59.1%

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

   10. sub-neg 59.1%

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

   11. log1p-define 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in u0 around 0 77.5%

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

6. Taylor expanded in cos2phi around inf 21.3%

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

   1. associate-/l* 21.3%

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

8. Simplified 21.3%

   \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{u0}{cos2phi}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024090 
(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)))))