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

Percentage Accurate: 60.5% → 98.4%

Time: 15.1s

Alternatives: 8

Speedup: 12.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.5% 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

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

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 62.1%

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

   1. sub-neg 62.1%

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

   2. log1p-def 98.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 2: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 4.600000102072954 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{t_0 + cos2phi \cdot {alphax}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{alphay}^{2}}{sin2phi \cdot \left(\frac{1}{u0} - \left(0.5 + u0 \cdot 0.08333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 4.600000102072954e-5)
     (/ u0 (+ t_0 (* cos2phi (pow alphax -2.0))))
     (/
      (pow alphay 2.0)
      (* sin2phi (- (/ 1.0 u0) (+ 0.5 (* u0 0.08333333333333333))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 4.600000102072954e-5f) {
		tmp = u0 / (t_0 + (cos2phi * powf(alphax, -2.0f)));
	} else {
		tmp = powf(alphay, 2.0f) / (sin2phi * ((1.0f / u0) - (0.5f + (u0 * 0.08333333333333333f))));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 4.600000102072954e-5) then
        tmp = u0 / (t_0 + (cos2phi * (alphax ** (-2.0e0))))
    else
        tmp = (alphay ** 2.0e0) / (sin2phi * ((1.0e0 / u0) - (0.5e0 + (u0 * 0.08333333333333333e0))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(4.600000102072954e-5))
		tmp = Float32(u0 / Float32(t_0 + Float32(cos2phi * (alphax ^ Float32(-2.0)))));
	else
		tmp = Float32((alphay ^ Float32(2.0)) / Float32(sin2phi * Float32(Float32(Float32(1.0) / u0) - Float32(Float32(0.5) + Float32(u0 * Float32(0.08333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(4.600000102072954e-5))
		tmp = u0 / (t_0 + (cos2phi * (alphax ^ single(-2.0))));
	else
		tmp = (alphay ^ single(2.0)) / (sin2phi * ((single(1.0) / u0) - (single(0.5) + (u0 * single(0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.6000001e-5

   1. Initial program 53.9%

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

      1. sub-neg 53.9%

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

      2. log1p-def 98.8%

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

   3. Simplified 98.8%

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

      1. clear-num 98.6%

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

      2. associate-/r/ 98.6%

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

      3. pow2 98.6%

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

      4. pow-flip 98.7%

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

      5. metadata-eval 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in u0 around 0 76.5%

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

      1. mul-1-neg 76.5%

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

   8. Simplified 76.5%

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

   if 4.6000001e-5 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.0%

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

   2. Taylor expanded in cos2phi around 0 67.1%

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

      1. mul-1-neg 67.1%

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

      2. associate-/l* 66.7%

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

      3. distribute-neg-frac 66.7%

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

      4. sub-neg 66.7%

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

      5. mul-1-neg 66.7%

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

      6. log1p-def 96.4%

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

      7. mul-1-neg 96.4%

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

   4. Simplified 96.4%

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

   5. Taylor expanded in u0 around 0 87.7%

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

      1. +-commutative 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. +-commutative 87.7%

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

      5. mul-1-neg 87.7%

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

      6. unsub-neg 87.7%

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

      7. *-commutative 87.7%

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

      8. distribute-rgt-out 87.7%

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

      9. metadata-eval 87.7%

         \[\leadsto \frac{-{alphay}^{2}}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot \color{blue}{-0.08333333333333333}\right)} \]

   7. Simplified 87.7%

      \[\leadsto \frac{-{alphay}^{2}}{\color{blue}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)}} \]

   8. Taylor expanded in sin2phi around -inf 87.7%

      \[\leadsto \color{blue}{\frac{{alphay}^{2}}{sin2phi \cdot \left(\frac{1}{u0} - \left(0.5 + 0.08333333333333333 \cdot u0\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 87.7%

         \[\leadsto \frac{{alphay}^{2}}{sin2phi \cdot \left(\frac{1}{u0} - \left(0.5 + \color{blue}{u0 \cdot 0.08333333333333333}\right)\right)} \]

   10. Simplified 87.7%

       \[\leadsto \color{blue}{\frac{{alphay}^{2}}{sin2phi \cdot \left(\frac{1}{u0} - \left(0.5 + u0 \cdot 0.08333333333333333\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.600000102072954 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + cos2phi \cdot {alphax}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{alphay}^{2}}{sin2phi \cdot \left(\frac{1}{u0} - \left(0.5 + u0 \cdot 0.08333333333333333\right)\right)}\\ \end{array} \]

Alternative 3: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0}{t_0 + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{{alphay}^{2}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.009999999776482582)
     (/ u0 (+ t_0 (/ (/ cos2phi alphax) alphax)))
     (- (/ (pow alphay 2.0) (- (* sin2phi 0.5) (/ sin2phi u0)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.009999999776482582f) {
		tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax));
	} else {
		tmp = -(powf(alphay, 2.0f) / ((sin2phi * 0.5f) - (sin2phi / u0)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.009999999776482582e0) then
        tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax))
    else
        tmp = -((alphay ** 2.0e0) / ((sin2phi * 0.5e0) - (sin2phi / u0)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.009999999776482582))
		tmp = Float32(u0 / Float32(t_0 + Float32(Float32(cos2phi / alphax) / alphax)));
	else
		tmp = Float32(-Float32((alphay ^ Float32(2.0)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.009999999776482582))
		tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax));
	else
		tmp = -((alphay ^ single(2.0)) / ((sin2phi * single(0.5)) - (sin2phi / u0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.00999999978

   1. Initial program 54.2%

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

   2. Taylor expanded in u0 around 0 76.2%

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

      1. mul-1-neg 76.2%

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

   4. Simplified 76.2%

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

      1. associate-/r* 76.2%

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

      2. div-inv 76.2%

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

   6. Applied egg-rr 76.2%

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

      1. un-div-inv 76.2%

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

   8. Applied egg-rr 76.2%

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

   if 0.00999999978 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.4%

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

   2. Taylor expanded in cos2phi around 0 67.6%

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

      1. mul-1-neg 67.6%

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

      2. associate-/l* 67.2%

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

      3. distribute-neg-frac 67.2%

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

      4. sub-neg 67.2%

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

      5. mul-1-neg 67.2%

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

      6. log1p-def 96.9%

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

      7. mul-1-neg 96.9%

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

   4. Simplified 96.9%

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

   5. Taylor expanded in u0 around 0 84.8%

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

      1. +-commutative 84.8%

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

      2. mul-1-neg 84.8%

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

      3. unsub-neg 84.8%

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

      4. *-commutative 84.8%

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

   7. Simplified 84.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{{alphay}^{2}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 4: 75.9% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

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

2. Taylor expanded in u0 around 0 74.1%

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

   1. mul-1-neg 74.1%

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

4. Simplified 74.1%

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

5. Final simplification 74.1%

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

Alternative 5: 75.9% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

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

2. Taylor expanded in u0 around 0 74.1%

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

   1. mul-1-neg 74.1%

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

4. Simplified 74.1%

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

   1. associate-/r* 74.2%

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

   2. div-inv 74.1%

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

6. Applied egg-rr 74.1%

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

   1. un-div-inv 74.2%

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

8. Applied egg-rr 74.2%

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

9. Final simplification 74.2%

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

Alternative 6: 65.6% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{\frac{\frac{sin2phi}{u0}}{alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 2.00000009162741e-18)
   (* (* alphax alphax) (/ u0 cos2phi))
   (/ alphay (/ (/ sin2phi u0) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 2.00000009162741e-18f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = alphay / ((sin2phi / u0) / alphay);
	}
	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 <= 2.00000009162741e-18) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = alphay / ((sin2phi / u0) / alphay)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(2.00000009162741e-18))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(alphay / Float32(Float32(sin2phi / u0) / alphay));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(2.00000009162741e-18))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = alphay / ((sin2phi / u0) / alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.00000009e-18

   1. Initial program 53.7%

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

   2. Taylor expanded in u0 around 0 75.9%

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

      1. mul-1-neg 75.9%

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

   4. Simplified 75.9%

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

      1. associate-/r* 75.9%

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

      2. div-inv 75.8%

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

   6. Applied egg-rr 75.8%

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

   7. Taylor expanded in cos2phi around inf 60.4%

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

      1. *-lft-identity 60.4%

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

      2. times-frac 60.4%

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

      3. /-rgt-identity 60.4%

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

   9. Simplified 60.4%

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

       1. pow2 60.4%

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

   11. Applied egg-rr 60.4%

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

   if 2.00000009e-18 < sin2phi

   1. Initial program 65.4%

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

   2. Taylor expanded in cos2phi around 0 61.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   3. 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* 61.5%

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

      3. distribute-neg-frac 61.5%

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

      4. sub-neg 61.5%

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

      5. mul-1-neg 61.5%

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

      6. log1p-def 90.5%

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

      7. mul-1-neg 90.5%

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

   4. Simplified 90.5%

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

      1. add-sqr-sqrt -0.0%

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

      2. *-un-lft-identity -0.0%

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

      3. times-frac -0.0%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod -0.0%

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

      6. sqr-neg -0.0%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt -0.0%

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

      9. pow2 -0.0%

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

      10. sqrt-prod -0.0%

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

      11. add-sqr-sqrt -0.0%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 15.3%

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

      14. sqr-neg 15.3%

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

      15. sqrt-unprod 15.3%

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

      16. add-sqr-sqrt 15.3%

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

      17. pow2 15.3%

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

      18. sqrt-prod 15.3%

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

      19. add-sqr-sqrt 15.3%

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

   6. Applied egg-rr 66.4%

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

   7. Taylor expanded in u0 around 0 68.7%

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

      1. associate-/l* 68.2%

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

   9. Simplified 68.2%

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

       1. clear-num 67.5%

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

       2. frac-times 67.6%

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

       3. metadata-eval 67.6%

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

       4. div-inv 67.6%

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

       5. /-rgt-identity 67.6%

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

       6. metadata-eval 67.6%

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

       7. times-frac 67.6%

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

       8. *-un-lft-identity 67.6%

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

       9. *-un-lft-identity 67.6%

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

   11. Applied egg-rr 67.6%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{\frac{\frac{sin2phi}{u0}}{alphay}}\\ \end{array} \]

Alternative 7: 66.7% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 2.00000009162741e-18)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* alphay (* u0 (/ alphay sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 2.00000009162741e-18f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = alphay * (u0 * (alphay / sin2phi));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 2.00000009162741e-18) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = alphay * (u0 * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(2.00000009162741e-18))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(alphay * Float32(u0 * Float32(alphay / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(2.00000009162741e-18))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = alphay * (u0 * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.00000009e-18

   1. Initial program 53.7%

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

   2. Taylor expanded in u0 around 0 75.9%

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

      1. mul-1-neg 75.9%

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

   4. Simplified 75.9%

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

      1. associate-/r* 75.9%

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

      2. div-inv 75.8%

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

   6. Applied egg-rr 75.8%

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

   7. Taylor expanded in cos2phi around inf 60.4%

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

      1. *-lft-identity 60.4%

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

      2. times-frac 60.4%

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

      3. /-rgt-identity 60.4%

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

   9. Simplified 60.4%

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

       1. pow2 60.4%

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

   11. Applied egg-rr 60.4%

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

   if 2.00000009e-18 < sin2phi

   1. Initial program 65.4%

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

   2. Taylor expanded in cos2phi around 0 61.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   3. 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* 61.5%

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

      3. distribute-neg-frac 61.5%

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

      4. sub-neg 61.5%

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

      5. mul-1-neg 61.5%

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

      6. log1p-def 90.5%

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

      7. mul-1-neg 90.5%

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

   4. Simplified 90.5%

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

      1. add-sqr-sqrt -0.0%

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

      2. *-un-lft-identity -0.0%

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

      3. times-frac -0.0%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod -0.0%

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

      6. sqr-neg -0.0%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt -0.0%

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

      9. pow2 -0.0%

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

      10. sqrt-prod -0.0%

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

      11. add-sqr-sqrt -0.0%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 15.3%

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

      14. sqr-neg 15.3%

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

      15. sqrt-unprod 15.3%

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

      16. add-sqr-sqrt 15.3%

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

      17. pow2 15.3%

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

      18. sqrt-prod 15.3%

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

      19. add-sqr-sqrt 15.3%

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

   6. Applied egg-rr 66.4%

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

   7. Taylor expanded in u0 around 0 68.7%

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

      1. associate-/l* 68.2%

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

   9. Simplified 68.2%

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

       1. associate-/r/ 68.8%

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

   11. Applied egg-rr 68.8%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 8: 23.7% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

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

2. Taylor expanded in u0 around 0 74.1%

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

   1. mul-1-neg 74.1%

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

4. Simplified 74.1%

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

   1. associate-/r* 74.2%

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

   2. div-inv 74.1%

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

6. Applied egg-rr 74.1%

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

7. Taylor expanded in cos2phi around inf 25.5%

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

   1. *-lft-identity 25.5%

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

   2. times-frac 25.5%

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

   3. /-rgt-identity 25.5%

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

9. Simplified 25.5%

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

    1. pow2 25.5%

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

11. Applied egg-rr 25.5%

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

12. Final simplification 25.5%

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

Reproduce

herbie shell --seed 2023318 
(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)))))