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

Percentage Accurate: 61.0% → 98.8%

Time: 11.1s

Alternatives: 25

Speedup: 2.1×

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

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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)))))

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)
use fmin_fmax_functions
    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: 61.0% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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)))))

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)
use fmin_fmax_functions
    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: 98.8% accurate, 0.8× speedup

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

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (*
 (/
  (- (log1p (- u0)))
  (fma (* alphay alphay) (/ cos2phi (* alphax alphax)) sin2phi))
 (* alphay alphay)))

C

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

Julia

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

Derivation

1. Initial program 61.0%

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

3. Applied rewrites 61.4%

   \[\leadsto \frac{-\log \left(1 - u0\right)}{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)} \cdot \left(alphay \cdot alphay\right) \]
4. Step-by-step derivation

5. Applied rewrites 98.8%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)} \cdot \left(alphay \cdot alphay\right) \]
6. Add Preprocessing

Alternative 2: 98.4% accurate, 0.9× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.0%

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

3. Applied rewrites 98.4%

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

5. Applied rewrites 98.4%

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

Alternative 3: 96.4% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.996399998664856:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t\_0 - \frac{-\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (/ sin2phi (* alphay alphay))))
  (if (<= (- 1.0 u0) 0.996399998664856)
    (/ (- (log (- 1.0 u0))) (- t_0 (/ (- (/ cos2phi alphax)) alphax)))
    (/ (fma (* 0.5 u0) u0 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 ((1.0f - u0) <= 0.996399998664856f) {
		tmp = -logf((1.0f - u0)) / (t_0 - (-(cos2phi / alphax) / alphax));
	} else {
		tmp = fmaf((0.5f * u0), u0, 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 (Float32(Float32(1.0) - u0) <= Float32(0.996399998664856))
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(t_0 - Float32(Float32(-Float32(cos2phi / alphax)) / alphax)));
	else
		tmp = Float32(fma(Float32(Float32(0.5) * u0), u0, u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996399999

   1. Initial program 61.0%

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

   3. Applied rewrites 61.0%

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

   5. Applied rewrites 61.0%

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

   if 0.996399999 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 87.4%

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

   6. Applied rewrites 87.5%

      \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.4% accurate, 0.8× speedup

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

Math

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.996399998664856:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{t\_0}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0
        (+
         (/ cos2phi (* alphax alphax))
         (/ sin2phi (* alphay alphay)))))
  (if (<= (- 1.0 u0) 0.996399998664856)
    (/ (- (log (- 1.0 u0))) t_0)
    (/ (fma (* 0.5 u0) u0 u0) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay));
	float tmp;
	if ((1.0f - u0) <= 0.996399998664856f) {
		tmp = -logf((1.0f - u0)) / t_0;
	} else {
		tmp = fmaf((0.5f * u0), u0, u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.996399998664856))
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / t_0);
	else
		tmp = Float32(fma(Float32(Float32(0.5) * u0), u0, u0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996399999

   1. Initial program 61.0%

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

   if 0.996399999 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 87.4%

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

   6. Applied rewrites 87.5%

      \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.0% accurate, 0.9× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 7.659865514142439 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 7.659865514142439e-5)
  (/
   (fma (* 0.5 u0) u0 u0)
   (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
  (* (/ (- (log1p (- u0))) sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 7.659865514142439e-5f) {
		tmp = fmaf((0.5f * u0), u0, u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (-log1pf(-u0) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(7.659865514142439e-5))
		tmp = Float32(fma(Float32(Float32(0.5) * u0), u0, u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(Float32(-log1p(Float32(-u0))) / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 7.65986551e-5

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 87.4%

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

   6. Applied rewrites 87.5%

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

   if 7.65986551e-5 < sin2phi

   1. Initial program 61.0%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 73.9%

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

Alternative 6: 92.9% accurate, 0.9× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 7.659865514142439 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 7.659865514142439e-5)
  (*
   (/
    (fma 0.5 u0 1.0)
    (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
   u0)
  (* (/ (- (log1p (- u0))) sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 7.659865514142439e-5f) {
		tmp = (fmaf(0.5f, u0, 1.0f) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))) * u0;
	} else {
		tmp = (-log1pf(-u0) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(7.659865514142439e-5))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax)))) * u0);
	else
		tmp = Float32(Float32(Float32(-log1p(Float32(-u0))) / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 7.65986551e-5

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 87.4%

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

   6. Applied rewrites 87.4%

      \[\leadsto \frac{\mathsf{fma}\left(0.5, u0, 1\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot u0 \]

   if 7.65986551e-5 < sin2phi

   1. Initial program 61.0%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 73.9%

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

Alternative 7: 87.2% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.234215751821921 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{-\left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)}{-u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.234215751821921e-8)
  (/
   1.0
   (/
    (-
     (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
    (- u0)))
  (* (/ (- (log1p (- u0))) sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.234215751821921e-8f) {
		tmp = 1.0f / (-((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax))) / -u0);
	} else {
		tmp = (-log1pf(-u0) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.234215751821921e-8))
		tmp = Float32(Float32(1.0) / Float32(Float32(-Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax)))) / Float32(-u0)));
	else
		tmp = Float32(Float32(Float32(-log1p(Float32(-u0))) / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.23421575e-8

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.6%

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

   6. Applied rewrites 73.6%

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

   if 1.23421575e-8 < sin2phi

   1. Initial program 61.0%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 73.9%

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

Alternative 8: 87.2% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.234215751821921 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\mathsf{fma}\left(cos2phi, \frac{1}{alphax \cdot alphax}, \frac{sin2phi}{alphay \cdot alphay}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.234215751821921e-8)
  (/
   (- (- u0))
   (fma
    cos2phi
    (/ 1.0 (* alphax alphax))
    (/ sin2phi (* alphay alphay))))
  (* (/ (- (log1p (- u0))) sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.234215751821921e-8f) {
		tmp = -(-u0) / fmaf(cos2phi, (1.0f / (alphax * alphax)), (sin2phi / (alphay * alphay)));
	} else {
		tmp = (-log1pf(-u0) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.234215751821921e-8))
		tmp = Float32(Float32(-Float32(-u0)) / fma(cos2phi, Float32(Float32(1.0) / Float32(alphax * alphax)), Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(Float32(-log1p(Float32(-u0))) / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.23421575e-8

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.6%

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

   6. Applied rewrites 75.6%

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

   8. Applied rewrites 75.6%

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

   if 1.23421575e-8 < sin2phi

   1. Initial program 61.0%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 73.9%

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

Alternative 9: 87.2% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.234215751821921 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.234215751821921e-8)
  (/
   (- (- u0))
   (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
  (* (/ (- (log1p (- u0))) sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.234215751821921e-8f) {
		tmp = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = (-log1pf(-u0) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.234215751821921e-8))
		tmp = Float32(Float32(-Float32(-u0)) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(-log1p(Float32(-u0))) / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.23421575e-8

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.6%

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

   6. Applied rewrites 75.6%

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

   if 1.23421575e-8 < sin2phi

   1. Initial program 61.0%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   8. Applied rewrites 73.9%

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

Alternative 10: 87.2% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.234215751821921 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot alphay\right) \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.234215751821921e-8)
  (/
   (- (- u0))
   (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
  (* (* (/ (- (log1p (- u0))) sin2phi) alphay) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.234215751821921e-8f) {
		tmp = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = ((-log1pf(-u0) / sin2phi) * alphay) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.234215751821921e-8))
		tmp = Float32(Float32(-Float32(-u0)) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(Float32(-log1p(Float32(-u0))) / sin2phi) * alphay) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.23421575e-8

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.6%

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

   6. Applied rewrites 75.6%

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

   if 1.23421575e-8 < sin2phi

   1. Initial program 61.0%

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

   3. Applied rewrites 61.4%

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)} \cdot \left(alphay \cdot alphay\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 49.1%

      \[\leadsto \left(\frac{-\log \left(1 - u0\right)}{sin2phi} \cdot alphay\right) \cdot alphay \]
   9. Step-by-step derivation

   10. Applied rewrites 73.9%

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

Alternative 11: 87.2% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.234215751821921 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.234215751821921e-8)
  (/
   (- (- u0))
   (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
  (* (- (log1p (- u0))) (/ (* alphay alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.234215751821921e-8f) {
		tmp = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = -log1pf(-u0) * ((alphay * alphay) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.234215751821921e-8))
		tmp = Float32(Float32(-Float32(-u0)) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) * Float32(Float32(alphay * alphay) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.23421575e-8

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.6%

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

   6. Applied rewrites 75.6%

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

   if 1.23421575e-8 < sin2phi

   1. Initial program 61.0%

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

   2. Taylor expanded in alphax around inf

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

   4. Applied rewrites 49.1%

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

   6. Applied rewrites 73.9%

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

   8. Applied rewrites 49.1%

      \[\leadsto \left(-\log \left(1 - u0\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.9%

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

Alternative 12: 82.7% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9983999729156494:\\ \;\;\;\;\left(\frac{-\log \left(1 - u0\right)}{sin2phi} \cdot alphay\right) \cdot alphay\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= (- 1.0 u0) 0.9983999729156494)
  (* (* (/ (- (log (- 1.0 u0))) sin2phi) alphay) alphay)
  (/
   (- (- u0))
   (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((1.0f - u0) <= 0.9983999729156494f) {
		tmp = ((-logf((1.0f - u0)) / sin2phi) * alphay) * alphay;
	} else {
		tmp = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    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 ((1.0e0 - u0) <= 0.9983999729156494e0) then
        tmp = ((-log((1.0e0 - u0)) / sin2phi) * alphay) * alphay
    else
        tmp = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9983999729156494))
		tmp = Float32(Float32(Float32(Float32(-log(Float32(Float32(1.0) - u0))) / sin2phi) * alphay) * alphay);
	else
		tmp = Float32(Float32(-Float32(-u0)) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9983999729156494))
		tmp = ((-log((single(1.0) - u0)) / sin2phi) * alphay) * alphay;
	else
		tmp = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.998399973

   1. Initial program 61.0%

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

   3. Applied rewrites 61.4%

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)} \cdot \left(alphay \cdot alphay\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 49.1%

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

   if 0.998399973 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.6%

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

   6. Applied rewrites 75.6%

      \[\leadsto \frac{-\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 75.7% accurate, 1.2× speedup

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

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)\\ \mathbf{if}\;sin2phi \leq 1.030062684406026 \cdot 10^{-18}:\\ \;\;\;\;\frac{t\_0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (fma (* 0.5 u0) u0 u0)))
  (if (<= sin2phi 1.030062684406026e-18)
    (/ t_0 (* cos2phi (/ 1.0 (* alphax alphax))))
    (* (/ t_0 sin2phi) (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = fmaf((0.5f * u0), u0, u0);
	float tmp;
	if (sin2phi <= 1.030062684406026e-18f) {
		tmp = t_0 / (cos2phi * (1.0f / (alphax * alphax)));
	} else {
		tmp = (t_0 / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = fma(Float32(Float32(0.5) * u0), u0, u0)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.030062684406026e-18))
		tmp = Float32(t_0 / Float32(cos2phi * Float32(Float32(1.0) / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(t_0 / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.03006268e-18

   1. Initial program 61.0%

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in alphax around 0

      \[\leadsto \frac{-\log \left(1 - u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   5. Step-by-step derivation

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 26.4%

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

   11. Applied rewrites 26.4%

       \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]

   if 1.03006268e-18 < sin2phi

   1. Initial program 61.0%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 66.4%

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

   11. Applied rewrites 66.5%

       \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 75.7% accurate, 1.2× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.030062684406026 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.030062684406026e-18)
  (/ (* (fma 0.5 u0 1.0) u0) (* cos2phi (/ 1.0 (* alphax alphax))))
  (* (/ (fma (* 0.5 u0) u0 u0) sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.030062684406026e-18f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) / (cos2phi * (1.0f / (alphax * alphax)));
	} else {
		tmp = (fmaf((0.5f * u0), u0, u0) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.030062684406026e-18))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / Float32(cos2phi * Float32(Float32(1.0) / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(fma(Float32(Float32(0.5) * u0), u0, u0) / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.03006268e-18

   1. Initial program 61.0%

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in alphax around 0

      \[\leadsto \frac{-\log \left(1 - u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   5. Step-by-step derivation

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 26.4%

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

   11. Applied rewrites 26.4%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]

   if 1.03006268e-18 < sin2phi

   1. Initial program 61.0%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 66.4%

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

   11. Applied rewrites 66.5%

       \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 75.7% accurate, 1.4× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.030062684406026 \cdot 10^{-18}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.030062684406026e-18)
  (* (* (fma 0.5 u0 1.0) u0) (/ (* alphax alphax) cos2phi))
  (* (/ (fma (* 0.5 u0) u0 u0) sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.030062684406026e-18f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) * ((alphax * alphax) / cos2phi);
	} else {
		tmp = (fmaf((0.5f * u0), u0, u0) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.030062684406026e-18))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) * Float32(Float32(alphax * alphax) / cos2phi));
	else
		tmp = Float32(Float32(fma(Float32(Float32(0.5) * u0), u0, u0) / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.03006268e-18

   1. Initial program 61.0%

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in alphax around 0

      \[\leadsto \frac{-\log \left(1 - u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   5. Step-by-step derivation

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 26.4%

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

   11. Applied rewrites 26.4%

       \[\leadsto \left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \frac{alphax \cdot alphax}{cos2phi} \]

   if 1.03006268e-18 < sin2phi

   1. Initial program 61.0%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 66.4%

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

   11. Applied rewrites 66.5%

       \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 75.7% accurate, 1.4× speedup

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

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\\ \mathbf{if}\;sin2phi \leq 1.030062684406026 \cdot 10^{-18}:\\ \;\;\;\;t\_0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (* (fma 0.5 u0 1.0) u0)))
  (if (<= sin2phi 1.030062684406026e-18)
    (* t_0 (/ (* alphax alphax) cos2phi))
    (* (/ t_0 sin2phi) (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = fmaf(0.5f, u0, 1.0f) * u0;
	float tmp;
	if (sin2phi <= 1.030062684406026e-18f) {
		tmp = t_0 * ((alphax * alphax) / cos2phi);
	} else {
		tmp = (t_0 / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.030062684406026e-18))
		tmp = Float32(t_0 * Float32(Float32(alphax * alphax) / cos2phi));
	else
		tmp = Float32(Float32(t_0 / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.03006268e-18

   1. Initial program 61.0%

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in alphax around 0

      \[\leadsto \frac{-\log \left(1 - u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   5. Step-by-step derivation

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 26.4%

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

   11. Applied rewrites 26.4%

       \[\leadsto \left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \frac{alphax \cdot alphax}{cos2phi} \]

   if 1.03006268e-18 < sin2phi

   1. Initial program 61.0%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 66.4%

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

   11. Applied rewrites 66.4%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{sin2phi} \cdot \left(alphay \cdot alphay\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 75.7% accurate, 1.4× speedup

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

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\\ \mathbf{if}\;sin2phi \leq 1.030062684406026 \cdot 10^{-18}:\\ \;\;\;\;t\_0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t\_0}{sin2phi} \cdot alphay\right) \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (* (fma 0.5 u0 1.0) u0)))
  (if (<= sin2phi 1.030062684406026e-18)
    (* t_0 (/ (* alphax alphax) cos2phi))
    (* (* (/ t_0 sin2phi) alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = fmaf(0.5f, u0, 1.0f) * u0;
	float tmp;
	if (sin2phi <= 1.030062684406026e-18f) {
		tmp = t_0 * ((alphax * alphax) / cos2phi);
	} else {
		tmp = ((t_0 / sin2phi) * alphay) * alphay;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.030062684406026e-18))
		tmp = Float32(t_0 * Float32(Float32(alphax * alphax) / cos2phi));
	else
		tmp = Float32(Float32(Float32(t_0 / sin2phi) * alphay) * alphay);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.03006268e-18

   1. Initial program 61.0%

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in alphax around 0

      \[\leadsto \frac{-\log \left(1 - u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   5. Step-by-step derivation

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 26.4%

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

   11. Applied rewrites 26.4%

       \[\leadsto \left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \frac{alphax \cdot alphax}{cos2phi} \]

   if 1.03006268e-18 < sin2phi

   1. Initial program 61.0%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in alphax around inf

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 66.4%

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

   11. Applied rewrites 66.4%

       \[\leadsto \left(\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{sin2phi} \cdot alphay\right) \cdot alphay \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 75.7% accurate, 1.4× speedup

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

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\\ \mathbf{if}\;sin2phi \leq 1.030062684406026 \cdot 10^{-18}:\\ \;\;\;\;t\_0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (* (fma 0.5 u0 1.0) u0)))
  (if (<= sin2phi 1.030062684406026e-18)
    (* t_0 (/ (* alphax alphax) cos2phi))
    (/ (* t_0 (* alphay alphay)) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = fmaf(0.5f, u0, 1.0f) * u0;
	float tmp;
	if (sin2phi <= 1.030062684406026e-18f) {
		tmp = t_0 * ((alphax * alphax) / cos2phi);
	} else {
		tmp = (t_0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.030062684406026e-18))
		tmp = Float32(t_0 * Float32(Float32(alphax * alphax) / cos2phi));
	else
		tmp = Float32(Float32(t_0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.03006268e-18

   1. Initial program 61.0%

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in alphax around 0

      \[\leadsto \frac{-\log \left(1 - u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   5. Step-by-step derivation

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 26.4%

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

   11. Applied rewrites 26.4%

       \[\leadsto \left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \frac{alphax \cdot alphax}{cos2phi} \]

   if 1.03006268e-18 < sin2phi

   1. Initial program 61.0%

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

   2. Taylor expanded in alphax around inf

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

   4. Applied rewrites 49.1%

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 66.4%

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

   11. Applied rewrites 66.4%

       \[\leadsto \frac{\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \left(alphay \cdot alphay\right)}{sin2phi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 73.6% accurate, 1.4× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.030062684406026 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax}}{-\left(-u0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.030062684406026e-18)
  (/ 1.0 (/ (/ cos2phi (* alphax alphax)) (- (- u0))))
  (/ (* (* (fma 0.5 u0 1.0) u0) (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.030062684406026e-18f) {
		tmp = 1.0f / ((cos2phi / (alphax * alphax)) / -(-u0));
	} else {
		tmp = ((fmaf(0.5f, u0, 1.0f) * u0) * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.030062684406026e-18))
		tmp = Float32(Float32(1.0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) / Float32(-Float32(-u0))));
	else
		tmp = Float32(Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.03006268e-18

   1. Initial program 61.0%

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in alphax around 0

      \[\leadsto \frac{-\log \left(1 - u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   5. Step-by-step derivation

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{--1 \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   8. Step-by-step derivation

   9. Applied rewrites 23.8%

      \[\leadsto \frac{--1 \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.8%

       \[\leadsto \frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax}}{-\left(-u0\right)}} \]

   if 1.03006268e-18 < sin2phi

   1. Initial program 61.0%

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

   2. Taylor expanded in alphax around inf

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

   4. Applied rewrites 49.1%

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 66.4%

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

   11. Applied rewrites 66.4%

       \[\leadsto \frac{\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \left(alphay \cdot alphay\right)}{sin2phi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 66.3% accurate, 1.4× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.030062684406026 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax}}{-\left(-u0\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\left(-1 \cdot u0\right) \cdot \left(\frac{alphay}{sin2phi} \cdot alphay\right)\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.030062684406026e-18)
  (/ 1.0 (/ (/ cos2phi (* alphax alphax)) (- (- u0))))
  (* -1.0 (* (* -1.0 u0) (* (/ alphay sin2phi) alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.030062684406026e-18f) {
		tmp = 1.0f / ((cos2phi / (alphax * alphax)) / -(-u0));
	} else {
		tmp = -1.0f * ((-1.0f * u0) * ((alphay / sin2phi) * alphay));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.030062684406026e-18) then
        tmp = 1.0e0 / ((cos2phi / (alphax * alphax)) / -(-u0))
    else
        tmp = (-1.0e0) * (((-1.0e0) * u0) * ((alphay / sin2phi) * alphay))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.030062684406026e-18))
		tmp = Float32(Float32(1.0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) / Float32(-Float32(-u0))));
	else
		tmp = Float32(Float32(-1.0) * Float32(Float32(Float32(-1.0) * u0) * Float32(Float32(alphay / sin2phi) * alphay)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.030062684406026e-18))
		tmp = single(1.0) / ((cos2phi / (alphax * alphax)) / -(-u0));
	else
		tmp = single(-1.0) * ((single(-1.0) * u0) * ((alphay / sin2phi) * alphay));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.03006268e-18

   1. Initial program 61.0%

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in alphax around 0

      \[\leadsto \frac{-\log \left(1 - u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   5. Step-by-step derivation

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{--1 \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   8. Step-by-step derivation

   9. Applied rewrites 23.8%

      \[\leadsto \frac{--1 \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.8%

       \[\leadsto \frac{1}{\frac{\frac{cos2phi}{alphax \cdot alphax}}{-\left(-u0\right)}} \]

   if 1.03006268e-18 < sin2phi

   1. Initial program 61.0%

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

   2. Taylor expanded in alphax around inf

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

   4. Applied rewrites 49.1%

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 58.6%

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

   11. Applied rewrites 58.6%

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

Alternative 21: 66.3% accurate, 1.5× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.030062684406026 \cdot 10^{-18}:\\ \;\;\;\;\frac{-\left(-u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\left(-1 \cdot u0\right) \cdot \left(\frac{alphay}{sin2phi} \cdot alphay\right)\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.030062684406026e-18)
  (/ (- (- u0)) (* cos2phi (/ 1.0 (* alphax alphax))))
  (* -1.0 (* (* -1.0 u0) (* (/ alphay sin2phi) alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.030062684406026e-18f) {
		tmp = -(-u0) / (cos2phi * (1.0f / (alphax * alphax)));
	} else {
		tmp = -1.0f * ((-1.0f * u0) * ((alphay / sin2phi) * alphay));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.030062684406026e-18) then
        tmp = -(-u0) / (cos2phi * (1.0e0 / (alphax * alphax)))
    else
        tmp = (-1.0e0) * (((-1.0e0) * u0) * ((alphay / sin2phi) * alphay))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.030062684406026e-18))
		tmp = Float32(Float32(-Float32(-u0)) / Float32(cos2phi * Float32(Float32(1.0) / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(-1.0) * Float32(Float32(Float32(-1.0) * u0) * Float32(Float32(alphay / sin2phi) * alphay)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.030062684406026e-18))
		tmp = -(-u0) / (cos2phi * (single(1.0) / (alphax * alphax)));
	else
		tmp = single(-1.0) * ((single(-1.0) * u0) * ((alphay / sin2phi) * alphay));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.03006268e-18

   1. Initial program 61.0%

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in alphax around 0

      \[\leadsto \frac{-\log \left(1 - u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   5. Step-by-step derivation

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{--1 \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   8. Step-by-step derivation

   9. Applied rewrites 23.8%

      \[\leadsto \frac{--1 \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.8%

       \[\leadsto \frac{-\left(-u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]

   if 1.03006268e-18 < sin2phi

   1. Initial program 61.0%

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

   2. Taylor expanded in alphax around inf

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

   4. Applied rewrites 49.1%

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 58.6%

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

   11. Applied rewrites 58.6%

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

Alternative 22: 66.3% accurate, 1.5× speedup

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

Math

\[\begin{array}{l} t_0 := -\left(-u0\right)\\ \mathbf{if}\;sin2phi \leq 1.030062684406026 \cdot 10^{-18}:\\ \;\;\;\;\frac{t\_0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (let* ((t_0 (- (- u0))))
  (if (<= sin2phi 1.030062684406026e-18)
    (/ t_0 (* cos2phi (/ 1.0 (* alphax alphax))))
    (* (/ t_0 sin2phi) (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = -(-u0);
	float tmp;
	if (sin2phi <= 1.030062684406026e-18f) {
		tmp = t_0 / (cos2phi * (1.0f / (alphax * alphax)));
	} else {
		tmp = (t_0 / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    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 = -(-u0)
    if (sin2phi <= 1.030062684406026e-18) then
        tmp = t_0 / (cos2phi * (1.0e0 / (alphax * alphax)))
    else
        tmp = (t_0 / sin2phi) * (alphay * alphay)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(-Float32(-u0))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.030062684406026e-18))
		tmp = Float32(t_0 / Float32(cos2phi * Float32(Float32(1.0) / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(t_0 / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = -(-u0);
	tmp = single(0.0);
	if (sin2phi <= single(1.030062684406026e-18))
		tmp = t_0 / (cos2phi * (single(1.0) / (alphax * alphax)));
	else
		tmp = (t_0 / sin2phi) * (alphay * alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.03006268e-18

   1. Initial program 61.0%

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in alphax around 0

      \[\leadsto \frac{-\log \left(1 - u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   5. Step-by-step derivation

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{--1 \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   8. Step-by-step derivation

   9. Applied rewrites 23.8%

      \[\leadsto \frac{--1 \cdot u0}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.8%

       \[\leadsto \frac{-\left(-u0\right)}{cos2phi \cdot \frac{1}{alphax \cdot alphax}} \]

   if 1.03006268e-18 < sin2phi

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.6%

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

   6. Applied rewrites 75.9%

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

   7. Taylor expanded in alphax around inf

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

   9. Applied rewrites 58.6%

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

Alternative 23: 66.3% accurate, 1.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.030062684406026 \cdot 10^{-18}:\\ \;\;\;\;\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.030062684406026e-18)
  (* (/ u0 cos2phi) (* alphax alphax))
  (* (/ (- (- u0)) sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.030062684406026e-18f) {
		tmp = (u0 / cos2phi) * (alphax * alphax);
	} else {
		tmp = (-(-u0) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.030062684406026e-18) then
        tmp = (u0 / cos2phi) * (alphax * alphax)
    else
        tmp = (-(-u0) / sin2phi) * (alphay * alphay)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.030062684406026e-18))
		tmp = Float32(Float32(u0 / cos2phi) * Float32(alphax * alphax));
	else
		tmp = Float32(Float32(Float32(-Float32(-u0)) / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.030062684406026e-18))
		tmp = (u0 / cos2phi) * (alphax * alphax);
	else
		tmp = (-(-u0) / sin2phi) * (alphay * alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.03006268e-18

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.6%

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

   6. Applied rewrites 75.6%

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

   7. Taylor expanded in alphax around 0

      \[\leadsto \left(-1 \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right) \cdot \left(alphax \cdot alphax\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 22.6%

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

   10. Taylor expanded in u0 around 0

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

   12. Applied rewrites 23.8%

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

   if 1.03006268e-18 < sin2phi

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.6%

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

   6. Applied rewrites 75.9%

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

   7. Taylor expanded in alphax around inf

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

   9. Applied rewrites 58.6%

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

Alternative 24: 66.3% accurate, 2.1× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.030062684406026 \cdot 10^{-18}:\\ \;\;\;\;\frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot u0}{sin2phi} \cdot alphay\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
  :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))
  (if (<= sin2phi 1.030062684406026e-18)
  (* (/ u0 cos2phi) (* alphax alphax))
  (* (/ (* alphay u0) sin2phi) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.030062684406026e-18f) {
		tmp = (u0 / cos2phi) * (alphax * alphax);
	} else {
		tmp = ((alphay * u0) / sin2phi) * alphay;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.030062684406026e-18) then
        tmp = (u0 / cos2phi) * (alphax * alphax)
    else
        tmp = ((alphay * u0) / sin2phi) * alphay
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.030062684406026e-18))
		tmp = Float32(Float32(u0 / cos2phi) * Float32(alphax * alphax));
	else
		tmp = Float32(Float32(Float32(alphay * u0) / sin2phi) * alphay);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.030062684406026e-18))
		tmp = (u0 / cos2phi) * (alphax * alphax);
	else
		tmp = ((alphay * u0) / sin2phi) * alphay;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.03006268e-18

   1. Initial program 61.0%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.6%

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

   6. Applied rewrites 75.6%

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

   7. Taylor expanded in alphax around 0

      \[\leadsto \left(-1 \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right) \cdot \left(alphax \cdot alphax\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 22.6%

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

   10. Taylor expanded in u0 around 0

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

   12. Applied rewrites 23.8%

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

   if 1.03006268e-18 < sin2phi

   1. Initial program 61.0%

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

   3. Applied rewrites 61.4%

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)} \cdot \left(alphay \cdot alphay\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 49.1%

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

   9. Taylor expanded in u0 around 0

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

   11. Applied rewrites 58.6%

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

Alternative 25: 58.6% accurate, 2.8× speedup

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

Math

\[\frac{alphay \cdot u0}{sin2phi} \cdot alphay \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.0%

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

3. Applied rewrites 61.4%

   \[\leadsto \frac{-\log \left(1 - u0\right)}{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)} \cdot \left(alphay \cdot alphay\right) \]
4. Step-by-step derivation

5. Applied rewrites 61.4%

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

6. Taylor expanded in alphax around inf

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

8. Applied rewrites 49.1%

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

9. Taylor expanded in u0 around 0

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

11. Applied rewrites 58.6%

    \[\leadsto \frac{alphay \cdot u0}{sin2phi} \cdot alphay \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2026070 
(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)))))