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

Percentage Accurate: 61.1% → 98.8%

Time: 53.8s

Alternatives: 19

Speedup: 1.4×

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: 61.1% 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

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

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 (/ cos2phi (* alphax alphax)) (* alphay alphay) sin2phi))
 (* alphay alphay)))

C

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

Julia

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

Derivation

1. Initial program 61.1%

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

3. Applied rewrites 61.6%

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

5. Applied rewrites 98.8%

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

Alternative 2: 98.7% 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

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

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)))
  (fma (/ cos2phi (* alphax alphax)) alphay (/ sin2phi alphay)))
 alphay))

C

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

Julia

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

Derivation

1. Initial program 61.1%

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

3. Applied rewrites 61.5%

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

5. Applied rewrites 61.5%

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

7. Applied rewrites 98.7%

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

Alternative 3: 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 61.1%

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

3. Applied rewrites 98.4%

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

Alternative 4: 96.3% 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

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;1 - u0 \leq 0.996999979019165:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\mathsf{fma}\left(sin2phi, \frac{1}{alphay \cdot alphay}, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, u0\right)}{t\_0 + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

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 (/ cos2phi (* alphax alphax))))
  (if (<= (- 1.0 u0) 0.996999979019165)
    (/
     (- (log (- 1.0 u0)))
     (fma sin2phi (/ 1.0 (* alphay alphay)) t_0))
    (/ (fma u0 (* 0.5 u0) u0) (+ t_0 (/ sin2phi (* alphay alphay)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float tmp;
	if ((1.0f - u0) <= 0.996999979019165f) {
		tmp = -logf((1.0f - u0)) / fmaf(sin2phi, (1.0f / (alphay * alphay)), t_0);
	} else {
		tmp = fmaf(u0, (0.5f * u0), u0) / (t_0 + (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.996999979019165))
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / fma(sin2phi, Float32(Float32(1.0) / Float32(alphay * alphay)), t_0));
	else
		tmp = Float32(fma(u0, Float32(Float32(0.5) * u0), u0) / Float32(t_0 + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996999979

   1. Initial program 61.1%

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

   3. Applied rewrites 61.1%

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

   if 0.996999979 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 61.1%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 87.3%

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

   6. Applied rewrites 87.4%

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

Alternative 5: 96.3% accurate, 0.7× 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

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ t_1 := -\log \left(1 - u0\right)\\ \mathbf{if}\;t\_1 \leq 0.003000000026077032:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, u0\right)}{t\_0 + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_0 + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]

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 (/ cos2phi (* alphax alphax))) (t_1 (- (log (- 1.0 u0)))))
  (if (<= t_1 0.003000000026077032)
    (/ (fma u0 (* 0.5 u0) u0) (+ t_0 (/ sin2phi (* alphay alphay))))
    (/ t_1 (+ t_0 (/ (/ sin2phi alphay) alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float t_1 = -logf((1.0f - u0));
	float tmp;
	if (t_1 <= 0.003000000026077032f) {
		tmp = fmaf(u0, (0.5f * u0), u0) / (t_0 + (sin2phi / (alphay * alphay)));
	} else {
		tmp = t_1 / (t_0 + ((sin2phi / alphay) / alphay));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	t_1 = Float32(-log(Float32(Float32(1.0) - u0)))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.003000000026077032))
		tmp = Float32(fma(u0, Float32(Float32(0.5) * u0), u0) / Float32(t_0 + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(t_1 / Float32(t_0 + Float32(Float32(sin2phi / alphay) / alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0))) < 0.00300000003

   1. Initial program 61.1%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 87.3%

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

   6. Applied rewrites 87.4%

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

   if 0.00300000003 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u0)))

   1. Initial program 61.1%

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

   3. Applied rewrites 61.1%

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.3% 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

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.996999979019165:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, u0\right)}{t\_0}\\ \end{array} \]

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) 0.996999979019165)
    (/ (- (log (- 1.0 u0))) t_0)
    (/ (fma u0 (* 0.5 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) <= 0.996999979019165f) {
		tmp = -logf((1.0f - u0)) / t_0;
	} else {
		tmp = fmaf(u0, (0.5f * 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(0.996999979019165))
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / t_0);
	else
		tmp = Float32(fma(u0, Float32(Float32(0.5) * u0), u0) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996999979

   1. Initial program 61.1%

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

   if 0.996999979 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 61.1%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 87.3%

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

   6. Applied rewrites 87.4%

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

Alternative 7: 92.4% 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

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 18.478857040405273:\\ \;\;\;\;u0 \cdot \frac{\mathsf{fma}\left(u0, 0.5, 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

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 18.478857040405273)
  (*
   u0
   (/
    (fma u0 0.5 1.0)
    (+ (/ 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 <= 18.478857040405273f) {
		tmp = u0 * (fmaf(u0, 0.5f, 1.0f) / ((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(18.478857040405273))
		tmp = Float32(u0 * Float32(fma(u0, Float32(0.5), Float32(1.0)) / 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 < 18.478857

   1. Initial program 61.1%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 87.3%

      \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}, \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) \]

   6. Applied rewrites 87.2%

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

   if 18.478857 < sin2phi

   1. Initial program 61.1%

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

   3. Applied rewrites 61.6%

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 74.4%

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

Alternative 8: 87.2% accurate, 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

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.400000004814217 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\mathsf{fma}\left(\frac{1}{alphay}, \frac{sin2phi}{alphay}, \frac{cos2phi}{alphax \cdot alphax}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

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 (* alphay alphay)) 1.400000004814217e-9)
  (/
   (- (- u0))
   (fma
    (/ 1.0 alphay)
    (/ sin2phi 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 / (alphay * alphay)) <= 1.400000004814217e-9f) {
		tmp = -(-u0) / fmaf((1.0f / alphay), (sin2phi / 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 (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.400000004814217e-9))
		tmp = Float32(Float32(-Float32(-u0)) / fma(Float32(Float32(1.0) / alphay), Float32(sin2phi / 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 (/.f32 sin2phi (*.f32 alphay alphay)) < 1.4e-9

   1. Initial program 61.1%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.7%

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

   6. Applied rewrites 75.7%

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

   8. Applied rewrites 75.7%

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

   if 1.4e-9 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.1%

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

   3. Applied rewrites 61.6%

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 74.4%

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

Alternative 9: 87.2% accurate, 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

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 1.400000004814217 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

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))
  (let* ((t_0 (/ sin2phi (* alphay alphay))))
  (if (<= t_0 1.400000004814217e-9)
    (/ (- (- u0)) (+ (/ cos2phi (* alphax alphax)) t_0))
    (* (/ (- (log1p (- u0))) sin2phi) (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 1.400000004814217e-9f) {
		tmp = -(-u0) / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (-log1pf(-u0) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.400000004814217e-9))
		tmp = Float32(Float32(-Float32(-u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	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 (/.f32 sin2phi (*.f32 alphay alphay)) < 1.4e-9

   1. Initial program 61.1%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.7%

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

   6. Applied rewrites 75.7%

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

   if 1.4e-9 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.1%

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

   3. Applied rewrites 61.6%

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 74.4%

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

Alternative 10: 82.8% 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

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.996999979019165:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{sin2phi}{alphay}} \cdot alphay\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

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 (<= (- 1.0 u0) 0.996999979019165)
  (* (/ (- (log (- 1.0 u0))) (/ sin2phi alphay)) alphay)
  (/
   (- (- u0))
   (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((1.0f - u0) <= 0.996999979019165f) {
		tmp = (-logf((1.0f - u0)) / (sin2phi / alphay)) * alphay;
	} else {
		tmp = -(-u0) / ((cos2phi / (alphax * alphax)) + (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 ((1.0e0 - u0) <= 0.996999979019165e0) then
        tmp = (-log((1.0e0 - u0)) / (sin2phi / alphay)) * alphay
    else
        tmp = -(-u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    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(0.996999979019165))
		tmp = Float32(Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(sin2phi / alphay)) * alphay);
	else
		tmp = Float32(Float32(-Float32(-u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.996999979019165))
		tmp = (-log((single(1.0) - u0)) / (sin2phi / alphay)) * alphay;
	else
		tmp = -(-u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996999979

   1. Initial program 61.1%

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

   3. Applied rewrites 61.5%

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 49.4%

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

   if 0.996999979 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 61.1%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 75.7%

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

   6. Applied rewrites 75.7%

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

Alternative 11: 76.1% 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

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\left(u0 \cdot \mathsf{fma}\left(0.5, \frac{alphax \cdot u0}{cos2phi}, \frac{alphax}{cos2phi}\right)\right) \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, 1 \cdot u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

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 (* alphay alphay)) 2.00000009162741e-18)
  (*
   (* u0 (fma 0.5 (/ (* alphax u0) cos2phi) (/ alphax cos2phi)))
   alphax)
  (* (/ (fma u0 (* 0.5 u0) (* 1.0 u0)) sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.00000009162741e-18f) {
		tmp = (u0 * fmaf(0.5f, ((alphax * u0) / cos2phi), (alphax / cos2phi))) * alphax;
	} else {
		tmp = (fmaf(u0, (0.5f * u0), (1.0f * u0)) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.00000009162741e-18))
		tmp = Float32(Float32(u0 * fma(Float32(0.5), Float32(Float32(alphax * u0) / cos2phi), Float32(alphax / cos2phi))) * alphax);
	else
		tmp = Float32(Float32(fma(u0, Float32(Float32(0.5) * u0), Float32(Float32(1.0) * u0)) / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.00000009e-18

   1. Initial program 61.1%

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

   3. Applied rewrites 61.5%

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 22.5%

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

   9. Taylor expanded in u0 around 0

      \[\leadsto \left(u0 \cdot \left(\frac{1}{2} \cdot \frac{alphax \cdot u0}{cos2phi} + \frac{alphax}{cos2phi}\right)\right) \cdot alphax \]

   11. Applied rewrites 25.9%

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

   if 2.00000009e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.1%

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

   3. Applied rewrites 61.6%

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

   4. Taylor expanded in u0 around 0

      \[\leadsto \frac{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay \cdot alphay, sin2phi\right)} \cdot \left(alphay \cdot alphay\right) \]

   6. Applied rewrites 87.6%

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

   7. Taylor expanded in alphax around inf

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

   9. Applied rewrites 66.9%

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

   11. Applied rewrites 66.9%

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

Alternative 12: 76.1% 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

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\left(-1 \cdot \frac{alphax \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right)}{cos2phi}\right) \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, 1 \cdot u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

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 (* alphay alphay)) 2.00000009162741e-18)
  (*
   (* -1.0 (/ (* alphax (* u0 (- (* -0.5 u0) 1.0))) cos2phi))
   alphax)
  (* (/ (fma u0 (* 0.5 u0) (* 1.0 u0)) sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.00000009162741e-18f) {
		tmp = (-1.0f * ((alphax * (u0 * ((-0.5f * u0) - 1.0f))) / cos2phi)) * alphax;
	} else {
		tmp = (fmaf(u0, (0.5f * u0), (1.0f * u0)) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.00000009162741e-18))
		tmp = Float32(Float32(Float32(-1.0) * Float32(Float32(alphax * Float32(u0 * Float32(Float32(Float32(-0.5) * u0) - Float32(1.0)))) / cos2phi)) * alphax);
	else
		tmp = Float32(Float32(fma(u0, Float32(Float32(0.5) * u0), Float32(Float32(1.0) * u0)) / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.00000009e-18

   1. Initial program 61.1%

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

   3. Applied rewrites 61.5%

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 22.5%

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

   9. Taylor expanded in u0 around 0

      \[\leadsto \left(-1 \cdot \frac{alphax \cdot \left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)}{cos2phi}\right) \cdot alphax \]

   11. Applied rewrites 25.9%

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

   if 2.00000009e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.1%

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

   3. Applied rewrites 61.6%

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

   4. Taylor expanded in u0 around 0

      \[\leadsto \frac{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay \cdot alphay, sin2phi\right)} \cdot \left(alphay \cdot alphay\right) \]

   6. Applied rewrites 87.6%

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

   7. Taylor expanded in alphax around inf

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

   9. Applied rewrites 66.9%

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

   11. Applied rewrites 66.9%

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

Alternative 13: 76.1% 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

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, 0.5 \cdot u0, 1 \cdot u0\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

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 (* alphay alphay)) 2.00000009162741e-18)
  (/ (* u0 (fma u0 0.5 1.0)) (/ cos2phi (* alphax alphax)))
  (* (/ (fma u0 (* 0.5 u0) (* 1.0 u0)) sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.00000009162741e-18f) {
		tmp = (u0 * fmaf(u0, 0.5f, 1.0f)) / (cos2phi / (alphax * alphax));
	} else {
		tmp = (fmaf(u0, (0.5f * u0), (1.0f * u0)) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.00000009162741e-18))
		tmp = Float32(Float32(u0 * fma(u0, Float32(0.5), Float32(1.0))) / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(fma(u0, Float32(Float32(0.5) * u0), Float32(Float32(1.0) * u0)) / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.00000009e-18

   1. Initial program 61.1%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 22.5%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax}} \]

   9. Applied rewrites 25.9%

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

   11. Applied rewrites 25.9%

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

   if 2.00000009e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.1%

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

   3. Applied rewrites 61.6%

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

   4. Taylor expanded in u0 around 0

      \[\leadsto \frac{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay \cdot alphay, sin2phi\right)} \cdot \left(alphay \cdot alphay\right) \]

   6. Applied rewrites 87.6%

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

   7. Taylor expanded in alphax around inf

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

   9. Applied rewrites 66.9%

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

   11. Applied rewrites 66.9%

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

Alternative 14: 76.1% 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

\[\begin{array}{l} t_0 := u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\frac{t\_0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

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 (fma u0 0.5 1.0))))
  (if (<= (/ sin2phi (* alphay alphay)) 2.00000009162741e-18)
    (/ t_0 (/ cos2phi (* alphax alphax)))
    (* (/ t_0 sin2phi) (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = u0 * fmaf(u0, 0.5f, 1.0f);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.00000009162741e-18f) {
		tmp = t_0 / (cos2phi / (alphax * alphax));
	} else {
		tmp = (t_0 / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(u0 * fma(u0, Float32(0.5), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.00000009162741e-18))
		tmp = Float32(t_0 / Float32(cos2phi / 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 (/.f32 sin2phi (*.f32 alphay alphay)) < 2.00000009e-18

   1. Initial program 61.1%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 22.5%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax}} \]

   9. Applied rewrites 25.9%

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

   11. Applied rewrites 25.9%

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

   if 2.00000009e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.1%

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

   3. Applied rewrites 61.6%

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

   4. Taylor expanded in u0 around 0

      \[\leadsto \frac{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay \cdot alphay, sin2phi\right)} \cdot \left(alphay \cdot alphay\right) \]

   6. Applied rewrites 87.6%

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

   7. Taylor expanded in alphax around inf

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

   9. Applied rewrites 66.9%

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

   11. Applied rewrites 66.9%

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

Alternative 15: 74.3% 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

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\frac{-\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

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 (* alphay alphay)) 2.00000009162741e-18)
  (/ (- (- u0)) (/ cos2phi (* alphax alphax)))
  (* (/ (* u0 (fma u0 0.5 1.0)) sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.00000009162741e-18f) {
		tmp = -(-u0) / (cos2phi / (alphax * alphax));
	} else {
		tmp = ((u0 * fmaf(u0, 0.5f, 1.0f)) / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.00000009162741e-18))
		tmp = Float32(Float32(-Float32(-u0)) / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(Float32(u0 * fma(u0, Float32(0.5), Float32(1.0))) / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.00000009e-18

   1. Initial program 61.1%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 22.5%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{--1 \cdot u0}{\frac{cos2phi}{alphax \cdot alphax}} \]

   9. Applied rewrites 23.5%

      \[\leadsto \frac{--1 \cdot u0}{\frac{cos2phi}{alphax \cdot alphax}} \]

   11. Applied rewrites 23.5%

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

   if 2.00000009e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.1%

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

   3. Applied rewrites 61.6%

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

   4. Taylor expanded in u0 around 0

      \[\leadsto \frac{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay \cdot alphay, sin2phi\right)} \cdot \left(alphay \cdot alphay\right) \]

   6. Applied rewrites 87.6%

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

   7. Taylor expanded in alphax around inf

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

   9. Applied rewrites 66.9%

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

   11. Applied rewrites 66.9%

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

Alternative 16: 66.8% accurate, 1.3× 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

\[\begin{array}{l} t_0 := -\left(-u0\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\frac{t\_0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

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 (* alphay alphay)) 2.00000009162741e-18)
    (/ t_0 (/ cos2phi (* 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 / (alphay * alphay)) <= 2.00000009162741e-18f) {
		tmp = t_0 / (cos2phi / (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 / (alphay * alphay)) <= 2.00000009162741e-18) then
        tmp = t_0 / (cos2phi / (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 (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.00000009162741e-18))
		tmp = Float32(t_0 / Float32(cos2phi / 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 / (alphay * alphay)) <= single(2.00000009162741e-18))
		tmp = t_0 / (cos2phi / (alphax * alphax));
	else
		tmp = (t_0 / sin2phi) * (alphay * alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.00000009e-18

   1. Initial program 61.1%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in alphax around 0

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

   6. Applied rewrites 22.5%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \frac{--1 \cdot u0}{\frac{cos2phi}{alphax \cdot alphax}} \]

   9. Applied rewrites 23.5%

      \[\leadsto \frac{--1 \cdot u0}{\frac{cos2phi}{alphax \cdot alphax}} \]

   11. Applied rewrites 23.5%

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

   if 2.00000009e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.1%

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

   3. Applied rewrites 61.6%

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

   4. Taylor expanded in u0 around 0

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

   6. Applied rewrites 76.0%

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

   7. Taylor expanded in alphax around inf

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

   9. Applied rewrites 59.1%

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

   11. Applied rewrites 59.1%

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

Alternative 17: 66.8% accurate, 1.3× 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

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

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 (* alphay alphay)) 2.00000009162741e-18)
  (* (/ (* alphax u0) cos2phi) alphax)
  (* (/ (- (- u0)) sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.00000009162741e-18f) {
		tmp = ((alphax * u0) / cos2phi) * 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 / (alphay * alphay)) <= 2.00000009162741e-18) then
        tmp = ((alphax * u0) / cos2phi) * 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 (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.00000009162741e-18))
		tmp = Float32(Float32(Float32(alphax * u0) / cos2phi) * 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 / (alphay * alphay)) <= single(2.00000009162741e-18))
		tmp = ((alphax * u0) / cos2phi) * alphax;
	else
		tmp = (-(-u0) / sin2phi) * (alphay * alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.00000009e-18

   1. Initial program 61.1%

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

   3. Applied rewrites 61.5%

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 22.5%

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

   9. Taylor expanded in u0 around 0

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

   11. Applied rewrites 23.5%

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

   if 2.00000009e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.1%

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

   3. Applied rewrites 61.6%

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

   4. Taylor expanded in u0 around 0

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

   6. Applied rewrites 76.0%

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

   7. Taylor expanded in alphax around inf

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

   9. Applied rewrites 59.1%

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

   11. Applied rewrites 59.1%

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

Alternative 18: 66.8% 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

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

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 (* alphay alphay)) 2.00000009162741e-18)
  (* (/ (* alphax u0) cos2phi) alphax)
  (* (/ (* alphay u0) sin2phi) alphay)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.00000009162741e-18f) {
		tmp = ((alphax * u0) / cos2phi) * 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 / (alphay * alphay)) <= 2.00000009162741e-18) then
        tmp = ((alphax * u0) / cos2phi) * 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 (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.00000009162741e-18))
		tmp = Float32(Float32(Float32(alphax * u0) / cos2phi) * 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 / (alphay * alphay)) <= single(2.00000009162741e-18))
		tmp = ((alphax * u0) / cos2phi) * alphax;
	else
		tmp = ((alphay * u0) / sin2phi) * alphay;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.00000009e-18

   1. Initial program 61.1%

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

   3. Applied rewrites 61.5%

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 22.5%

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

   9. Taylor expanded in u0 around 0

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

   11. Applied rewrites 23.5%

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

   if 2.00000009e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.1%

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

   3. Applied rewrites 61.5%

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 49.5%

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

   9. Taylor expanded in u0 around 0

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

   11. Applied rewrites 59.1%

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

Alternative 19: 59.1% 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 61.1%

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

3. Applied rewrites 61.5%

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

5. Applied rewrites 61.5%

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

6. Taylor expanded in alphax around inf

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

8. Applied rewrites 49.5%

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

9. Taylor expanded in u0 around 0

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

11. Applied rewrites 59.1%

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

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(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)))))