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

Percentage Accurate: 60.7% → 98.3%

Time: 6.0s

Alternatives: 10

Speedup: 1.3×

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
 (/
  (- (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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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: 60.7% accurate, 1.0× speedup

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
 (/
  (- (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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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: 98.3% accurate, 0.9× speedup

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
 (/
  (- (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 60.7%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3%

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

3. Applied rewrites 98.3%

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

Alternative 2: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;u0 \leq 0.0029899999499320984:\\ \;\;\;\;\frac{u0 \cdot \left(1 + 0.5 \cdot u0\right)}{t\_0 + \frac{1}{\frac{alphay \cdot alphay}{sin2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{\mathsf{fma}\left(t\_0, alphay, \frac{sin2phi}{alphay}\right)} \cdot \left(-\log \left(1 - u0\right)\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ cos2phi (* alphax alphax))))
   (if (<= u0 0.0029899999499320984)
     (/
      (* u0 (+ 1.0 (* 0.5 u0)))
      (+ t_0 (/ 1.0 (/ (* alphay alphay) sin2phi))))
     (* (/ alphay (fma t_0 alphay (/ sin2phi alphay))) (- (log (- 1.0 u0)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float tmp;
	if (u0 <= 0.0029899999499320984f) {
		tmp = (u0 * (1.0f + (0.5f * u0))) / (t_0 + (1.0f / ((alphay * alphay) / sin2phi)));
	} else {
		tmp = (alphay / fmaf(t_0, alphay, (sin2phi / alphay))) * -logf((1.0f - u0));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0029899999499320984))
		tmp = Float32(Float32(u0 * Float32(Float32(1.0) + Float32(Float32(0.5) * u0))) / Float32(t_0 + Float32(Float32(1.0) / Float32(Float32(alphay * alphay) / sin2phi))));
	else
		tmp = Float32(Float32(alphay / fma(t_0, alphay, Float32(sin2phi / alphay))) * Float32(-log(Float32(Float32(1.0) - u0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00298999995

   1. Initial program 60.7%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3%

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

   3. Applied rewrites 98.3%

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

      1. lift-/.f32 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f32 N/A

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

      4. lower-unsound-/.f32 98.2%

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 87.5%

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

   8. Applied rewrites 87.5%

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

   if 0.00298999995 < u0

   1. Initial program 60.7%

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

      1. lift-/.f32 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

   3. Applied rewrites 61.0%

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

Alternative 3: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;u0 \leq 0.0029899999499320984:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{\mathsf{fma}\left(t\_0, alphay, \frac{sin2phi}{alphay}\right)} \cdot \left(-\log \left(1 - u0\right)\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ cos2phi (* alphax alphax))))
   (if (<= u0 0.0029899999499320984)
     (/ (* (fma 0.5 u0 1.0) u0) (+ (/ sin2phi (* alphay alphay)) t_0))
     (* (/ alphay (fma t_0 alphay (/ sin2phi alphay))) (- (log (- 1.0 u0)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float tmp;
	if (u0 <= 0.0029899999499320984f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) / ((sin2phi / (alphay * alphay)) + t_0);
	} else {
		tmp = (alphay / fmaf(t_0, alphay, (sin2phi / alphay))) * -logf((1.0f - u0));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0029899999499320984))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + t_0));
	else
		tmp = Float32(Float32(alphay / fma(t_0, alphay, Float32(sin2phi / alphay))) * Float32(-log(Float32(Float32(1.0) - u0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00298999995

   1. Initial program 60.7%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

   6. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 87.5%

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

   if 0.00298999995 < u0

   1. Initial program 60.7%

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

      1. lift-/.f32 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

   3. Applied rewrites 61.0%

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

Alternative 4: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;u0 \leq 0.0029899999499320984:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{-1}{t\_0}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))
   (if (<= u0 0.0029899999499320984)
     (/ (* (fma 0.5 u0 1.0) u0) t_0)
     (* (log (- 1.0 u0)) (/ -1.0 t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax));
	float tmp;
	if (u0 <= 0.0029899999499320984f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) / t_0;
	} else {
		tmp = logf((1.0f - u0)) * (-1.0f / t_0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax)))
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0029899999499320984))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / t_0);
	else
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(-1.0) / t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00298999995

   1. Initial program 60.7%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

   6. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 87.5%

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

   if 0.00298999995 < u0

   1. Initial program 60.7%

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

      1. lift-/.f32 N/A

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

      2. frac-2neg N/A

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

      3. mult-flip N/A

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

      4. lift-neg.f32 N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f32 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lift-+.f32 N/A

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

      10. add-flip N/A

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

      11. sub-negate N/A

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

      12. sub-negate-rev N/A

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

      13. add-flip N/A

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

      14. lift-+.f32 N/A

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

      15. lower-/.f32 60.6%

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

   3. Applied rewrites 60.6%

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

Alternative 5: 96.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ t_1 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;u0 \leq 0.0029899999499320984:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{t\_1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t\_0 + t\_1}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ cos2phi (* alphax alphax)))
        (t_1 (/ sin2phi (* alphay alphay))))
   (if (<= u0 0.0029899999499320984)
     (/ (* (fma 0.5 u0 1.0) u0) (+ t_1 t_0))
     (/ (- (log (- 1.0 u0))) (+ t_0 t_1)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float t_1 = sin2phi / (alphay * alphay);
	float tmp;
	if (u0 <= 0.0029899999499320984f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) / (t_1 + t_0);
	} else {
		tmp = -logf((1.0f - u0)) / (t_0 + t_1);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	t_1 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0029899999499320984))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / Float32(t_1 + t_0));
	else
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(t_0 + t_1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00298999995

   1. Initial program 60.7%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

   6. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 87.5%

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

   if 0.00298999995 < u0

   1. Initial program 60.7%

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

Alternative 6: 92.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;sin2phi \leq 65:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{t\_0}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= sin2phi 65.0)
     (/ (* (fma 0.5 u0 1.0) u0) (+ t_0 (/ cos2phi (* alphax alphax))))
     (/ (- (log1p (- u0))) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (sin2phi <= 65.0f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = -log1pf(-u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(65.0))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 65

   1. Initial program 60.7%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \color{blue}{\left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. lower-+.f32 N/A

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. lower-*.f32 93.1%

         \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   5. Step-by-step derivation

   6. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   7. Taylor expanded in u0 around 0

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

   9. Applied rewrites 87.5%

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

   if 65 < sin2phi

   1. Initial program 60.7%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3%

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

   3. Applied rewrites 98.3%

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

      1. lift-+.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 98.3%

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 73.7%

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

Alternative 7: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 12000:\\ \;\;\;\;\frac{-\left(-u0\right)}{t\_0 + \frac{1}{\frac{alphax}{\frac{cos2phi}{alphax}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{t\_0}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 12000.0)
     (/ (- (- u0)) (+ t_0 (/ 1.0 (/ alphax (/ cos2phi alphax)))))
     (/ (- (log1p (- u0))) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 12000.0f) {
		tmp = -(-u0) / (t_0 + (1.0f / (alphax / (cos2phi / alphax))));
	} else {
		tmp = -log1pf(-u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(12000.0))
		tmp = Float32(Float32(-Float32(-u0)) / Float32(t_0 + Float32(Float32(1.0) / Float32(alphax / Float32(cos2phi / alphax)))));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 12000

   1. Initial program 60.7%

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

   2. Taylor expanded in u0 around 0

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

      1. lower-*.f32 76.0%

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

   4. Applied rewrites 76.0%

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

      1. lift-*.f32 N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f32 76.0%

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

      4. lift-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(lift-+.f32, \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

      5. lift-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite<=}\left(+-commutative, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite<=}\left(lift-+.f32, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

   6. Applied rewrites 76.0%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f32 N/A

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

      5. div-flip N/A

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

      6. lower-unsound-/.f32 N/A

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

      7. lower-unsound-/.f32 76.0%

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

   8. Applied rewrites 76.0%

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

   if 12000 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.7%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3%

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

   3. Applied rewrites 98.3%

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

      1. lift-+.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 98.3%

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 73.7%

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

Alternative 8: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 12000:\\ \;\;\;\;\frac{-\left(-u0\right)}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{t\_0}\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 12000.0)
     (/ (- (- u0)) (+ t_0 (/ cos2phi (* alphax alphax))))
     (/ (- (log1p (- u0))) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 12000.0f) {
		tmp = -(-u0) / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = -log1pf(-u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(12000.0))
		tmp = Float32(Float32(-Float32(-u0)) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 12000

   1. Initial program 60.7%

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

   2. Taylor expanded in u0 around 0

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

      1. lower-*.f32 76.0%

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

   4. Applied rewrites 76.0%

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

      1. lift-*.f32 N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f32 76.0%

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

      4. lift-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(lift-+.f32, \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

      5. lift-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite<=}\left(+-commutative, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite<=}\left(lift-+.f32, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

   6. Applied rewrites 76.0%

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

   if 12000 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.7%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3%

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

   3. Applied rewrites 98.3%

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

      1. lift-+.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. add-to-fraction N/A

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

      4. lower-/.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 98.3%

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 73.7%

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

Alternative 9: 76.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 = -(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

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

2. Taylor expanded in u0 around 0

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

   1. lower-*.f32 76.0%

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

4. Applied rewrites 76.0%

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

   1. lift-*.f32 N/A

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

   2. mul-1-neg N/A

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

   3. lift-neg.f32 76.0%

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

   4. lift-neg.f32 N/A

      \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite=>}\left(lift-+.f32, \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

   5. lift-neg.f32 N/A

      \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite<=}\left(+-commutative, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

   6. lift-neg.f32 N/A

      \[\leadsto \frac{-\left(-u0\right)}{\mathsf{Rewrite<=}\left(lift-+.f32, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

6. Applied rewrites 76.0%

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

Alternative 10: 21.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 = ((-1.0e0) * ((alphax * log((1.0e0 - u0))) / cos2phi)) * alphax
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3%

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

3. Applied rewrites 98.3%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. add-to-fraction N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 98.3%

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

5. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. add-to-fraction-rev N/A

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

   6. lift-/.f32 N/A

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

   7. +-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lift-/.f32 N/A

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

   12. add-to-fraction-rev N/A

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

   13. lift-fma.f32 N/A

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

   14. associate-/r/ N/A

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

   15. lower-*.f32 N/A

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

7. Applied rewrites 60.6%

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

8. Taylor expanded in alphax around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lower--.f32 21.9%

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

10. Applied rewrites 21.9%

    \[\leadsto \color{blue}{\left(-1 \cdot \frac{alphax \cdot \log \left(1 - u0\right)}{cos2phi}\right)} \cdot alphax \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025207 
(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)))))