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

Percentage Accurate: 60.9% → 97.8%

Time: 9.5s

Alternatives: 10

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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: 60.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 97.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;u0 \leq 0.035599999129772186:\\ \;\;\;\;\frac{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(\log \left(1 - u0 \cdot u0\right) - \mathsf{log1p}\left(u0\right)\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))
   (if (<= u0 0.035599999129772186)
     (/
      (*
       -1.0
       (*
        (- (* (- (* (- (* -0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0)
        u0))
      t_0)
     (/ (* -1.0 (- (log (- 1.0 (* u0 u0))) (log1p 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 (u0 <= 0.035599999129772186f) {
		tmp = (-1.0f * (((((((-0.25f * u0) - 0.3333333333333333f) * u0) - 0.5f) * u0) - 1.0f) * u0)) / t_0;
	} else {
		tmp = (-1.0f * (logf((1.0f - (u0 * u0))) - log1pf(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 (u0 <= Float32(0.035599999129772186))
		tmp = Float32(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333)) * u0) - Float32(0.5)) * u0) - Float32(1.0)) * u0)) / t_0);
	else
		tmp = Float32(Float32(Float32(-1.0) * Float32(log(Float32(Float32(1.0) - Float32(u0 * u0))) - log1p(u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.0355999991

   1. Initial program 54.3%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 98.1

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

   5. Applied rewrites 98.1%

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

   if 0.0355999991 < u0

   1. Initial program 93.6%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. lower-log.f32 N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. lower--.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-log1p.f32 94.4

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

   4. Applied rewrites 94.4%

      \[\leadsto \frac{-\color{blue}{\left(\log \left(1 - u0 \cdot u0\right) - \mathsf{log1p}\left(u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.035599999129772186:\\ \;\;\;\;\frac{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(\log \left(1 - u0 \cdot u0\right) - \mathsf{log1p}\left(u0\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;u0 \leq 0.03200000151991844:\\ \;\;\;\;\frac{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \log \left(1 - u0\right)}{\frac{cos2phi}{e^{\log alphax \cdot 2}} + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= u0 0.03200000151991844)
     (/
      (*
       -1.0
       (*
        (- (* (- (* (- (* -0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0)
        u0))
      (+ (/ cos2phi (* alphax alphax)) t_0))
     (/
      (* -1.0 (log (- 1.0 u0)))
      (+ (/ cos2phi (exp (* (log alphax) 2.0))) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (u0 <= 0.03200000151991844f) {
		tmp = (-1.0f * (((((((-0.25f * u0) - 0.3333333333333333f) * u0) - 0.5f) * u0) - 1.0f) * u0)) / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (-1.0f * logf((1.0f - u0))) / ((cos2phi / expf((logf(alphax) * 2.0f))) + t_0);
	}
	return tmp;
}

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
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (u0 <= 0.03200000151991844e0) then
        tmp = ((-1.0e0) * ((((((((-0.25e0) * u0) - 0.3333333333333333e0) * u0) - 0.5e0) * u0) - 1.0e0) * u0)) / ((cos2phi / (alphax * alphax)) + t_0)
    else
        tmp = ((-1.0e0) * log((1.0e0 - u0))) / ((cos2phi / exp((log(alphax) * 2.0e0))) + t_0)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (u0 <= Float32(0.03200000151991844))
		tmp = Float32(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333)) * u0) - Float32(0.5)) * u0) - Float32(1.0)) * u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(Float32(-1.0) * log(Float32(Float32(1.0) - u0))) / Float32(Float32(cos2phi / exp(Float32(log(alphax) * Float32(2.0)))) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (u0 <= single(0.03200000151991844))
		tmp = (single(-1.0) * (((((((single(-0.25) * u0) - single(0.3333333333333333)) * u0) - single(0.5)) * u0) - single(1.0)) * u0)) / ((cos2phi / (alphax * alphax)) + t_0);
	else
		tmp = (single(-1.0) * log((single(1.0) - u0))) / ((cos2phi / exp((log(alphax) * single(2.0)))) + t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.0320000015

   1. Initial program 54.0%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.0320000015 < u0

   1. Initial program 92.8%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-log.f32 93.0

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

   4. Applied rewrites 93.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.03200000151991844:\\ \;\;\;\;\frac{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \log \left(1 - u0\right)}{\frac{cos2phi}{e^{\log alphax \cdot 2}} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u0\right) \leq -0.04500000178813934:\\ \;\;\;\;\frac{\log \left(\frac{1}{1 - u0}\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (log (- 1.0 u0)) -0.04500000178813934)
   (/
    (log (/ 1.0 (- 1.0 u0)))
    (/
     (fma (/ sin2phi alphay) alphax (* alphay (/ cos2phi alphax)))
     (* alphay alphax)))
   (/
    (*
     -1.0
     (* (- (* (- (* (- (* -0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0) u0))
    (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (logf((1.0f - u0)) <= -0.04500000178813934f) {
		tmp = logf((1.0f / (1.0f - u0))) / (fmaf((sin2phi / alphay), alphax, (alphay * (cos2phi / alphax))) / (alphay * alphax));
	} else {
		tmp = (-1.0f * (((((((-0.25f * u0) - 0.3333333333333333f) * u0) - 0.5f) * u0) - 1.0f) * u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u0)) <= Float32(-0.04500000178813934))
		tmp = Float32(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u0))) / Float32(fma(Float32(sin2phi / alphay), alphax, Float32(alphay * Float32(cos2phi / alphax))) / Float32(alphay * alphax)));
	else
		tmp = Float32(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333)) * u0) - Float32(0.5)) * u0) - Float32(1.0)) * u0)) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0450000018

   1. Initial program 93.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 92.8

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

      8. lift-+.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-/.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-/.f32 N/A

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

      13. +-commutative N/A

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

      14. associate-/r* N/A

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

      15. associate-/r* N/A

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

      16. frac-add N/A

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

      17. lower-/.f32 N/A

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

      18. lower-fma.f32 N/A

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

      19. lower-/.f32 N/A

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

      20. lower-*.f32 N/A

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

      21. lower-/.f32 N/A

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

      22. lower-*.f32 92.4

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

   4. Applied rewrites 92.4%

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

   if -0.0450000018 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 54.9%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \frac{-\color{blue}{\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u0\right) \leq -0.04500000178813934:\\ \;\;\;\;\frac{\log \left(\frac{1}{1 - u0}\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u0\right) \leq -0.04500000178813934:\\ \;\;\;\;\frac{\log \left(\frac{1}{1 - u0}\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{\left(alphay \cdot alphay\right) \cdot cos2phi}{alphax \cdot alphax}}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (log (- 1.0 u0)) -0.04500000178813934)
   (/
    (log (/ 1.0 (- 1.0 u0)))
    (/
     (fma (/ sin2phi alphay) alphax (* alphay (/ cos2phi alphax)))
     (* alphay alphax)))
   (*
    -1.0
    (/
     (* (- (* (- (* (- (* -0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0) u0)
     (+
      (/ sin2phi (* alphay alphay))
      (/
       (/ (* (* alphay alphay) cos2phi) (* alphax alphax))
       (* alphay alphay)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (logf((1.0f - u0)) <= -0.04500000178813934f) {
		tmp = logf((1.0f / (1.0f - u0))) / (fmaf((sin2phi / alphay), alphax, (alphay * (cos2phi / alphax))) / (alphay * alphax));
	} else {
		tmp = -1.0f * ((((((((-0.25f * u0) - 0.3333333333333333f) * u0) - 0.5f) * u0) - 1.0f) * u0) / ((sin2phi / (alphay * alphay)) + ((((alphay * alphay) * cos2phi) / (alphax * alphax)) / (alphay * alphay))));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u0)) <= Float32(-0.04500000178813934))
		tmp = Float32(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u0))) / Float32(fma(Float32(sin2phi / alphay), alphax, Float32(alphay * Float32(cos2phi / alphax))) / Float32(alphay * alphax)));
	else
		tmp = Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333)) * u0) - Float32(0.5)) * u0) - Float32(1.0)) * u0) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(Float32(Float32(alphay * alphay) * cos2phi) / Float32(alphax * alphax)) / Float32(alphay * alphay)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0450000018

   1. Initial program 93.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 92.8

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

      8. lift-+.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-/.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-/.f32 N/A

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

      13. +-commutative N/A

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

      14. associate-/r* N/A

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

      15. associate-/r* N/A

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

      16. frac-add N/A

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

      17. lower-/.f32 N/A

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

      18. lower-fma.f32 N/A

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

      19. lower-/.f32 N/A

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

      20. lower-*.f32 N/A

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

      21. lower-/.f32 N/A

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

      22. lower-*.f32 92.4

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

   4. Applied rewrites 92.4%

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

   if -0.0450000018 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 54.9%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in alphay around 0

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

      1. pow2 N/A

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

      2. rem-exp-log N/A

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

      3. div-add N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. pow2 N/A

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

      12. lift-*.f32 N/A

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

      13. pow2 N/A

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

      14. lift-*.f32 N/A

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

      15. pow2 N/A

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

      16. lift-*.f32 98.0

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

   8. Applied rewrites 98.0%

      \[\leadsto \frac{-\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{\left(alphay \cdot alphay\right) \cdot cos2phi}{alphax \cdot alphax}}{alphay \cdot alphay}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u0\right) \leq -0.04500000178813934:\\ \;\;\;\;\frac{\log \left(\frac{1}{1 - u0}\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{\left(alphay \cdot alphay\right) \cdot cos2phi}{alphax \cdot alphax}}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphay \cdot alphax}\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;u0 \leq 0.041999999433755875:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_1, \frac{0.25 \cdot u0}{t\_0}\right), u0, \frac{0.5}{t\_0}\right), u0, t\_1\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{1 - u0}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0
         (/
          (fma (/ sin2phi alphay) alphax (* alphay (/ cos2phi alphax)))
          (* alphay alphax)))
        (t_1 (/ 1.0 t_0)))
   (if (<= u0 0.041999999433755875)
     (*
      (fma
       (fma (fma 0.3333333333333333 t_1 (/ (* 0.25 u0) t_0)) u0 (/ 0.5 t_0))
       u0
       t_1)
      u0)
     (/ (log (/ 1.0 (- 1.0 u0))) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = fmaf((sin2phi / alphay), alphax, (alphay * (cos2phi / alphax))) / (alphay * alphax);
	float t_1 = 1.0f / t_0;
	float tmp;
	if (u0 <= 0.041999999433755875f) {
		tmp = fmaf(fmaf(fmaf(0.3333333333333333f, t_1, ((0.25f * u0) / t_0)), u0, (0.5f / t_0)), u0, t_1) * u0;
	} else {
		tmp = logf((1.0f / (1.0f - u0))) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(fma(Float32(sin2phi / alphay), alphax, Float32(alphay * Float32(cos2phi / alphax))) / Float32(alphay * alphax))
	t_1 = Float32(Float32(1.0) / t_0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.041999999433755875))
		tmp = Float32(fma(fma(fma(Float32(0.3333333333333333), t_1, Float32(Float32(Float32(0.25) * u0) / t_0)), u0, Float32(Float32(0.5) / t_0)), u0, t_1) * u0);
	else
		tmp = Float32(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.0419999994

   1. Initial program 54.9%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 97.8%

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

   if 0.0419999994 < u0

   1. Initial program 93.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 92.8

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

      8. lift-+.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-/.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-/.f32 N/A

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

      13. +-commutative N/A

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

      14. associate-/r* N/A

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

      15. associate-/r* N/A

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

      16. frac-add N/A

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

      17. lower-/.f32 N/A

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

      18. lower-fma.f32 N/A

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

      19. lower-/.f32 N/A

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

      20. lower-*.f32 N/A

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

      21. lower-/.f32 N/A

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

      22. lower-*.f32 92.4

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

   4. Applied rewrites 92.4%

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

Alternative 6: 92.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphay \cdot alphax}\\ t_1 := \frac{1}{t\_0}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_1, \frac{0.25 \cdot u0}{t\_0}\right), u0, \frac{0.5}{t\_0}\right), u0, t\_1\right) \cdot u0 \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0
         (/
          (fma (/ sin2phi alphay) alphax (* alphay (/ cos2phi alphax)))
          (* alphay alphax)))
        (t_1 (/ 1.0 t_0)))
   (*
    (fma
     (fma (fma 0.3333333333333333 t_1 (/ (* 0.25 u0) t_0)) u0 (/ 0.5 t_0))
     u0
     t_1)
    u0)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = fmaf((sin2phi / alphay), alphax, (alphay * (cos2phi / alphax))) / (alphay * alphax);
	float t_1 = 1.0f / t_0;
	return fmaf(fmaf(fmaf(0.3333333333333333f, t_1, ((0.25f * u0) / t_0)), u0, (0.5f / t_0)), u0, t_1) * u0;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(fma(Float32(sin2phi / alphay), alphax, Float32(alphay * Float32(cos2phi / alphax))) / Float32(alphay * alphax))
	t_1 = Float32(Float32(1.0) / t_0)
	return Float32(fma(fma(fma(Float32(0.3333333333333333), t_1, Float32(Float32(Float32(0.25) * u0) / t_0)), u0, Float32(Float32(0.5) / t_0)), u0, t_1) * u0)
end

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 92.7%

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

Alternative 7: 71.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{cos2phi \cdot cos2phi}{sin2phi \cdot sin2phi}\\ t_1 := \log \left(1 - u0\right)\\ t_2 := {alphay}^{4} \cdot cos2phi\\ t_3 := \frac{t\_2}{\left(alphax \cdot alphax\right) \cdot sin2phi}\\ \mathbf{if}\;t\_1 \leq -0.02500000037252903:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\left(\left(cos2phi \cdot cos2phi\right) \cdot t\_1\right) \cdot {\left(alphay \cdot alphay\right)}^{3}}{{\left(alphax \cdot alphax\right)}^{2} \cdot sin2phi}, -1, \frac{\left({\left(alphay \cdot alphay\right)}^{2} \cdot cos2phi\right) \cdot t\_1}{alphax \cdot alphax}\right)}{sin2phi}, -1, \left(alphay \cdot alphay\right) \cdot t\_1\right)}{sin2phi} \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{t\_2}{alphax \cdot alphax}, \frac{{alphay}^{6} \cdot \left(cos2phi \cdot cos2phi\right)}{{alphax}^{4} \cdot sin2phi}\right)}{sin2phi}, \mathsf{fma}\left(-1, alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(-1, -0.5 \cdot t\_3 + 0.5 \cdot t\_0, \mathsf{fma}\left(-0.5, alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(-1, -0.3333333333333333 \cdot t\_3 + 0.3333333333333333 \cdot t\_0, \mathsf{fma}\left(-0.3333333333333333, alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(-1, -0.25 \cdot t\_3 + 0.25 \cdot t\_0, -0.25 \cdot \left(alphay \cdot alphay\right)\right)\right)\right)\right)\right)\right)\right)}{sin2phi} \cdot -1\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0
         (*
          (/ (pow alphay 6.0) (pow alphax 4.0))
          (/ (* cos2phi cos2phi) (* sin2phi sin2phi))))
        (t_1 (log (- 1.0 u0)))
        (t_2 (* (pow alphay 4.0) cos2phi))
        (t_3 (/ t_2 (* (* alphax alphax) sin2phi))))
   (if (<= t_1 -0.02500000037252903)
     (*
      (/
       (fma
        (/
         (fma
          (/
           (* (* (* cos2phi cos2phi) t_1) (pow (* alphay alphay) 3.0))
           (* (pow (* alphax alphax) 2.0) sin2phi))
          -1.0
          (/
           (* (* (pow (* alphay alphay) 2.0) cos2phi) t_1)
           (* alphax alphax)))
         sin2phi)
        -1.0
        (* (* alphay alphay) t_1))
       sin2phi)
      -1.0)
     (*
      (/
       (*
        u0
        (fma
         -1.0
         (/
          (fma
           -1.0
           (/ t_2 (* alphax alphax))
           (/
            (* (pow alphay 6.0) (* cos2phi cos2phi))
            (* (pow alphax 4.0) sin2phi)))
          sin2phi)
         (fma
          -1.0
          (* alphay alphay)
          (*
           u0
           (fma
            -1.0
            (+ (* -0.5 t_3) (* 0.5 t_0))
            (fma
             -0.5
             (* alphay alphay)
             (*
              u0
              (fma
               -1.0
               (+ (* -0.3333333333333333 t_3) (* 0.3333333333333333 t_0))
               (fma
                -0.3333333333333333
                (* alphay alphay)
                (*
                 u0
                 (fma
                  -1.0
                  (+ (* -0.25 t_3) (* 0.25 t_0))
                  (* -0.25 (* alphay alphay)))))))))))))
       sin2phi)
      -1.0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (powf(alphay, 6.0f) / powf(alphax, 4.0f)) * ((cos2phi * cos2phi) / (sin2phi * sin2phi));
	float t_1 = logf((1.0f - u0));
	float t_2 = powf(alphay, 4.0f) * cos2phi;
	float t_3 = t_2 / ((alphax * alphax) * sin2phi);
	float tmp;
	if (t_1 <= -0.02500000037252903f) {
		tmp = (fmaf((fmaf(((((cos2phi * cos2phi) * t_1) * powf((alphay * alphay), 3.0f)) / (powf((alphax * alphax), 2.0f) * sin2phi)), -1.0f, (((powf((alphay * alphay), 2.0f) * cos2phi) * t_1) / (alphax * alphax))) / sin2phi), -1.0f, ((alphay * alphay) * t_1)) / sin2phi) * -1.0f;
	} else {
		tmp = ((u0 * fmaf(-1.0f, (fmaf(-1.0f, (t_2 / (alphax * alphax)), ((powf(alphay, 6.0f) * (cos2phi * cos2phi)) / (powf(alphax, 4.0f) * sin2phi))) / sin2phi), fmaf(-1.0f, (alphay * alphay), (u0 * fmaf(-1.0f, ((-0.5f * t_3) + (0.5f * t_0)), fmaf(-0.5f, (alphay * alphay), (u0 * fmaf(-1.0f, ((-0.3333333333333333f * t_3) + (0.3333333333333333f * t_0)), fmaf(-0.3333333333333333f, (alphay * alphay), (u0 * fmaf(-1.0f, ((-0.25f * t_3) + (0.25f * t_0)), (-0.25f * (alphay * alphay))))))))))))) / sin2phi) * -1.0f;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32((alphay ^ Float32(6.0)) / (alphax ^ Float32(4.0))) * Float32(Float32(cos2phi * cos2phi) / Float32(sin2phi * sin2phi)))
	t_1 = log(Float32(Float32(1.0) - u0))
	t_2 = Float32((alphay ^ Float32(4.0)) * cos2phi)
	t_3 = Float32(t_2 / Float32(Float32(alphax * alphax) * sin2phi))
	tmp = Float32(0.0)
	if (t_1 <= Float32(-0.02500000037252903))
		tmp = Float32(Float32(fma(Float32(fma(Float32(Float32(Float32(Float32(cos2phi * cos2phi) * t_1) * (Float32(alphay * alphay) ^ Float32(3.0))) / Float32((Float32(alphax * alphax) ^ Float32(2.0)) * sin2phi)), Float32(-1.0), Float32(Float32(Float32((Float32(alphay * alphay) ^ Float32(2.0)) * cos2phi) * t_1) / Float32(alphax * alphax))) / sin2phi), Float32(-1.0), Float32(Float32(alphay * alphay) * t_1)) / sin2phi) * Float32(-1.0));
	else
		tmp = Float32(Float32(Float32(u0 * fma(Float32(-1.0), Float32(fma(Float32(-1.0), Float32(t_2 / Float32(alphax * alphax)), Float32(Float32((alphay ^ Float32(6.0)) * Float32(cos2phi * cos2phi)) / Float32((alphax ^ Float32(4.0)) * sin2phi))) / sin2phi), fma(Float32(-1.0), Float32(alphay * alphay), Float32(u0 * fma(Float32(-1.0), Float32(Float32(Float32(-0.5) * t_3) + Float32(Float32(0.5) * t_0)), fma(Float32(-0.5), Float32(alphay * alphay), Float32(u0 * fma(Float32(-1.0), Float32(Float32(Float32(-0.3333333333333333) * t_3) + Float32(Float32(0.3333333333333333) * t_0)), fma(Float32(-0.3333333333333333), Float32(alphay * alphay), Float32(u0 * fma(Float32(-1.0), Float32(Float32(Float32(-0.25) * t_3) + Float32(Float32(0.25) * t_0)), Float32(Float32(-0.25) * Float32(alphay * alphay))))))))))))) / sin2phi) * Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0250000004

   1. Initial program 92.8%

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

   3. Taylor expanded in sin2phi around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{{alphay}^{6} \cdot \left({cos2phi}^{2} \cdot \log \left(1 - u0\right)\right)}{{alphax}^{4} \cdot sin2phi} + \frac{{alphay}^{4} \cdot \left(cos2phi \cdot \log \left(1 - u0\right)\right)}{{alphax}^{2}}}{sin2phi} + {alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]

   5. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\left(\left(cos2phi \cdot cos2phi\right) \cdot \log \left(1 - u0\right)\right) \cdot {\left(alphay \cdot alphay\right)}^{3}}{{\left(alphax \cdot alphax\right)}^{2} \cdot sin2phi}, -1, \frac{\left({\left(alphay \cdot alphay\right)}^{2} \cdot cos2phi\right) \cdot \log \left(1 - u0\right)}{alphax \cdot alphax}\right)}{sin2phi}, -1, \left(alphay \cdot alphay\right) \cdot \log \left(1 - u0\right)\right)}{sin2phi} \cdot -1} \]

   if -0.0250000004 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 53.3%

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

   3. Taylor expanded in sin2phi around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{{alphay}^{6} \cdot \left({cos2phi}^{2} \cdot \log \left(1 - u0\right)\right)}{{alphax}^{4} \cdot sin2phi} + \frac{{alphay}^{4} \cdot \left(cos2phi \cdot \log \left(1 - u0\right)\right)}{{alphax}^{2}}}{sin2phi} + {alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]

   5. Applied rewrites 41.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\left(\left(cos2phi \cdot cos2phi\right) \cdot \log \left(1 - u0\right)\right) \cdot {\left(alphay \cdot alphay\right)}^{3}}{{\left(alphax \cdot alphax\right)}^{2} \cdot sin2phi}, -1, \frac{\left({\left(alphay \cdot alphay\right)}^{2} \cdot cos2phi\right) \cdot \log \left(1 - u0\right)}{alphax \cdot alphax}\right)}{sin2phi}, -1, \left(alphay \cdot alphay\right) \cdot \log \left(1 - u0\right)\right)}{sin2phi} \cdot -1} \]

   6. Taylor expanded in u0 around 0

      \[\leadsto \frac{u0 \cdot \left(-1 \cdot \frac{-1 \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2}} + \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot sin2phi}}{sin2phi} + \left(-1 \cdot {alphay}^{2} + u0 \cdot \left(-1 \cdot \left(\frac{-1}{2} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi} + \frac{1}{2} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{2}}\right) + \left(\frac{-1}{2} \cdot {alphay}^{2} + u0 \cdot \left(-1 \cdot \left(\frac{-1}{3} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi} + \frac{1}{3} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{2}}\right) + \left(\frac{-1}{3} \cdot {alphay}^{2} + u0 \cdot \left(-1 \cdot \left(\frac{-1}{4} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi} + \frac{1}{4} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{2}}\right) + \frac{-1}{4} \cdot {alphay}^{2}\right)\right)\right)\right)\right)\right)\right)}{sin2phi} \cdot -1 \]

   8. Applied rewrites 68.8%

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{{alphay}^{4} \cdot cos2phi}{alphax \cdot alphax}, \frac{{alphay}^{6} \cdot \left(cos2phi \cdot cos2phi\right)}{{alphax}^{4} \cdot sin2phi}\right)}{sin2phi}, \mathsf{fma}\left(-1, alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(-1, -0.5 \cdot \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi} + 0.5 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{cos2phi \cdot cos2phi}{sin2phi \cdot sin2phi}\right), \mathsf{fma}\left(-0.5, alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(-1, -0.3333333333333333 \cdot \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi} + 0.3333333333333333 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{cos2phi \cdot cos2phi}{sin2phi \cdot sin2phi}\right), \mathsf{fma}\left(-0.3333333333333333, alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(-1, -0.25 \cdot \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi} + 0.25 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{cos2phi \cdot cos2phi}{sin2phi \cdot sin2phi}\right), -0.25 \cdot \left(alphay \cdot alphay\right)\right)\right)\right)\right)\right)\right)\right)}{sin2phi} \cdot -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi}\\ t_1 := \frac{alphay \cdot alphay}{sin2phi}\\ t_2 := \left(u0 \cdot u0\right) \cdot u0\\ t_3 := \frac{{alphay}^{6}}{{alphax}^{4}}\\ t_4 := t\_3 \cdot \left(\frac{cos2phi}{sin2phi} \cdot \frac{cos2phi}{sin2phi}\right)\\ u0 \cdot \left(t\_2 \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(-1, \frac{-0.25 \cdot t\_0 + 0.25 \cdot t\_4}{sin2phi}, -0.25 \cdot t\_1\right), \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{{alphay}^{4}}{alphax} \cdot \frac{cos2phi}{alphax}, t\_3 \cdot \frac{cos2phi \cdot cos2phi}{sin2phi}\right)}{sin2phi}, -1 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi \cdot t\_2}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-0.5, t\_0, 0.5 \cdot t\_4\right)}{sin2phi}, -0.5 \cdot t\_1\right)}{u0 \cdot u0}, \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-0.3333333333333333, t\_0, 0.3333333333333333 \cdot t\_4\right)}{sin2phi}, -0.3333333333333333 \cdot t\_1\right)}{-1 \cdot u0}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ (* (pow alphay 4.0) cos2phi) (* (* alphax alphax) sin2phi)))
        (t_1 (/ (* alphay alphay) sin2phi))
        (t_2 (* (* u0 u0) u0))
        (t_3 (/ (pow alphay 6.0) (pow alphax 4.0)))
        (t_4 (* t_3 (* (/ cos2phi sin2phi) (/ cos2phi sin2phi)))))
   (*
    u0
    (*
     t_2
     (fma
      -1.0
      (fma -1.0 (/ (+ (* -0.25 t_0) (* 0.25 t_4)) sin2phi) (* -0.25 t_1))
      (fma
       -1.0
       (/
        (fma
         -1.0
         (/
          (fma
           -1.0
           (* (/ (pow alphay 4.0) alphax) (/ cos2phi alphax))
           (* t_3 (/ (* cos2phi cos2phi) sin2phi)))
          sin2phi)
         (* -1.0 (* alphay alphay)))
        (* sin2phi t_2))
       (fma
        -1.0
        (/
         (fma -1.0 (/ (fma -0.5 t_0 (* 0.5 t_4)) sin2phi) (* -0.5 t_1))
         (* u0 u0))
        (/
         (fma
          -1.0
          (/ (fma -0.3333333333333333 t_0 (* 0.3333333333333333 t_4)) sin2phi)
          (* -0.3333333333333333 t_1))
         (* -1.0 u0)))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (powf(alphay, 4.0f) * cos2phi) / ((alphax * alphax) * sin2phi);
	float t_1 = (alphay * alphay) / sin2phi;
	float t_2 = (u0 * u0) * u0;
	float t_3 = powf(alphay, 6.0f) / powf(alphax, 4.0f);
	float t_4 = t_3 * ((cos2phi / sin2phi) * (cos2phi / sin2phi));
	return u0 * (t_2 * fmaf(-1.0f, fmaf(-1.0f, (((-0.25f * t_0) + (0.25f * t_4)) / sin2phi), (-0.25f * t_1)), fmaf(-1.0f, (fmaf(-1.0f, (fmaf(-1.0f, ((powf(alphay, 4.0f) / alphax) * (cos2phi / alphax)), (t_3 * ((cos2phi * cos2phi) / sin2phi))) / sin2phi), (-1.0f * (alphay * alphay))) / (sin2phi * t_2)), fmaf(-1.0f, (fmaf(-1.0f, (fmaf(-0.5f, t_0, (0.5f * t_4)) / sin2phi), (-0.5f * t_1)) / (u0 * u0)), (fmaf(-1.0f, (fmaf(-0.3333333333333333f, t_0, (0.3333333333333333f * t_4)) / sin2phi), (-0.3333333333333333f * t_1)) / (-1.0f * u0))))));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32((alphay ^ Float32(4.0)) * cos2phi) / Float32(Float32(alphax * alphax) * sin2phi))
	t_1 = Float32(Float32(alphay * alphay) / sin2phi)
	t_2 = Float32(Float32(u0 * u0) * u0)
	t_3 = Float32((alphay ^ Float32(6.0)) / (alphax ^ Float32(4.0)))
	t_4 = Float32(t_3 * Float32(Float32(cos2phi / sin2phi) * Float32(cos2phi / sin2phi)))
	return Float32(u0 * Float32(t_2 * fma(Float32(-1.0), fma(Float32(-1.0), Float32(Float32(Float32(Float32(-0.25) * t_0) + Float32(Float32(0.25) * t_4)) / sin2phi), Float32(Float32(-0.25) * t_1)), fma(Float32(-1.0), Float32(fma(Float32(-1.0), Float32(fma(Float32(-1.0), Float32(Float32((alphay ^ Float32(4.0)) / alphax) * Float32(cos2phi / alphax)), Float32(t_3 * Float32(Float32(cos2phi * cos2phi) / sin2phi))) / sin2phi), Float32(Float32(-1.0) * Float32(alphay * alphay))) / Float32(sin2phi * t_2)), fma(Float32(-1.0), Float32(fma(Float32(-1.0), Float32(fma(Float32(-0.5), t_0, Float32(Float32(0.5) * t_4)) / sin2phi), Float32(Float32(-0.5) * t_1)) / Float32(u0 * u0)), Float32(fma(Float32(-1.0), Float32(fma(Float32(-0.3333333333333333), t_0, Float32(Float32(0.3333333333333333) * t_4)) / sin2phi), Float32(Float32(-0.3333333333333333) * t_1)) / Float32(Float32(-1.0) * u0)))))))
end

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in sin2phi around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{{alphay}^{6} \cdot \left({cos2phi}^{2} \cdot \log \left(1 - u0\right)\right)}{{alphax}^{4} \cdot sin2phi} + \frac{{alphay}^{4} \cdot \left(cos2phi \cdot \log \left(1 - u0\right)\right)}{{alphax}^{2}}}{sin2phi} + {alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]

5. Applied rewrites 47.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\left(\left(cos2phi \cdot cos2phi\right) \cdot \log \left(1 - u0\right)\right) \cdot {\left(alphay \cdot alphay\right)}^{3}}{{\left(alphax \cdot alphax\right)}^{2} \cdot sin2phi}, -1, \frac{\left({\left(alphay \cdot alphay\right)}^{2} \cdot cos2phi\right) \cdot \log \left(1 - u0\right)}{alphax \cdot alphax}\right)}{sin2phi}, -1, \left(alphay \cdot alphay\right) \cdot \log \left(1 - u0\right)\right)}{sin2phi} \cdot -1} \]

6. Taylor expanded in u0 around 0

   \[\leadsto u0 \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2}} + \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot sin2phi}}{sin2phi} + -1 \cdot {alphay}^{2}}{sin2phi} + u0 \cdot \left(-1 \cdot \left(-1 \cdot \frac{\frac{-1}{2} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi} + \frac{1}{2} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{2}}}{sin2phi} + \frac{-1}{2} \cdot \frac{{alphay}^{2}}{sin2phi}\right) + u0 \cdot \left(-1 \cdot \left(u0 \cdot \left(-1 \cdot \frac{\frac{-1}{4} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi} + \frac{1}{4} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{2}}}{sin2phi} + \frac{-1}{4} \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) + -1 \cdot \left(-1 \cdot \frac{\frac{-1}{3} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi} + \frac{1}{3} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{2}}}{sin2phi} + \frac{-1}{3} \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right)\right)\right)} \]

8. Applied rewrites 65.3%

   \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{{alphay}^{4} \cdot cos2phi}{alphax \cdot alphax}, \frac{{alphay}^{6} \cdot \left(cos2phi \cdot cos2phi\right)}{{alphax}^{4} \cdot sin2phi}\right)}{sin2phi}, -1 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}, u0 \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(-1, \frac{-0.5 \cdot \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi} + 0.5 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{cos2phi \cdot cos2phi}{sin2phi \cdot sin2phi}\right)}{sin2phi}, -0.5 \cdot \frac{alphay \cdot alphay}{sin2phi}\right), u0 \cdot \mathsf{fma}\left(-1, u0 \cdot \mathsf{fma}\left(-1, \frac{-0.25 \cdot \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi} + 0.25 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{cos2phi \cdot cos2phi}{sin2phi \cdot sin2phi}\right)}{sin2phi}, -0.25 \cdot \frac{alphay \cdot alphay}{sin2phi}\right), -1 \cdot \mathsf{fma}\left(-1, \frac{-0.3333333333333333 \cdot \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi} + 0.3333333333333333 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{cos2phi \cdot cos2phi}{sin2phi \cdot sin2phi}\right)}{sin2phi}, -0.3333333333333333 \cdot \frac{alphay \cdot alphay}{sin2phi}\right)\right)\right)\right)} \]

9. Taylor expanded in u0 around inf

   \[\leadsto u0 \cdot \left({u0}^{3} \cdot \left(-1 \cdot \left(-1 \cdot \frac{\frac{-1}{4} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi} + \frac{1}{4} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{2}}}{sin2phi} + \frac{-1}{4} \cdot \frac{{alphay}^{2}}{sin2phi}\right) + \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2}} + \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot sin2phi}}{sin2phi} + -1 \cdot {alphay}^{2}}{sin2phi \cdot {u0}^{3}} + \left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{2} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi} + \frac{1}{2} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{2}}}{sin2phi} + \frac{-1}{2} \cdot \frac{{alphay}^{2}}{sin2phi}}{{u0}^{2}} + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{3} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi} + \frac{1}{3} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{2}}}{sin2phi} + \frac{-1}{3} \cdot \frac{{alphay}^{2}}{sin2phi}}{u0}\right)\right)}\right)\right) \]

11. Applied rewrites 66.0%

    \[\leadsto u0 \cdot \left(\left(\left(u0 \cdot u0\right) \cdot u0\right) \cdot \mathsf{fma}\left(-1, \color{blue}{\mathsf{fma}\left(-1, \frac{-0.25 \cdot \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi} + 0.25 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \left(\frac{cos2phi}{sin2phi} \cdot \frac{cos2phi}{sin2phi}\right)\right)}{sin2phi}, -0.25 \cdot \frac{alphay \cdot alphay}{sin2phi}\right)}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{{alphay}^{4}}{alphax} \cdot \frac{cos2phi}{alphax}, \frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{cos2phi \cdot cos2phi}{sin2phi}\right)}{sin2phi}, -1 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi \cdot \left(\left(u0 \cdot u0\right) \cdot u0\right)}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-0.5, \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi}, 0.5 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \left(\frac{cos2phi}{sin2phi} \cdot \frac{cos2phi}{sin2phi}\right)\right)\right)}{sin2phi}, -0.5 \cdot \frac{alphay \cdot alphay}{sin2phi}\right)}{u0 \cdot u0}, -1 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi}, 0.3333333333333333 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \left(\frac{cos2phi}{sin2phi} \cdot \frac{cos2phi}{sin2phi}\right)\right)\right)}{sin2phi}, -0.3333333333333333 \cdot \frac{alphay \cdot alphay}{sin2phi}\right)}{u0}\right)\right)\right)\right) \]

12. Final simplification 66.0%

    \[\leadsto u0 \cdot \left(\left(\left(u0 \cdot u0\right) \cdot u0\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(-1, \frac{-0.25 \cdot \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi} + 0.25 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \left(\frac{cos2phi}{sin2phi} \cdot \frac{cos2phi}{sin2phi}\right)\right)}{sin2phi}, -0.25 \cdot \frac{alphay \cdot alphay}{sin2phi}\right), \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{{alphay}^{4}}{alphax} \cdot \frac{cos2phi}{alphax}, \frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{cos2phi \cdot cos2phi}{sin2phi}\right)}{sin2phi}, -1 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi \cdot \left(\left(u0 \cdot u0\right) \cdot u0\right)}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-0.5, \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi}, 0.5 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \left(\frac{cos2phi}{sin2phi} \cdot \frac{cos2phi}{sin2phi}\right)\right)\right)}{sin2phi}, -0.5 \cdot \frac{alphay \cdot alphay}{sin2phi}\right)}{u0 \cdot u0}, \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi}, 0.3333333333333333 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \left(\frac{cos2phi}{sin2phi} \cdot \frac{cos2phi}{sin2phi}\right)\right)\right)}{sin2phi}, -0.3333333333333333 \cdot \frac{alphay \cdot alphay}{sin2phi}\right)}{-1 \cdot u0}\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 9: 62.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(alphax \cdot sin2phi\right)}^{2}\\ t_1 := \frac{1}{1 - u0}\\ t_2 := \log t\_1\\ t_3 := \left(sin2phi \cdot sin2phi\right) \cdot sin2phi\\ t_4 := {alphax}^{4} \cdot t\_3\\ t_5 := \frac{cos2phi}{t\_0}\\ t_6 := \frac{cos2phi \cdot cos2phi}{t\_4}\\ \mathbf{if}\;\log \left(1 - u0\right) \leq -0.04500000178813934:\\ \;\;\;\;\mathsf{fma}\left(cos2phi, \frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{\log \left({t\_1}^{cos2phi}\right)}{t\_3} - \frac{{alphay}^{4} \cdot t\_2}{t\_0}, \frac{\left(alphay \cdot alphay\right) \cdot t\_2}{sin2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;{alphay}^{6} \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.5 \cdot t\_6 + u0 \cdot \mathsf{fma}\left(0.25, \frac{\left(cos2phi \cdot cos2phi\right) \cdot u0}{t\_4}, 0.3333333333333333 \cdot t\_6\right), t\_6\right), \frac{\mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(-1, t\_5, u0 \cdot \mathsf{fma}\left(-0.5, t\_5, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, t\_5, -0.25 \cdot \left(\frac{cos2phi}{alphax \cdot alphax} \cdot \frac{u0}{sin2phi \cdot sin2phi}\right)\right)\right)\right), {alphay}^{4}, \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(0.25, \frac{u0}{sin2phi}, 0.3333333333333333 \cdot \frac{1}{sin2phi}\right), 0.5 \cdot \frac{1}{sin2phi}\right), \frac{1}{sin2phi}\right)\right)\right)}{\left(alphay \cdot alphay\right) \cdot {alphay}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (pow (* alphax sin2phi) 2.0))
        (t_1 (/ 1.0 (- 1.0 u0)))
        (t_2 (log t_1))
        (t_3 (* (* sin2phi sin2phi) sin2phi))
        (t_4 (* (pow alphax 4.0) t_3))
        (t_5 (/ cos2phi t_0))
        (t_6 (/ (* cos2phi cos2phi) t_4)))
   (if (<= (log (- 1.0 u0)) -0.04500000178813934)
     (fma
      cos2phi
      (-
       (*
        (/ (pow alphay 6.0) (pow alphax 4.0))
        (/ (log (pow t_1 cos2phi)) t_3))
       (/ (* (pow alphay 4.0) t_2) t_0))
      (/ (* (* alphay alphay) t_2) sin2phi))
     (*
      (pow alphay 6.0)
      (fma
       u0
       (fma
        u0
        (+
         (* 0.5 t_6)
         (*
          u0
          (fma
           0.25
           (/ (* (* cos2phi cos2phi) u0) t_4)
           (* 0.3333333333333333 t_6))))
        t_6)
       (/
        (fma
         (*
          u0
          (fma
           -1.0
           t_5
           (*
            u0
            (fma
             -0.5
             t_5
             (*
              u0
              (fma
               -0.3333333333333333
               t_5
               (*
                -0.25
                (*
                 (/ cos2phi (* alphax alphax))
                 (/ u0 (* sin2phi sin2phi))))))))))
         (pow alphay 4.0)
         (*
          (* alphay alphay)
          (*
           u0
           (fma
            u0
            (fma
             u0
             (fma 0.25 (/ u0 sin2phi) (* 0.3333333333333333 (/ 1.0 sin2phi)))
             (* 0.5 (/ 1.0 sin2phi)))
            (/ 1.0 sin2phi)))))
        (* (* alphay alphay) (pow alphay 4.0))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = powf((alphax * sin2phi), 2.0f);
	float t_1 = 1.0f / (1.0f - u0);
	float t_2 = logf(t_1);
	float t_3 = (sin2phi * sin2phi) * sin2phi;
	float t_4 = powf(alphax, 4.0f) * t_3;
	float t_5 = cos2phi / t_0;
	float t_6 = (cos2phi * cos2phi) / t_4;
	float tmp;
	if (logf((1.0f - u0)) <= -0.04500000178813934f) {
		tmp = fmaf(cos2phi, (((powf(alphay, 6.0f) / powf(alphax, 4.0f)) * (logf(powf(t_1, cos2phi)) / t_3)) - ((powf(alphay, 4.0f) * t_2) / t_0)), (((alphay * alphay) * t_2) / sin2phi));
	} else {
		tmp = powf(alphay, 6.0f) * fmaf(u0, fmaf(u0, ((0.5f * t_6) + (u0 * fmaf(0.25f, (((cos2phi * cos2phi) * u0) / t_4), (0.3333333333333333f * t_6)))), t_6), (fmaf((u0 * fmaf(-1.0f, t_5, (u0 * fmaf(-0.5f, t_5, (u0 * fmaf(-0.3333333333333333f, t_5, (-0.25f * ((cos2phi / (alphax * alphax)) * (u0 / (sin2phi * sin2phi)))))))))), powf(alphay, 4.0f), ((alphay * alphay) * (u0 * fmaf(u0, fmaf(u0, fmaf(0.25f, (u0 / sin2phi), (0.3333333333333333f * (1.0f / sin2phi))), (0.5f * (1.0f / sin2phi))), (1.0f / sin2phi))))) / ((alphay * alphay) * powf(alphay, 4.0f))));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(alphax * sin2phi) ^ Float32(2.0)
	t_1 = Float32(Float32(1.0) / Float32(Float32(1.0) - u0))
	t_2 = log(t_1)
	t_3 = Float32(Float32(sin2phi * sin2phi) * sin2phi)
	t_4 = Float32((alphax ^ Float32(4.0)) * t_3)
	t_5 = Float32(cos2phi / t_0)
	t_6 = Float32(Float32(cos2phi * cos2phi) / t_4)
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u0)) <= Float32(-0.04500000178813934))
		tmp = fma(cos2phi, Float32(Float32(Float32((alphay ^ Float32(6.0)) / (alphax ^ Float32(4.0))) * Float32(log((t_1 ^ cos2phi)) / t_3)) - Float32(Float32((alphay ^ Float32(4.0)) * t_2) / t_0)), Float32(Float32(Float32(alphay * alphay) * t_2) / sin2phi));
	else
		tmp = Float32((alphay ^ Float32(6.0)) * fma(u0, fma(u0, Float32(Float32(Float32(0.5) * t_6) + Float32(u0 * fma(Float32(0.25), Float32(Float32(Float32(cos2phi * cos2phi) * u0) / t_4), Float32(Float32(0.3333333333333333) * t_6)))), t_6), Float32(fma(Float32(u0 * fma(Float32(-1.0), t_5, Float32(u0 * fma(Float32(-0.5), t_5, Float32(u0 * fma(Float32(-0.3333333333333333), t_5, Float32(Float32(-0.25) * Float32(Float32(cos2phi / Float32(alphax * alphax)) * Float32(u0 / Float32(sin2phi * sin2phi)))))))))), (alphay ^ Float32(4.0)), Float32(Float32(alphay * alphay) * Float32(u0 * fma(u0, fma(u0, fma(Float32(0.25), Float32(u0 / sin2phi), Float32(Float32(0.3333333333333333) * Float32(Float32(1.0) / sin2phi))), Float32(Float32(0.5) * Float32(Float32(1.0) / sin2phi))), Float32(Float32(1.0) / sin2phi))))) / Float32(Float32(alphay * alphay) * (alphay ^ Float32(4.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0450000018

   1. Initial program 93.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 92.8

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

      8. lift-+.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-/.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-/.f32 N/A

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

      13. +-commutative N/A

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

      14. associate-/r* N/A

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

      15. associate-/r* N/A

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

      16. frac-add N/A

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

      17. lower-/.f32 N/A

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

      18. lower-fma.f32 N/A

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

      19. lower-/.f32 N/A

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

      20. lower-*.f32 N/A

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

      21. lower-/.f32 N/A

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

      22. lower-*.f32 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{cos2phi \cdot \left(\frac{{alphay}^{6} \cdot \left(cos2phi \cdot \log \left(\frac{1}{1 - u0}\right)\right)}{{alphax}^{4} \cdot {sin2phi}^{3}} - \frac{{alphay}^{4} \cdot \log \left(\frac{1}{1 - u0}\right)}{{alphax}^{2} \cdot {sin2phi}^{2}}\right) + \frac{{alphay}^{2} \cdot \log \left(\frac{1}{1 - u0}\right)}{sin2phi}} \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(cos2phi, \color{blue}{\frac{{alphay}^{6} \cdot \left(cos2phi \cdot \log \left(\frac{1}{1 - u0}\right)\right)}{{alphax}^{4} \cdot {sin2phi}^{3}} - \frac{{alphay}^{4} \cdot \log \left(\frac{1}{1 - u0}\right)}{{alphax}^{2} \cdot {sin2phi}^{2}}}, \frac{{alphay}^{2} \cdot \log \left(\frac{1}{1 - u0}\right)}{sin2phi}\right) \]

   7. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(cos2phi, \frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{\log \left({\left(\frac{1}{1 - u0}\right)}^{cos2phi}\right)}{\left(sin2phi \cdot sin2phi\right) \cdot sin2phi} - \frac{{alphay}^{4} \cdot \log \left(\frac{1}{1 - u0}\right)}{{\left(alphax \cdot sin2phi\right)}^{2}}, \frac{\left(alphay \cdot alphay\right) \cdot \log \left(\frac{1}{1 - u0}\right)}{sin2phi}\right)} \]

   if -0.0450000018 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 54.9%

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

   3. Taylor expanded in sin2phi around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{{alphay}^{6} \cdot \left({cos2phi}^{2} \cdot \log \left(1 - u0\right)\right)}{{alphax}^{4} \cdot sin2phi} + \frac{{alphay}^{4} \cdot \left(cos2phi \cdot \log \left(1 - u0\right)\right)}{{alphax}^{2}}}{sin2phi} + {alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]

   5. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\left(\left(cos2phi \cdot cos2phi\right) \cdot \log \left(1 - u0\right)\right) \cdot {\left(alphay \cdot alphay\right)}^{3}}{{\left(alphax \cdot alphax\right)}^{2} \cdot sin2phi}, -1, \frac{\left({\left(alphay \cdot alphay\right)}^{2} \cdot cos2phi\right) \cdot \log \left(1 - u0\right)}{alphax \cdot alphax}\right)}{sin2phi}, -1, \left(alphay \cdot alphay\right) \cdot \log \left(1 - u0\right)\right)}{sin2phi} \cdot -1} \]

   6. Taylor expanded in u0 around 0

      \[\leadsto u0 \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2}} + \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot sin2phi}}{sin2phi} + -1 \cdot {alphay}^{2}}{sin2phi} + u0 \cdot \left(-1 \cdot \left(-1 \cdot \frac{\frac{-1}{2} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi} + \frac{1}{2} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{2}}}{sin2phi} + \frac{-1}{2} \cdot \frac{{alphay}^{2}}{sin2phi}\right) + u0 \cdot \left(-1 \cdot \left(u0 \cdot \left(-1 \cdot \frac{\frac{-1}{4} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi} + \frac{1}{4} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{2}}}{sin2phi} + \frac{-1}{4} \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) + -1 \cdot \left(-1 \cdot \frac{\frac{-1}{3} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi} + \frac{1}{3} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{2}}}{sin2phi} + \frac{-1}{3} \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right)\right)\right)} \]

   8. Applied rewrites 68.5%

      \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{{alphay}^{4} \cdot cos2phi}{alphax \cdot alphax}, \frac{{alphay}^{6} \cdot \left(cos2phi \cdot cos2phi\right)}{{alphax}^{4} \cdot sin2phi}\right)}{sin2phi}, -1 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}, u0 \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(-1, \frac{-0.5 \cdot \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi} + 0.5 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{cos2phi \cdot cos2phi}{sin2phi \cdot sin2phi}\right)}{sin2phi}, -0.5 \cdot \frac{alphay \cdot alphay}{sin2phi}\right), u0 \cdot \mathsf{fma}\left(-1, u0 \cdot \mathsf{fma}\left(-1, \frac{-0.25 \cdot \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi} + 0.25 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{cos2phi \cdot cos2phi}{sin2phi \cdot sin2phi}\right)}{sin2phi}, -0.25 \cdot \frac{alphay \cdot alphay}{sin2phi}\right), -1 \cdot \mathsf{fma}\left(-1, \frac{-0.3333333333333333 \cdot \frac{{alphay}^{4} \cdot cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi} + 0.3333333333333333 \cdot \left(\frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{cos2phi \cdot cos2phi}{sin2phi \cdot sin2phi}\right)}{sin2phi}, -0.3333333333333333 \cdot \frac{alphay \cdot alphay}{sin2phi}\right)\right)\right)\right)} \]

   9. Taylor expanded in alphay around inf

      \[\leadsto {alphay}^{6} \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot \frac{{cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{3}} + u0 \cdot \left(\frac{1}{4} \cdot \frac{{cos2phi}^{2} \cdot u0}{{alphax}^{4} \cdot {sin2phi}^{3}} + \frac{1}{3} \cdot \frac{{cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{3}}\right)\right) + \frac{{cos2phi}^{2}}{{alphax}^{4} \cdot {sin2phi}^{3}}\right) + \color{blue}{\left(\frac{u0 \cdot \left(-1 \cdot \frac{cos2phi}{{alphax}^{2} \cdot {sin2phi}^{2}} + u0 \cdot \left(\frac{-1}{2} \cdot \frac{cos2phi}{{alphax}^{2} \cdot {sin2phi}^{2}} + u0 \cdot \left(\frac{-1}{3} \cdot \frac{cos2phi}{{alphax}^{2} \cdot {sin2phi}^{2}} + \frac{-1}{4} \cdot \frac{cos2phi \cdot u0}{{alphax}^{2} \cdot {sin2phi}^{2}}\right)\right)\right)}{{alphay}^{2}} + \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{sin2phi} + \frac{1}{3} \cdot \frac{1}{sin2phi}\right) + \frac{1}{2} \cdot \frac{1}{sin2phi}\right) + \frac{1}{sin2phi}\right)}{{alphay}^{4}}\right)}\right) \]

   11. Applied rewrites 60.9%

       \[\leadsto {alphay}^{6} \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, 0.5 \cdot \frac{cos2phi \cdot cos2phi}{{alphax}^{4} \cdot \left(\left(sin2phi \cdot sin2phi\right) \cdot sin2phi\right)} + u0 \cdot \mathsf{fma}\left(0.25, \frac{\left(cos2phi \cdot cos2phi\right) \cdot u0}{{alphax}^{4} \cdot \left(\left(sin2phi \cdot sin2phi\right) \cdot sin2phi\right)}, 0.3333333333333333 \cdot \frac{cos2phi \cdot cos2phi}{{alphax}^{4} \cdot \left(\left(sin2phi \cdot sin2phi\right) \cdot sin2phi\right)}\right), \frac{cos2phi \cdot cos2phi}{{alphax}^{4} \cdot \left(\left(sin2phi \cdot sin2phi\right) \cdot sin2phi\right)}\right)}, \frac{\mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(-1, \frac{cos2phi}{{\left(alphax \cdot sin2phi\right)}^{2}}, u0 \cdot \mathsf{fma}\left(-0.5, \frac{cos2phi}{{\left(alphax \cdot sin2phi\right)}^{2}}, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{cos2phi}{{\left(alphax \cdot sin2phi\right)}^{2}}, -0.25 \cdot \left(\frac{cos2phi}{alphax \cdot alphax} \cdot \frac{u0}{sin2phi \cdot sin2phi}\right)\right)\right)\right), {alphay}^{4}, \left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(0.25, \frac{u0}{sin2phi}, 0.3333333333333333 \cdot \frac{1}{sin2phi}\right), 0.5 \cdot \frac{1}{sin2phi}\right), \frac{1}{sin2phi}\right)\right)\right)}{\left(alphay \cdot alphay\right) \cdot {alphay}^{4}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 40.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 - u0}\\ t_1 := \log t\_0\\ \mathsf{fma}\left(cos2phi, \frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{\log \left({t\_0}^{cos2phi}\right)}{\left(sin2phi \cdot sin2phi\right) \cdot sin2phi} - \frac{{alphay}^{4} \cdot t\_1}{{\left(alphax \cdot sin2phi\right)}^{2}}, \frac{\left(alphay \cdot alphay\right) \cdot t\_1}{sin2phi}\right) \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ 1.0 (- 1.0 u0))) (t_1 (log t_0)))
   (fma
    cos2phi
    (-
     (*
      (/ (pow alphay 6.0) (pow alphax 4.0))
      (/ (log (pow t_0 cos2phi)) (* (* sin2phi sin2phi) sin2phi)))
     (/ (* (pow alphay 4.0) t_1) (pow (* alphax sin2phi) 2.0)))
    (/ (* (* alphay alphay) t_1) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = 1.0f / (1.0f - u0);
	float t_1 = logf(t_0);
	return fmaf(cos2phi, (((powf(alphay, 6.0f) / powf(alphax, 4.0f)) * (logf(powf(t_0, cos2phi)) / ((sin2phi * sin2phi) * sin2phi))) - ((powf(alphay, 4.0f) * t_1) / powf((alphax * sin2phi), 2.0f))), (((alphay * alphay) * t_1) / sin2phi));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) - u0))
	t_1 = log(t_0)
	return fma(cos2phi, Float32(Float32(Float32((alphay ^ Float32(6.0)) / (alphax ^ Float32(4.0))) * Float32(log((t_0 ^ cos2phi)) / Float32(Float32(sin2phi * sin2phi) * sin2phi))) - Float32(Float32((alphay ^ Float32(4.0)) * t_1) / (Float32(alphax * sin2phi) ^ Float32(2.0)))), Float32(Float32(Float32(alphay * alphay) * t_1) / sin2phi))
end

Derivation

1. Initial program 61.0%

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

   1. lift-neg.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. neg-log N/A

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

   5. lower-log.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lift--.f32 58.9

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

   8. lift-+.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-/.f32 N/A

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

   13. +-commutative N/A

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

   14. associate-/r* N/A

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

   15. associate-/r* N/A

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

   16. frac-add N/A

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

   17. lower-/.f32 N/A

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

   18. lower-fma.f32 N/A

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

   19. lower-/.f32 N/A

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

   20. lower-*.f32 N/A

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

   21. lower-/.f32 N/A

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

   22. lower-*.f32 58.8

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

4. Applied rewrites 58.8%

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

5. Taylor expanded in cos2phi around 0

   \[\leadsto \color{blue}{cos2phi \cdot \left(\frac{{alphay}^{6} \cdot \left(cos2phi \cdot \log \left(\frac{1}{1 - u0}\right)\right)}{{alphax}^{4} \cdot {sin2phi}^{3}} - \frac{{alphay}^{4} \cdot \log \left(\frac{1}{1 - u0}\right)}{{alphax}^{2} \cdot {sin2phi}^{2}}\right) + \frac{{alphay}^{2} \cdot \log \left(\frac{1}{1 - u0}\right)}{sin2phi}} \]
6. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(cos2phi, \color{blue}{\frac{{alphay}^{6} \cdot \left(cos2phi \cdot \log \left(\frac{1}{1 - u0}\right)\right)}{{alphax}^{4} \cdot {sin2phi}^{3}} - \frac{{alphay}^{4} \cdot \log \left(\frac{1}{1 - u0}\right)}{{alphax}^{2} \cdot {sin2phi}^{2}}}, \frac{{alphay}^{2} \cdot \log \left(\frac{1}{1 - u0}\right)}{sin2phi}\right) \]

7. Applied rewrites 41.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(cos2phi, \frac{{alphay}^{6}}{{alphax}^{4}} \cdot \frac{\log \left({\left(\frac{1}{1 - u0}\right)}^{cos2phi}\right)}{\left(sin2phi \cdot sin2phi\right) \cdot sin2phi} - \frac{{alphay}^{4} \cdot \log \left(\frac{1}{1 - u0}\right)}{{\left(alphax \cdot sin2phi\right)}^{2}}, \frac{\left(alphay \cdot alphay\right) \cdot \log \left(\frac{1}{1 - u0}\right)}{sin2phi}\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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)))))