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

Percentage Accurate: 60.0% → 97.9%

Time: 1.7min

Alternatives: 13

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: 60.0% 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: 97.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;u0 \leq 0.05900000035762787:\\ \;\;\;\;\frac{\left(-{u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, u0\right)\right)}{\frac{cos2phi}{\left(-alphax\right) \cdot alphax} - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= u0 0.05900000035762787)
     (/
      (- (- (pow u0 3.0)) (log1p (fma u0 u0 u0)))
      (- (/ cos2phi (* (- alphax) alphax)) t_0))
     (/ (- (log (- 1.0 u0))) (+ (/ (/ cos2phi alphax) alphax) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (u0 <= 0.05900000035762787f) {
		tmp = (-powf(u0, 3.0f) - log1pf(fmaf(u0, u0, u0))) / ((cos2phi / (-alphax * alphax)) - t_0);
	} else {
		tmp = -logf((1.0f - u0)) / (((cos2phi / alphax) / alphax) + t_0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (u0 <= Float32(0.05900000035762787))
		tmp = Float32(Float32(Float32(-(u0 ^ Float32(3.0))) - log1p(fma(u0, u0, u0))) / Float32(Float32(cos2phi / Float32(Float32(-alphax) * alphax)) - t_0));
	else
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.0590000004

   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. 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. flip3-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. lower-log.f32 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. metadata-eval N/A

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

      11. lower-log1p.f32 N/A

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \color{blue}{\mathsf{log1p}\left(u0 \cdot u0 + 1 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      12. lower-fma.f32 N/A

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      13. lower-*.f32 95.9

          \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, \color{blue}{1 \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Applied rewrites 95.9%

      \[\leadsto \frac{-\color{blue}{\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{-\left(\color{blue}{-1 \cdot {u0}^{3}} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{-\left(-1 \cdot \color{blue}{{u0}^{3}} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. lift-pow.f32 98.5

         \[\leadsto \frac{-\left(-1 \cdot {u0}^{\color{blue}{3}} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. Applied rewrites 98.5%

      \[\leadsto \frac{-\left(\color{blue}{-1 \cdot {u0}^{3}} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   if 0.0590000004 < u0

   1. Initial program 96.0%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 96.0

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

   4. Applied rewrites 96.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.05900000035762787:\\ \;\;\;\;\frac{\left(-{u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, u0\right)\right)}{\frac{cos2phi}{\left(-alphax\right) \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{{u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, u0\right)\right)}{\frac{cos2phi}{\left(-alphax\right) \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (-
   (*
    (pow u0 3.0)
    (-
     (*
      (pow u0 3.0)
      (- (* (pow u0 3.0) (- (* -0.25 (pow u0 3.0)) 0.3333333333333333)) 0.5))
     1.0))
   (log1p (fma u0 u0 u0)))
  (- (/ cos2phi (* (- alphax) alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return ((powf(u0, 3.0f) * ((powf(u0, 3.0f) * ((powf(u0, 3.0f) * ((-0.25f * powf(u0, 3.0f)) - 0.3333333333333333f)) - 0.5f)) - 1.0f)) - log1pf(fmaf(u0, u0, u0))) / ((cos2phi / (-alphax * alphax)) - (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32((u0 ^ Float32(3.0)) * Float32(Float32((u0 ^ Float32(3.0)) * Float32(Float32((u0 ^ Float32(3.0)) * Float32(Float32(Float32(-0.25) * (u0 ^ Float32(3.0))) - Float32(0.3333333333333333))) - Float32(0.5))) - Float32(1.0))) - log1p(fma(u0, u0, u0))) / Float32(Float32(cos2phi / Float32(Float32(-alphax) * alphax)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. flip3-- N/A

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

   4. log-div N/A

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

   5. lower--.f32 N/A

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

   6. lower-log.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower--.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. metadata-eval N/A

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

   11. lower-log1p.f32 N/A

       \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \color{blue}{\mathsf{log1p}\left(u0 \cdot u0 + 1 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   12. lower-fma.f32 N/A

       \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   13. lower-*.f32 95.8

       \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, \color{blue}{1 \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

4. Applied rewrites 95.8%

   \[\leadsto \frac{-\color{blue}{\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Taylor expanded in u0 around 0

   \[\leadsto \frac{-\left(\color{blue}{{u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \color{blue}{\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lift-pow.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left(\color{blue}{{u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right)} - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. lower--.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - \color{blue}{1}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lift-pow.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. lower--.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. lower-*.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lift-pow.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. lower--.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   11. lift-pow.f32 97.1

       \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

7. Applied rewrites 97.1%

   \[\leadsto \frac{-\left(\color{blue}{{u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Final simplification 97.1%

   \[\leadsto \frac{{u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, u0\right)\right)}{\frac{cos2phi}{\left(-alphax\right) \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}} \]
9. Add Preprocessing

Alternative 3: 96.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{{u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right) - \mathsf{log1p}\left({u0}^{2} \cdot \left(1 + \frac{1}{u0}\right)\right)}{\frac{cos2phi}{\left(-alphax\right) \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (-
   (*
    (pow u0 3.0)
    (-
     (*
      (pow u0 3.0)
      (- (* (pow u0 3.0) (- (* -0.25 (pow u0 3.0)) 0.3333333333333333)) 0.5))
     1.0))
   (log1p (* (pow u0 2.0) (+ 1.0 (/ 1.0 u0)))))
  (- (/ cos2phi (* (- alphax) alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return ((powf(u0, 3.0f) * ((powf(u0, 3.0f) * ((powf(u0, 3.0f) * ((-0.25f * powf(u0, 3.0f)) - 0.3333333333333333f)) - 0.5f)) - 1.0f)) - log1pf((powf(u0, 2.0f) * (1.0f + (1.0f / u0))))) / ((cos2phi / (-alphax * alphax)) - (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32((u0 ^ Float32(3.0)) * Float32(Float32((u0 ^ Float32(3.0)) * Float32(Float32((u0 ^ Float32(3.0)) * Float32(Float32(Float32(-0.25) * (u0 ^ Float32(3.0))) - Float32(0.3333333333333333))) - Float32(0.5))) - Float32(1.0))) - log1p(Float32((u0 ^ Float32(2.0)) * Float32(Float32(1.0) + Float32(Float32(1.0) / u0))))) / Float32(Float32(cos2phi / Float32(Float32(-alphax) * alphax)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. flip3-- N/A

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

   4. log-div N/A

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

   5. lower--.f32 N/A

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

   6. lower-log.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower--.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. metadata-eval N/A

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

   11. lower-log1p.f32 N/A

       \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \color{blue}{\mathsf{log1p}\left(u0 \cdot u0 + 1 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   12. lower-fma.f32 N/A

       \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   13. lower-*.f32 95.8

       \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, \color{blue}{1 \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

4. Applied rewrites 95.8%

   \[\leadsto \frac{-\color{blue}{\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Taylor expanded in u0 around 0

   \[\leadsto \frac{-\left(\color{blue}{{u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \color{blue}{\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lift-pow.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left(\color{blue}{{u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right)} - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. lower--.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - \color{blue}{1}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lift-pow.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. lower--.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. lower-*.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lift-pow.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. lower--.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   11. lift-pow.f32 97.1

       \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

7. Applied rewrites 97.1%

   \[\leadsto \frac{-\left(\color{blue}{{u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Taylor expanded in u0 around inf

   \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\color{blue}{{u0}^{2} \cdot \left(1 + \frac{1}{u0}\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left({u0}^{2} \cdot \color{blue}{\left(1 + \frac{1}{u0}\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-pow.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left({u0}^{2} \cdot \left(\color{blue}{1} + \frac{1}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left({u0}^{2} \cdot \left(1 + \color{blue}{\frac{1}{u0}}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lower-/.f32 97.0

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right) - \mathsf{log1p}\left({u0}^{2} \cdot \left(1 + \frac{1}{\color{blue}{u0}}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

10. Applied rewrites 97.0%

    \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right) - \mathsf{log1p}\left(\color{blue}{{u0}^{2} \cdot \left(1 + \frac{1}{u0}\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

11. Final simplification 97.0%

    \[\leadsto \frac{{u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right) - \mathsf{log1p}\left({u0}^{2} \cdot \left(1 + \frac{1}{u0}\right)\right)}{\frac{cos2phi}{\left(-alphax\right) \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}} \]
12. Add Preprocessing

Alternative 4: 96.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{{u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right) - \mathsf{log1p}\left(u0 \cdot \left(1 + u0\right)\right)}{\frac{cos2phi}{\left(-alphax\right) \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (-
   (*
    (pow u0 3.0)
    (-
     (*
      (pow u0 3.0)
      (- (* (pow u0 3.0) (- (* -0.25 (pow u0 3.0)) 0.3333333333333333)) 0.5))
     1.0))
   (log1p (* u0 (+ 1.0 u0))))
  (- (/ cos2phi (* (- alphax) alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return ((powf(u0, 3.0f) * ((powf(u0, 3.0f) * ((powf(u0, 3.0f) * ((-0.25f * powf(u0, 3.0f)) - 0.3333333333333333f)) - 0.5f)) - 1.0f)) - log1pf((u0 * (1.0f + u0)))) / ((cos2phi / (-alphax * alphax)) - (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32((u0 ^ Float32(3.0)) * Float32(Float32((u0 ^ Float32(3.0)) * Float32(Float32((u0 ^ Float32(3.0)) * Float32(Float32(Float32(-0.25) * (u0 ^ Float32(3.0))) - Float32(0.3333333333333333))) - Float32(0.5))) - Float32(1.0))) - log1p(Float32(u0 * Float32(Float32(1.0) + u0)))) / Float32(Float32(cos2phi / Float32(Float32(-alphax) * alphax)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. flip3-- N/A

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

   4. log-div N/A

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

   5. lower--.f32 N/A

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

   6. lower-log.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower--.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. metadata-eval N/A

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

   11. lower-log1p.f32 N/A

       \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \color{blue}{\mathsf{log1p}\left(u0 \cdot u0 + 1 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   12. lower-fma.f32 N/A

       \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   13. lower-*.f32 95.8

       \[\leadsto \frac{-\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, \color{blue}{1 \cdot u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

4. Applied rewrites 95.8%

   \[\leadsto \frac{-\color{blue}{\left(\log \left(1 - {u0}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Taylor expanded in u0 around 0

   \[\leadsto \frac{-\left(\color{blue}{{u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \color{blue}{\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lift-pow.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left(\color{blue}{{u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right)} - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. lower--.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - \color{blue}{1}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lift-pow.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. lower--.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. lower-*.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lift-pow.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. lower--.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   11. lift-pow.f32 97.1

       \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

7. Applied rewrites 97.1%

   \[\leadsto \frac{-\left(\color{blue}{{u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Taylor expanded in u0 around 0

   \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(\color{blue}{u0 \cdot \left(1 + u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(\frac{-1}{4} \cdot {u0}^{3} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) - \mathsf{log1p}\left(u0 \cdot \color{blue}{\left(1 + u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-+.f32 96.8

      \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right) - \mathsf{log1p}\left(u0 \cdot \left(1 + \color{blue}{u0}\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

10. Applied rewrites 96.8%

    \[\leadsto \frac{-\left({u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right) - \mathsf{log1p}\left(\color{blue}{u0 \cdot \left(1 + u0\right)}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

11. Final simplification 96.8%

    \[\leadsto \frac{{u0}^{3} \cdot \left({u0}^{3} \cdot \left({u0}^{3} \cdot \left(-0.25 \cdot {u0}^{3} - 0.3333333333333333\right) - 0.5\right) - 1\right) - \mathsf{log1p}\left(u0 \cdot \left(1 + u0\right)\right)}{\frac{cos2phi}{\left(-alphax\right) \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}} \]
12. Add Preprocessing

Alternative 5: 97.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;u0 \leq 0.03799999877810478:\\ \;\;\;\;\frac{-u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ (/ cos2phi alphax) alphax) (/ sin2phi (* alphay alphay)))))
   (if (<= u0 0.03799999877810478)
     (/
      (-
       (*
        u0
        (- (* u0 (- (* u0 (- (* -0.25 u0) 0.3333333333333333)) 0.5)) 1.0)))
      t_0)
     (/ (- (log (- 1.0 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 <= 0.03799999877810478f) {
		tmp = -(u0 * ((u0 * ((u0 * ((-0.25f * u0) - 0.3333333333333333f)) - 0.5f)) - 1.0f)) / t_0;
	} else {
		tmp = -logf((1.0f - u0)) / 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 = ((cos2phi / alphax) / alphax) + (sin2phi / (alphay * alphay))
    if (u0 <= 0.03799999877810478e0) then
        tmp = -(u0 * ((u0 * ((u0 * (((-0.25e0) * u0) - 0.3333333333333333e0)) - 0.5e0)) - 1.0e0)) / t_0
    else
        tmp = -log((1.0e0 - u0)) / t_0
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(sin2phi / Float32(alphay * alphay)))
	tmp = Float32(0.0)
	if (u0 <= Float32(0.03799999877810478))
		tmp = Float32(Float32(-Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333))) - Float32(0.5))) - Float32(1.0)))) / t_0);
	else
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = ((cos2phi / alphax) / alphax) + (sin2phi / (alphay * alphay));
	tmp = single(0.0);
	if (u0 <= single(0.03799999877810478))
		tmp = -(u0 * ((u0 * ((u0 * ((single(-0.25) * u0) - single(0.3333333333333333))) - single(0.5))) - single(1.0))) / t_0;
	else
		tmp = -log((single(1.0) - u0)) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.0379999988

   1. Initial program 53.6%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 53.6

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 98.4

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

   7. Applied rewrites 98.4%

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

   if 0.0379999988 < u0

   1. Initial program 95.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. lift-/.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 95.7

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

   4. Applied rewrites 95.7%

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

Alternative 6: 97.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;u0 \leq 0.03799999877810478:\\ \;\;\;\;\frac{-u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= u0 0.03799999877810478)
     (/
      (-
       (*
        u0
        (- (* u0 (- (* u0 (- (* -0.25 u0) 0.3333333333333333)) 0.5)) 1.0)))
      (+ (/ (/ cos2phi alphax) alphax) t_0))
     (/ (- (log (- 1.0 u0))) (+ (/ cos2phi (* alphax alphax)) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (u0 <= 0.03799999877810478f) {
		tmp = -(u0 * ((u0 * ((u0 * ((-0.25f * u0) - 0.3333333333333333f)) - 0.5f)) - 1.0f)) / (((cos2phi / alphax) / alphax) + t_0);
	} else {
		tmp = -logf((1.0f - u0)) / ((cos2phi / (alphax * alphax)) + 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 <= 0.03799999877810478e0) then
        tmp = -(u0 * ((u0 * ((u0 * (((-0.25e0) * u0) - 0.3333333333333333e0)) - 0.5e0)) - 1.0e0)) / (((cos2phi / alphax) / alphax) + t_0)
    else
        tmp = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + 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(0.03799999877810478))
		tmp = Float32(Float32(-Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333))) - Float32(0.5))) - Float32(1.0)))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	else
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + 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(0.03799999877810478))
		tmp = -(u0 * ((u0 * ((u0 * ((single(-0.25) * u0) - single(0.3333333333333333))) - single(0.5))) - single(1.0))) / (((cos2phi / alphax) / alphax) + t_0);
	else
		tmp = -log((single(1.0) - u0)) / ((cos2phi / (alphax * alphax)) + t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.0379999988

   1. Initial program 53.6%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 53.6

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 98.4

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

   7. Applied rewrites 98.4%

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

   if 0.0379999988 < u0

   1. Initial program 95.7%

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

Alternative 7: 93.2% accurate, N/A× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (* u0 (- (* u0 (- (* u0 (- (* -0.25 u0) 0.3333333333333333)) 0.5)) 1.0)))
  (+ (/ (/ cos2phi alphax) alphax) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -(u0 * ((u0 * ((u0 * ((-0.25f * u0) - 0.3333333333333333f)) - 0.5f)) - 1.0f)) / (((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 = -(u0 * ((u0 * ((u0 * (((-0.25e0) * u0) - 0.3333333333333333e0)) - 0.5e0)) - 1.0e0)) / (((cos2phi / alphax) / alphax) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333))) - Float32(0.5))) - Float32(1.0)))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = -(u0 * ((u0 * ((u0 * ((single(-0.25) * u0) - single(0.3333333333333333))) - single(0.5))) - single(1.0))) / (((cos2phi / alphax) / alphax) + (sin2phi / (alphay * alphay)));
end

Derivation

1. Initial program 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 59.9

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

4. Applied rewrites 59.9%

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

5. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-*.f32 93.1

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

7. Applied rewrites 93.1%

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

Alternative 8: 93.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 93.0%

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

Alternative 9: 92.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, \frac{u0}{t\_0}, \frac{0.3333333333333333}{t\_0}\right), u0, \frac{0.5}{t\_0}\right), u0, \frac{1}{sin2phi \cdot \left({alphay}^{-2} + \frac{cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi}\right)}\right) \cdot u0 \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))
   (*
    (fma
     (fma (fma 0.25 (/ u0 t_0) (/ 0.3333333333333333 t_0)) u0 (/ 0.5 t_0))
     u0
     (/
      1.0
      (*
       sin2phi
       (+ (pow alphay -2.0) (/ cos2phi (* (* alphax alphax) sin2phi))))))
    u0)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax));
	return fmaf(fmaf(fmaf(0.25f, (u0 / t_0), (0.3333333333333333f / t_0)), u0, (0.5f / t_0)), u0, (1.0f / (sin2phi * (powf(alphay, -2.0f) + (cos2phi / ((alphax * alphax) * sin2phi)))))) * u0;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax)))
	return Float32(fma(fma(fma(Float32(0.25), Float32(u0 / t_0), Float32(Float32(0.3333333333333333) / t_0)), u0, Float32(Float32(0.5) / t_0)), u0, Float32(Float32(1.0) / Float32(sin2phi * Float32((alphay ^ Float32(-2.0)) + Float32(cos2phi / Float32(Float32(alphax * alphax) * sin2phi)))))) * u0)
end

Derivation

1. Initial program 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 93.0%

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

6. Taylor expanded in sin2phi around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. pow-flip N/A

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

   4. metadata-eval N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift-*.f32 91.9

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}, \frac{0.3333333333333333}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right), u0, \frac{0.5}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right), u0, \frac{1}{sin2phi \cdot \left({alphay}^{-2} + \frac{cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi}\right)}\right) \cdot u0 \]

8. Applied rewrites 91.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}, \frac{0.3333333333333333}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right), u0, \frac{0.5}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right), u0, \frac{1}{sin2phi \cdot \left({alphay}^{-2} + \frac{cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi}\right)}\right) \cdot u0 \]
9. Add Preprocessing

Alternative 10: 91.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, \frac{u0}{t\_0}, \frac{0.3333333333333333}{t\_0}\right), u0, \frac{0.5}{t\_0}\right), u0, \frac{1}{sin2phi \cdot \frac{1 + \frac{{alphay}^{2} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi}}{{alphay}^{2}}}\right) \cdot u0 \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))
   (*
    (fma
     (fma (fma 0.25 (/ u0 t_0) (/ 0.3333333333333333 t_0)) u0 (/ 0.5 t_0))
     u0
     (/
      1.0
      (*
       sin2phi
       (/
        (+ 1.0 (/ (* (pow alphay 2.0) cos2phi) (* (pow alphax 2.0) sin2phi)))
        (pow alphay 2.0)))))
    u0)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax));
	return fmaf(fmaf(fmaf(0.25f, (u0 / t_0), (0.3333333333333333f / t_0)), u0, (0.5f / t_0)), u0, (1.0f / (sin2phi * ((1.0f + ((powf(alphay, 2.0f) * cos2phi) / (powf(alphax, 2.0f) * sin2phi))) / powf(alphay, 2.0f))))) * u0;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax)))
	return Float32(fma(fma(fma(Float32(0.25), Float32(u0 / t_0), Float32(Float32(0.3333333333333333) / t_0)), u0, Float32(Float32(0.5) / t_0)), u0, Float32(Float32(1.0) / Float32(sin2phi * Float32(Float32(Float32(1.0) + Float32(Float32((alphay ^ Float32(2.0)) * cos2phi) / Float32((alphax ^ Float32(2.0)) * sin2phi))) / (alphay ^ Float32(2.0)))))) * u0)
end

Derivation

1. Initial program 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 93.0%

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

6. Taylor expanded in sin2phi around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. pow-flip N/A

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

   4. metadata-eval N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift-*.f32 91.9

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}, \frac{0.3333333333333333}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right), u0, \frac{0.5}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right), u0, \frac{1}{sin2phi \cdot \left({alphay}^{-2} + \frac{cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi}\right)}\right) \cdot u0 \]

8. Applied rewrites 91.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}, \frac{0.3333333333333333}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right), u0, \frac{0.5}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right), u0, \frac{1}{sin2phi \cdot \left({alphay}^{-2} + \frac{cos2phi}{\left(alphax \cdot alphax\right) \cdot sin2phi}\right)}\right) \cdot u0 \]

9. Taylor expanded in alphay around 0

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

    1. lower-/.f32 N/A

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

    2. lower-+.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-pow.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lower-pow.f32 N/A

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

    8. lower-pow.f32 91.7

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}, \frac{0.3333333333333333}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right), u0, \frac{0.5}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right), u0, \frac{1}{sin2phi \cdot \frac{1 + \frac{{alphay}^{2} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi}}{{alphay}^{2}}}\right) \cdot u0 \]

11. Applied rewrites 91.7%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}, \frac{0.3333333333333333}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right), u0, \frac{0.5}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right), u0, \frac{1}{sin2phi \cdot \frac{1 + \frac{{alphay}^{2} \cdot cos2phi}{{alphax}^{2} \cdot sin2phi}}{{alphay}^{2}}}\right) \cdot u0 \]
12. Add Preprocessing

Alternative 11: 91.4% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 91.4%

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

Alternative 12: 87.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ {u0}^{4} \cdot \left(\frac{0.3333333333333333}{u0 \cdot t\_0} + \mathsf{fma}\left(0.5, \frac{1}{\left(u0 \cdot u0\right) \cdot t\_0}, \frac{1}{{u0}^{3} \cdot t\_0} + 0.25 \cdot \frac{1}{t\_0}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))
   (*
    (pow u0 4.0)
    (+
     (/ 0.3333333333333333 (* u0 t_0))
     (fma
      0.5
      (/ 1.0 (* (* u0 u0) t_0))
      (+ (/ 1.0 (* (pow u0 3.0) t_0)) (* 0.25 (/ 1.0 t_0))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay));
	return powf(u0, 4.0f) * ((0.3333333333333333f / (u0 * t_0)) + fmaf(0.5f, (1.0f / ((u0 * u0) * t_0)), ((1.0f / (powf(u0, 3.0f) * t_0)) + (0.25f * (1.0f / t_0)))));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))
	return Float32((u0 ^ Float32(4.0)) * Float32(Float32(Float32(0.3333333333333333) / Float32(u0 * t_0)) + fma(Float32(0.5), Float32(Float32(1.0) / Float32(Float32(u0 * u0) * t_0)), Float32(Float32(Float32(1.0) / Float32((u0 ^ Float32(3.0)) * t_0)) + Float32(Float32(0.25) * Float32(Float32(1.0) / t_0))))))
end

Derivation

1. Initial program 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 93.0%

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

6. Taylor expanded in u0 around inf

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

   1. lower-*.f32 N/A

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

8. Applied rewrites 89.0%

   \[\leadsto {u0}^{4} \cdot \color{blue}{\left(\frac{0.3333333333333333}{u0 \cdot \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} + \mathsf{fma}\left(0.5, \frac{1}{\left(u0 \cdot u0\right) \cdot \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}, \frac{1}{{u0}^{3} \cdot \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} + 0.25 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\right)} \]
9. Add Preprocessing

Alternative 13: 69.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{alphay}^{8} \cdot {cos2phi}^{3}}{{alphax}^{6}}\\ t_1 := \frac{{alphay}^{6} \cdot \left(cos2phi \cdot cos2phi\right)}{{alphax}^{4}}\\ t_2 := \frac{{alphay}^{4} \cdot cos2phi}{alphax \cdot alphax}\\ \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{u0 \cdot \left(u0 \cdot \mathsf{fma}\left(-0.5, t\_0, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, t\_0, -0.25 \cdot \frac{{alphay}^{8} \cdot \left({cos2phi}^{3} \cdot u0\right)}{{alphax}^{6}}\right)\right) - t\_0\right)}{sin2phi}, u0 \cdot \left(u0 \cdot \mathsf{fma}\left(-0.5, t\_1, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, t\_1, -0.25 \cdot \frac{{alphay}^{6} \cdot \left(\left(cos2phi \cdot cos2phi\right) \cdot u0\right)}{{alphax}^{4}}\right)\right) - t\_1\right)\right)}{sin2phi}, u0 \cdot \left(u0 \cdot \mathsf{fma}\left(-0.5, t\_2, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, t\_2, -0.25 \cdot \frac{{alphay}^{4} \cdot \left(cos2phi \cdot u0\right)}{alphax \cdot alphax}\right)\right) - t\_2\right)\right)}{sin2phi}, u0 \cdot \mathsf{fma}\left(-1, alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(-0.5, alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, alphay \cdot alphay, -0.25 \cdot \left(\left(alphay \cdot alphay\right) \cdot u0\right)\right)\right)\right)\right)}{-sin2phi} \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ (* (pow alphay 8.0) (pow cos2phi 3.0)) (pow alphax 6.0)))
        (t_1 (/ (* (pow alphay 6.0) (* cos2phi cos2phi)) (pow alphax 4.0)))
        (t_2 (/ (* (pow alphay 4.0) cos2phi) (* alphax alphax))))
   (/
    (fma
     -1.0
     (/
      (fma
       -1.0
       (/
        (fma
         -1.0
         (/
          (*
           u0
           (-
            (*
             u0
             (fma
              -0.5
              t_0
              (*
               u0
               (fma
                -0.3333333333333333
                t_0
                (*
                 -0.25
                 (/
                  (* (pow alphay 8.0) (* (pow cos2phi 3.0) u0))
                  (pow alphax 6.0)))))))
            t_0))
          sin2phi)
         (*
          u0
          (-
           (*
            u0
            (fma
             -0.5
             t_1
             (*
              u0
              (fma
               -0.3333333333333333
               t_1
               (*
                -0.25
                (/
                 (* (pow alphay 6.0) (* (* cos2phi cos2phi) u0))
                 (pow alphax 4.0)))))))
           t_1)))
        sin2phi)
       (*
        u0
        (-
         (*
          u0
          (fma
           -0.5
           t_2
           (*
            u0
            (fma
             -0.3333333333333333
             t_2
             (*
              -0.25
              (/ (* (pow alphay 4.0) (* cos2phi u0)) (* alphax alphax)))))))
         t_2)))
      sin2phi)
     (*
      u0
      (fma
       -1.0
       (* alphay alphay)
       (*
        u0
        (fma
         -0.5
         (* alphay alphay)
         (*
          u0
          (fma
           -0.3333333333333333
           (* alphay alphay)
           (* -0.25 (* (* alphay alphay) u0)))))))))
    (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (powf(alphay, 8.0f) * powf(cos2phi, 3.0f)) / powf(alphax, 6.0f);
	float t_1 = (powf(alphay, 6.0f) * (cos2phi * cos2phi)) / powf(alphax, 4.0f);
	float t_2 = (powf(alphay, 4.0f) * cos2phi) / (alphax * alphax);
	return fmaf(-1.0f, (fmaf(-1.0f, (fmaf(-1.0f, ((u0 * ((u0 * fmaf(-0.5f, t_0, (u0 * fmaf(-0.3333333333333333f, t_0, (-0.25f * ((powf(alphay, 8.0f) * (powf(cos2phi, 3.0f) * u0)) / powf(alphax, 6.0f))))))) - t_0)) / sin2phi), (u0 * ((u0 * fmaf(-0.5f, t_1, (u0 * fmaf(-0.3333333333333333f, t_1, (-0.25f * ((powf(alphay, 6.0f) * ((cos2phi * cos2phi) * u0)) / powf(alphax, 4.0f))))))) - t_1))) / sin2phi), (u0 * ((u0 * fmaf(-0.5f, t_2, (u0 * fmaf(-0.3333333333333333f, t_2, (-0.25f * ((powf(alphay, 4.0f) * (cos2phi * u0)) / (alphax * alphax))))))) - t_2))) / sin2phi), (u0 * fmaf(-1.0f, (alphay * alphay), (u0 * fmaf(-0.5f, (alphay * alphay), (u0 * fmaf(-0.3333333333333333f, (alphay * alphay), (-0.25f * ((alphay * alphay) * u0))))))))) / -sin2phi;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32((alphay ^ Float32(8.0)) * (cos2phi ^ Float32(3.0))) / (alphax ^ Float32(6.0)))
	t_1 = Float32(Float32((alphay ^ Float32(6.0)) * Float32(cos2phi * cos2phi)) / (alphax ^ Float32(4.0)))
	t_2 = Float32(Float32((alphay ^ Float32(4.0)) * cos2phi) / Float32(alphax * alphax))
	return Float32(fma(Float32(-1.0), Float32(fma(Float32(-1.0), Float32(fma(Float32(-1.0), Float32(Float32(u0 * Float32(Float32(u0 * fma(Float32(-0.5), t_0, Float32(u0 * fma(Float32(-0.3333333333333333), t_0, Float32(Float32(-0.25) * Float32(Float32((alphay ^ Float32(8.0)) * Float32((cos2phi ^ Float32(3.0)) * u0)) / (alphax ^ Float32(6.0)))))))) - t_0)) / sin2phi), Float32(u0 * Float32(Float32(u0 * fma(Float32(-0.5), t_1, Float32(u0 * fma(Float32(-0.3333333333333333), t_1, Float32(Float32(-0.25) * Float32(Float32((alphay ^ Float32(6.0)) * Float32(Float32(cos2phi * cos2phi) * u0)) / (alphax ^ Float32(4.0)))))))) - t_1))) / sin2phi), Float32(u0 * Float32(Float32(u0 * fma(Float32(-0.5), t_2, Float32(u0 * fma(Float32(-0.3333333333333333), t_2, Float32(Float32(-0.25) * Float32(Float32((alphay ^ Float32(4.0)) * Float32(cos2phi * u0)) / Float32(alphax * alphax))))))) - t_2))) / sin2phi), Float32(u0 * fma(Float32(-1.0), Float32(alphay * alphay), Float32(u0 * fma(Float32(-0.5), Float32(alphay * alphay), Float32(u0 * fma(Float32(-0.3333333333333333), Float32(alphay * alphay), Float32(Float32(-0.25) * Float32(Float32(alphay * alphay) * u0))))))))) / Float32(-sin2phi))
end

Derivation

1. Initial program 59.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 93.0%

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

6. Taylor expanded in sin2phi around -inf

   \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{u0 \cdot \left(u0 \cdot \left(\frac{-1}{2} \cdot \frac{{alphay}^{8} \cdot {cos2phi}^{3}}{{alphax}^{6}} + u0 \cdot \left(\frac{-1}{3} \cdot \frac{{alphay}^{8} \cdot {cos2phi}^{3}}{{alphax}^{6}} + \frac{-1}{4} \cdot \frac{{alphay}^{8} \cdot \left({cos2phi}^{3} \cdot u0\right)}{{alphax}^{6}}\right)\right) - \frac{{alphay}^{8} \cdot {cos2phi}^{3}}{{alphax}^{6}}\right)}{sin2phi} + u0 \cdot \left(u0 \cdot \left(\frac{-1}{2} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4}} + u0 \cdot \left(\frac{-1}{3} \cdot \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4}} + \frac{-1}{4} \cdot \frac{{alphay}^{6} \cdot \left({cos2phi}^{2} \cdot u0\right)}{{alphax}^{4}}\right)\right) - \frac{{alphay}^{6} \cdot {cos2phi}^{2}}{{alphax}^{4}}\right)}{sin2phi} + u0 \cdot \left(u0 \cdot \left(\frac{-1}{2} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2}} + u0 \cdot \left(\frac{-1}{3} \cdot \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2}} + \frac{-1}{4} \cdot \frac{{alphay}^{4} \cdot \left(cos2phi \cdot u0\right)}{{alphax}^{2}}\right)\right) - \frac{{alphay}^{4} \cdot cos2phi}{{alphax}^{2}}\right)}{sin2phi} + u0 \cdot \left(-1 \cdot {alphay}^{2} + u0 \cdot \left(\frac{-1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{-1}{3} \cdot {alphay}^{2} + \frac{-1}{4} \cdot \left({alphay}^{2} \cdot u0\right)\right)\right)\right)}{sin2phi}} \]

8. Applied rewrites 71.0%

   \[\leadsto -1 \cdot \color{blue}{\frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{u0 \cdot \left(u0 \cdot \mathsf{fma}\left(-0.5, \frac{{alphay}^{8} \cdot {cos2phi}^{3}}{{alphax}^{6}}, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{{alphay}^{8} \cdot {cos2phi}^{3}}{{alphax}^{6}}, -0.25 \cdot \frac{{alphay}^{8} \cdot \left({cos2phi}^{3} \cdot u0\right)}{{alphax}^{6}}\right)\right) - \frac{{alphay}^{8} \cdot {cos2phi}^{3}}{{alphax}^{6}}\right)}{sin2phi}, u0 \cdot \left(u0 \cdot \mathsf{fma}\left(-0.5, \frac{{alphay}^{6} \cdot \left(cos2phi \cdot cos2phi\right)}{{alphax}^{4}}, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{{alphay}^{6} \cdot \left(cos2phi \cdot cos2phi\right)}{{alphax}^{4}}, -0.25 \cdot \frac{{alphay}^{6} \cdot \left(\left(cos2phi \cdot cos2phi\right) \cdot u0\right)}{{alphax}^{4}}\right)\right) - \frac{{alphay}^{6} \cdot \left(cos2phi \cdot cos2phi\right)}{{alphax}^{4}}\right)\right)}{sin2phi}, u0 \cdot \left(u0 \cdot \mathsf{fma}\left(-0.5, \frac{{alphay}^{4} \cdot cos2phi}{alphax \cdot alphax}, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{{alphay}^{4} \cdot cos2phi}{alphax \cdot alphax}, -0.25 \cdot \frac{{alphay}^{4} \cdot \left(cos2phi \cdot u0\right)}{alphax \cdot alphax}\right)\right) - \frac{{alphay}^{4} \cdot cos2phi}{alphax \cdot alphax}\right)\right)}{sin2phi}, u0 \cdot \mathsf{fma}\left(-1, alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(-0.5, alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, alphay \cdot alphay, -0.25 \cdot \left(\left(alphay \cdot alphay\right) \cdot u0\right)\right)\right)\right)\right)}{sin2phi}} \]

9. Final simplification 71.0%

   \[\leadsto \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{u0 \cdot \left(u0 \cdot \mathsf{fma}\left(-0.5, \frac{{alphay}^{8} \cdot {cos2phi}^{3}}{{alphax}^{6}}, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{{alphay}^{8} \cdot {cos2phi}^{3}}{{alphax}^{6}}, -0.25 \cdot \frac{{alphay}^{8} \cdot \left({cos2phi}^{3} \cdot u0\right)}{{alphax}^{6}}\right)\right) - \frac{{alphay}^{8} \cdot {cos2phi}^{3}}{{alphax}^{6}}\right)}{sin2phi}, u0 \cdot \left(u0 \cdot \mathsf{fma}\left(-0.5, \frac{{alphay}^{6} \cdot \left(cos2phi \cdot cos2phi\right)}{{alphax}^{4}}, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{{alphay}^{6} \cdot \left(cos2phi \cdot cos2phi\right)}{{alphax}^{4}}, -0.25 \cdot \frac{{alphay}^{6} \cdot \left(\left(cos2phi \cdot cos2phi\right) \cdot u0\right)}{{alphax}^{4}}\right)\right) - \frac{{alphay}^{6} \cdot \left(cos2phi \cdot cos2phi\right)}{{alphax}^{4}}\right)\right)}{sin2phi}, u0 \cdot \left(u0 \cdot \mathsf{fma}\left(-0.5, \frac{{alphay}^{4} \cdot cos2phi}{alphax \cdot alphax}, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{{alphay}^{4} \cdot cos2phi}{alphax \cdot alphax}, -0.25 \cdot \frac{{alphay}^{4} \cdot \left(cos2phi \cdot u0\right)}{alphax \cdot alphax}\right)\right) - \frac{{alphay}^{4} \cdot cos2phi}{alphax \cdot alphax}\right)\right)}{sin2phi}, u0 \cdot \mathsf{fma}\left(-1, alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(-0.5, alphay \cdot alphay, u0 \cdot \mathsf{fma}\left(-0.3333333333333333, alphay \cdot alphay, -0.25 \cdot \left(\left(alphay \cdot alphay\right) \cdot u0\right)\right)\right)\right)\right)}{-sin2phi} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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)))))