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

Percentage Accurate: 60.0% → 97.8%

Time: 10.3s

Alternatives: 23

Speedup: 5.4×

Specification

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

Math

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

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: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ t_1 := \frac{cos2phi}{alphax \cdot alphax}\\ t_2 := t\_0 + t\_1\\ \mathbf{if}\;u0 \leq 0.026000000536441803:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, \frac{u0}{t\_2}, \frac{0.3333333333333333}{t\_2}\right), u0, \frac{0.5}{t\_2}\right), u0, \frac{1}{t\_2}\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(\log \left(1 - u0 \cdot u0\right) - \mathsf{log1p}\left(u0\right)\right)}{t\_1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay)))
        (t_1 (/ cos2phi (* alphax alphax)))
        (t_2 (+ t_0 t_1)))
   (if (<= u0 0.026000000536441803)
     (*
      (fma
       (fma (fma 0.25 (/ u0 t_2) (/ 0.3333333333333333 t_2)) u0 (/ 0.5 t_2))
       u0
       (/ 1.0 t_2))
      u0)
     (/ (- (- (log (- 1.0 (* u0 u0))) (log1p u0))) (+ t_1 t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float t_1 = cos2phi / (alphax * alphax);
	float t_2 = t_0 + t_1;
	float tmp;
	if (u0 <= 0.026000000536441803f) {
		tmp = fmaf(fmaf(fmaf(0.25f, (u0 / t_2), (0.3333333333333333f / t_2)), u0, (0.5f / t_2)), u0, (1.0f / t_2)) * u0;
	} else {
		tmp = -(logf((1.0f - (u0 * u0))) - log1pf(u0)) / (t_1 + t_0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	t_1 = Float32(cos2phi / Float32(alphax * alphax))
	t_2 = Float32(t_0 + t_1)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.026000000536441803))
		tmp = Float32(fma(fma(fma(Float32(0.25), Float32(u0 / t_2), Float32(Float32(0.3333333333333333) / t_2)), u0, Float32(Float32(0.5) / t_2)), u0, Float32(Float32(1.0) / t_2)) * u0);
	else
		tmp = Float32(Float32(-Float32(log(Float32(Float32(1.0) - Float32(u0 * u0))) - log1p(u0))) / Float32(t_1 + t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.0260000005

   1. Initial program 51.6%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.5%

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

   if 0.0260000005 < u0

   1. Initial program 94.6%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. lower-log.f32 N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. lower--.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-log1p.f32 95.5

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

   4. Applied rewrites 95.5%

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

Alternative 2: 95.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (* (pow u0 3.0) (- (* -0.5 (pow u0 3.0)) 1.0)) (log1p (+ (* 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) * ((-0.5f * powf(u0, 3.0f)) - 1.0f)) - log1pf(((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(Float32(-0.5) * (u0 ^ Float32(3.0))) - Float32(1.0))) - log1p(Float32(Float32(u0 * u0) + u0))) / Float32(Float32(Float32(-cos2phi) / Float32(alphax * alphax)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 57.3%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 96.0

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

4. Applied rewrites 96.0%

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

5. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-pow.f32 96.4

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

7. Applied rewrites 96.4%

   \[\leadsto \frac{-\left(\color{blue}{{u0}^{3} \cdot \left(-0.5 \cdot {u0}^{3} - 1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. pow2 N/A

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

   4. lower-+.f32 N/A

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

   5. pow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 96.4

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

9. Applied rewrites 96.4%

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

10. Final simplification 96.4%

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

Alternative 3: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u0\right)\\ t_1 := \frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;t\_0 \leq -0.03999999910593033:\\ \;\;\;\;\frac{-t\_0}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, \frac{u0}{t\_1}, \frac{0.3333333333333333}{t\_1}\right), u0, \frac{0.5}{t\_1}\right), u0, \frac{1}{t\_1}\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0)))
        (t_1 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))
   (if (<= t_0 -0.03999999910593033)
     (/
      (- t_0)
      (/
       (fma (/ sin2phi alphay) alphax (* alphay (/ cos2phi alphax)))
       (* alphay alphax)))
     (*
      (fma
       (fma (fma 0.25 (/ u0 t_1) (/ 0.3333333333333333 t_1)) u0 (/ 0.5 t_1))
       u0
       (/ 1.0 t_1))
      u0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = logf((1.0f - u0));
	float t_1 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax));
	float tmp;
	if (t_0 <= -0.03999999910593033f) {
		tmp = -t_0 / (fmaf((sin2phi / alphay), alphax, (alphay * (cos2phi / alphax))) / (alphay * alphax));
	} else {
		tmp = fmaf(fmaf(fmaf(0.25f, (u0 / t_1), (0.3333333333333333f / t_1)), u0, (0.5f / t_1)), u0, (1.0f / t_1)) * u0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	t_1 = Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.03999999910593033))
		tmp = Float32(Float32(-t_0) / Float32(fma(Float32(sin2phi / alphay), alphax, Float32(alphay * Float32(cos2phi / alphax))) / Float32(alphay * alphax)));
	else
		tmp = Float32(fma(fma(fma(Float32(0.25), Float32(u0 / t_1), Float32(Float32(0.3333333333333333) / t_1)), u0, Float32(Float32(0.5) / t_1)), u0, Float32(Float32(1.0) / t_1)) * u0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0399999991

   1. Initial program 94.8%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. frac-add N/A

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

      10. lower-/.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 95.1

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

   4. Applied rewrites 95.1%

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

   if -0.0399999991 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 51.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.5%

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

Alternative 4: 95.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (* (pow u0 3.0) (- (* -0.5 (* (* u0 u0) u0)) 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) * ((-0.5f * ((u0 * u0) * u0)) - 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(Float32(-0.5) * Float32(Float32(u0 * u0) * u0)) - Float32(1.0))) - log1p(fma(u0, u0, u0))) / Float32(Float32(Float32(-cos2phi) / Float32(alphax * alphax)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 57.3%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 96.0

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

4. Applied rewrites 96.0%

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

5. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-pow.f32 96.4

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

7. Applied rewrites 96.4%

   \[\leadsto \frac{-\left(\color{blue}{{u0}^{3} \cdot \left(-0.5 \cdot {u0}^{3} - 1\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(u0, u0, 1 \cdot u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
8. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. unpow3 N/A

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

   3. pow2 N/A

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

   4. lower-*.f32 N/A

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

   5. pow2 N/A

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

   6. lower-*.f32 96.4

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

9. Applied rewrites 96.4%

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

10. Final simplification 96.4%

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

Alternative 5: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u0\right)\\ \mathbf{if}\;t\_0 \leq -0.03999999910593033:\\ \;\;\;\;\frac{-t\_0}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax, alphay \cdot \frac{cos2phi}{alphax}\right)}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0))))
   (if (<= t_0 -0.03999999910593033)
     (/
      (- t_0)
      (/
       (fma (/ sin2phi alphay) alphax (* alphay (/ cos2phi alphax)))
       (* alphay alphax)))
     (/
      (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
      (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = logf((1.0f - u0));
	float tmp;
	if (t_0 <= -0.03999999910593033f) {
		tmp = -t_0 / (fmaf((sin2phi / alphay), alphax, (alphay * (cos2phi / alphax))) / (alphay * alphax));
	} else {
		tmp = (fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.03999999910593033))
		tmp = Float32(Float32(-t_0) / Float32(fma(Float32(sin2phi / alphay), alphax, Float32(alphay * Float32(cos2phi / alphax))) / Float32(alphay * alphax)));
	else
		tmp = Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0399999991

   1. Initial program 94.8%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. frac-add N/A

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

      10. lower-/.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 95.1

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

   4. Applied rewrites 95.1%

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

   if -0.0399999991 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 51.9%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 6: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u0\right)\\ \mathbf{if}\;t\_0 \leq -0.03999999910593033:\\ \;\;\;\;\frac{-t\_0}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, alphay \cdot cos2phi\right)}{alphay \cdot \left(alphax \cdot alphax\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0))))
   (if (<= t_0 -0.03999999910593033)
     (/
      (- t_0)
      (/
       (fma (/ sin2phi alphay) (* alphax alphax) (* alphay cos2phi))
       (* alphay (* alphax alphax))))
     (/
      (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
      (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = logf((1.0f - u0));
	float tmp;
	if (t_0 <= -0.03999999910593033f) {
		tmp = -t_0 / (fmaf((sin2phi / alphay), (alphax * alphax), (alphay * cos2phi)) / (alphay * (alphax * alphax)));
	} else {
		tmp = (fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.03999999910593033))
		tmp = Float32(Float32(-t_0) / Float32(fma(Float32(sin2phi / alphay), Float32(alphax * alphax), Float32(alphay * cos2phi)) / Float32(alphay * Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0399999991

   1. Initial program 94.8%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-/r* N/A

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

      8. pow2 N/A

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

      9. frac-add N/A

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

      10. lower-/.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. pow2 N/A

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

      14. lift-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 N/A

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

      17. pow2 N/A

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

      18. lift-*.f32 95.0

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

   4. Applied rewrites 95.0%

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

   if -0.0399999991 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 51.9%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 7: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u0\right)\\ t_1 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;t\_0 \leq -0.03999999910593033:\\ \;\;\;\;\frac{-t\_0}{t\_1 + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{t\_1 + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0))) (t_1 (/ cos2phi (* alphax alphax))))
   (if (<= t_0 -0.03999999910593033)
     (/ (- t_0) (+ t_1 (/ (/ sin2phi alphay) alphay)))
     (/
      (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
      (+ t_1 (/ sin2phi (* alphay alphay)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = logf((1.0f - u0));
	float t_1 = cos2phi / (alphax * alphax);
	float tmp;
	if (t_0 <= -0.03999999910593033f) {
		tmp = -t_0 / (t_1 + ((sin2phi / alphay) / alphay));
	} else {
		tmp = (fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) / (t_1 + (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	t_1 = Float32(cos2phi / Float32(alphax * alphax))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.03999999910593033))
		tmp = Float32(Float32(-t_0) / Float32(t_1 + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / Float32(t_1 + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0399999991

   1. Initial program 94.8%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 95.0

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

   4. Applied rewrites 95.0%

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

   if -0.0399999991 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 51.9%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 8: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u0\right)\\ t_1 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq -0.03999999910593033:\\ \;\;\;\;\frac{-t\_0}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right), u0, 1\right) \cdot u0}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0)))
        (t_1 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))
   (if (<= t_0 -0.03999999910593033)
     (/ (- t_0) t_1)
     (/
      (* (fma (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0 1.0) u0)
      t_1))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = logf((1.0f - u0));
	float t_1 = (cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay));
	float tmp;
	if (t_0 <= -0.03999999910593033f) {
		tmp = -t_0 / t_1;
	} else {
		tmp = (fmaf(fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f), u0, 1.0f) * u0) / t_1;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	t_1 = Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.03999999910593033))
		tmp = Float32(Float32(-t_0) / t_1);
	else
		tmp = Float32(Float32(fma(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)), u0, Float32(1.0)) * u0) / t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0399999991

   1. Initial program 94.8%

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

   if -0.0399999991 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 51.9%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 9: 93.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 96.0

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

4. Applied rewrites 96.0%

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

5. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lower-*.f32 94.1

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

7. Applied rewrites 94.1%

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

Alternative 10: 93.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 94.1

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

5. Applied rewrites 94.1%

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

Alternative 11: 91.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 92.4

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

5. Applied rewrites 92.4%

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

Alternative 12: 83.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.999999873689376e-6)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (/
    (*
     u0
     (fma
      u0
      (fma
       0.3333333333333333
       (* (* alphay alphay) u0)
       (* 0.5 (* alphay alphay)))
      (* alphay alphay)))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999873689376e-6f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (u0 * fmaf(u0, fmaf(0.3333333333333333f, ((alphay * alphay) * u0), (0.5f * (alphay * alphay))), (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.999999873689376e-6))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(u0 * fma(u0, fma(Float32(0.3333333333333333), Float32(Float32(alphay * alphay) * u0), Float32(Float32(0.5) * Float32(alphay * alphay))), Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999987e-6

   1. Initial program 52.2%

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

   3. Taylor expanded in u0 around 0

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

   5. Applied rewrites 75.9%

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

   if 4.99999987e-6 < sin2phi

   1. Initial program 62.0%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   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 92.4%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

   8. Applied rewrites 91.9%

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

Alternative 13: 87.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 89.0

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

5. Applied rewrites 89.0%

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

Alternative 14: 83.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.999999873689376e-6)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (*
    (/
     (fma
      u0
      (* (* alphay alphay) (+ 0.5 (* 0.3333333333333333 u0)))
      (* alphay alphay))
     sin2phi)
    u0)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999873689376e-6f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (fmaf(u0, ((alphay * alphay) * (0.5f + (0.3333333333333333f * u0))), (alphay * alphay)) / sin2phi) * u0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.999999873689376e-6))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(fma(u0, Float32(Float32(alphay * alphay) * Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u0))), Float32(alphay * alphay)) / sin2phi) * u0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999987e-6

   1. Initial program 52.2%

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

   3. Taylor expanded in u0 around 0

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

   5. Applied rewrites 75.9%

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

   if 4.99999987e-6 < sin2phi

   1. Initial program 62.0%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   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 92.4%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 91.8

          \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   8. Applied rewrites 91.8%

      \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   9. Taylor expanded in alphay around 0

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

       1. lower-*.f32 N/A

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

       2. pow2 N/A

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

       3. lift-*.f32 N/A

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

       4. lower-+.f32 N/A

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

       5. lower-*.f32 91.8

          \[\leadsto \frac{\mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   11. Applied rewrites 91.8%

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

Alternative 15: 79.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + 0.3333333333333333 \cdot u0\\ \mathbf{if}\;sin2phi \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \left(alphax \cdot alphax\right) \cdot t\_0, alphax \cdot alphax\right)}{cos2phi} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot t\_0, alphay \cdot alphay\right)}{sin2phi} \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ 0.5 (* 0.3333333333333333 u0))))
   (if (<= sin2phi 5.000000156871975e-23)
     (* (/ (fma u0 (* (* alphax alphax) t_0) (* alphax alphax)) cos2phi) u0)
     (* (/ (fma u0 (* (* alphay alphay) t_0) (* alphay alphay)) sin2phi) u0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = 0.5f + (0.3333333333333333f * u0);
	float tmp;
	if (sin2phi <= 5.000000156871975e-23f) {
		tmp = (fmaf(u0, ((alphax * alphax) * t_0), (alphax * alphax)) / cos2phi) * u0;
	} else {
		tmp = (fmaf(u0, ((alphay * alphay) * t_0), (alphay * alphay)) / sin2phi) * u0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u0))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000156871975e-23))
		tmp = Float32(Float32(fma(u0, Float32(Float32(alphax * alphax) * t_0), Float32(alphax * alphax)) / cos2phi) * u0);
	else
		tmp = Float32(Float32(fma(u0, Float32(Float32(alphay * alphay) * t_0), Float32(alphay * alphay)) / sin2phi) * u0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000016e-23

   1. Initial program 54.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 92.0%

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

   6. Taylor expanded in cos2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 75.0

          \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphax \cdot alphax\right) \cdot u0, 0.5 \cdot \left(alphax \cdot alphax\right)\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   8. Applied rewrites 75.0%

      \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphax \cdot alphax\right) \cdot u0, 0.5 \cdot \left(alphax \cdot alphax\right)\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   9. Taylor expanded in alphax around 0

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

       1. lower-*.f32 N/A

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

       2. pow2 N/A

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

       3. lift-*.f32 N/A

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

       4. lower-+.f32 N/A

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

       5. lower-*.f32 75.0

          \[\leadsto \frac{\mathsf{fma}\left(u0, \left(alphax \cdot alphax\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   11. Applied rewrites 75.0%

       \[\leadsto \frac{\mathsf{fma}\left(u0, \left(alphax \cdot alphax\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   if 5.00000016e-23 < sin2phi

   1. Initial program 58.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 92.7%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 81.4

          \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   8. Applied rewrites 81.4%

      \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   9. Taylor expanded in alphay around 0

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

       1. lower-*.f32 N/A

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

       2. pow2 N/A

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

       3. lift-*.f32 N/A

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

       4. lower-+.f32 N/A

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

       5. lower-*.f32 81.4

          \[\leadsto \frac{\mathsf{fma}\left(u0, \left(alphay \cdot alphay\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   11. Applied rewrites 81.4%

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

Alternative 16: 78.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + 0.3333333333333333 \cdot u0\\ \mathbf{if}\;sin2phi \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0, \left(alphax \cdot alphax\right) \cdot t\_0, alphax \cdot alphax\right)}{cos2phi} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot t\_0\right)}{sin2phi} \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ 0.5 (* 0.3333333333333333 u0))))
   (if (<= sin2phi 5.000000156871975e-23)
     (* (/ (fma u0 (* (* alphax alphax) t_0) (* alphax alphax)) cos2phi) u0)
     (* (/ (* (* alphay alphay) (+ 1.0 (* u0 t_0))) sin2phi) u0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = 0.5f + (0.3333333333333333f * u0);
	float tmp;
	if (sin2phi <= 5.000000156871975e-23f) {
		tmp = (fmaf(u0, ((alphax * alphax) * t_0), (alphax * alphax)) / cos2phi) * u0;
	} else {
		tmp = (((alphay * alphay) * (1.0f + (u0 * t_0))) / sin2phi) * u0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u0))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000156871975e-23))
		tmp = Float32(Float32(fma(u0, Float32(Float32(alphax * alphax) * t_0), Float32(alphax * alphax)) / cos2phi) * u0);
	else
		tmp = Float32(Float32(Float32(Float32(alphay * alphay) * Float32(Float32(1.0) + Float32(u0 * t_0))) / sin2phi) * u0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000016e-23

   1. Initial program 54.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 92.0%

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

   6. Taylor expanded in cos2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 75.0

          \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphax \cdot alphax\right) \cdot u0, 0.5 \cdot \left(alphax \cdot alphax\right)\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   8. Applied rewrites 75.0%

      \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphax \cdot alphax\right) \cdot u0, 0.5 \cdot \left(alphax \cdot alphax\right)\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   9. Taylor expanded in alphax around 0

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

       1. lower-*.f32 N/A

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

       2. pow2 N/A

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

       3. lift-*.f32 N/A

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

       4. lower-+.f32 N/A

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

       5. lower-*.f32 75.0

          \[\leadsto \frac{\mathsf{fma}\left(u0, \left(alphax \cdot alphax\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   11. Applied rewrites 75.0%

       \[\leadsto \frac{\mathsf{fma}\left(u0, \left(alphax \cdot alphax\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   if 5.00000016e-23 < sin2phi

   1. Initial program 58.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 92.7%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 81.4

          \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   8. Applied rewrites 81.4%

      \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   9. Taylor expanded in alphay around 0

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

       1. lower-*.f32 N/A

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

       2. pow2 N/A

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

       3. lift-*.f32 N/A

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

       4. lower-+.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower-+.f32 N/A

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

       7. lower-*.f32 81.3

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

   11. Applied rewrites 81.3%

       \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right)\right)}{sin2phi} \cdot u0 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 78.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + u0 \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right)\\ \mathbf{if}\;sin2phi \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot t\_0}{cos2phi} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot t\_0}{sin2phi} \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ 1.0 (* u0 (+ 0.5 (* 0.3333333333333333 u0))))))
   (if (<= sin2phi 5.000000156871975e-23)
     (* (/ (* (* alphax alphax) t_0) cos2phi) u0)
     (* (/ (* (* alphay alphay) t_0) sin2phi) u0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = 1.0f + (u0 * (0.5f + (0.3333333333333333f * u0)));
	float tmp;
	if (sin2phi <= 5.000000156871975e-23f) {
		tmp = (((alphax * alphax) * t_0) / cos2phi) * u0;
	} else {
		tmp = (((alphay * alphay) * t_0) / sin2phi) * u0;
	}
	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 = 1.0e0 + (u0 * (0.5e0 + (0.3333333333333333e0 * u0)))
    if (sin2phi <= 5.000000156871975e-23) then
        tmp = (((alphax * alphax) * t_0) / cos2phi) * u0
    else
        tmp = (((alphay * alphay) * t_0) / sin2phi) * u0
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u0))))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000156871975e-23))
		tmp = Float32(Float32(Float32(Float32(alphax * alphax) * t_0) / cos2phi) * u0);
	else
		tmp = Float32(Float32(Float32(Float32(alphay * alphay) * t_0) / sin2phi) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = single(1.0) + (u0 * (single(0.5) + (single(0.3333333333333333) * u0)));
	tmp = single(0.0);
	if (sin2phi <= single(5.000000156871975e-23))
		tmp = (((alphax * alphax) * t_0) / cos2phi) * u0;
	else
		tmp = (((alphay * alphay) * t_0) / sin2phi) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000016e-23

   1. Initial program 54.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 92.0%

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

   6. Taylor expanded in cos2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 75.0

          \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphax \cdot alphax\right) \cdot u0, 0.5 \cdot \left(alphax \cdot alphax\right)\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   8. Applied rewrites 75.0%

      \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphax \cdot alphax\right) \cdot u0, 0.5 \cdot \left(alphax \cdot alphax\right)\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   9. Taylor expanded in alphax around 0

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

       1. lower-*.f32 N/A

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

       2. pow2 N/A

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

       3. lift-*.f32 N/A

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

       4. lower-+.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower-+.f32 N/A

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

       7. lower-*.f32 74.8

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

   11. Applied rewrites 74.8%

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

   if 5.00000016e-23 < sin2phi

   1. Initial program 58.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 92.7%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 81.4

          \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   8. Applied rewrites 81.4%

      \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   9. Taylor expanded in alphay around 0

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

       1. lower-*.f32 N/A

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

       2. pow2 N/A

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

       3. lift-*.f32 N/A

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

       4. lower-+.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower-+.f32 N/A

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

       7. lower-*.f32 81.3

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

   11. Applied rewrites 81.3%

       \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right)\right)}{sin2phi} \cdot u0 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 76.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(alphax \cdot alphax\right) \cdot \left(1 + u0 \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right)\right)}{cos2phi} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphay \cdot alphay\right) \cdot u0, alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.000000156871975e-23)
   (*
    (/
     (* (* alphax alphax) (+ 1.0 (* u0 (+ 0.5 (* 0.3333333333333333 u0)))))
     cos2phi)
    u0)
   (/ (* u0 (fma 0.5 (* (* alphay alphay) u0) (* alphay alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.000000156871975e-23f) {
		tmp = (((alphax * alphax) * (1.0f + (u0 * (0.5f + (0.3333333333333333f * u0))))) / cos2phi) * u0;
	} else {
		tmp = (u0 * fmaf(0.5f, ((alphay * alphay) * u0), (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000156871975e-23))
		tmp = Float32(Float32(Float32(Float32(alphax * alphax) * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u0))))) / cos2phi) * u0);
	else
		tmp = Float32(Float32(u0 * fma(Float32(0.5), Float32(Float32(alphay * alphay) * u0), Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000016e-23

   1. Initial program 54.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 92.0%

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

   6. Taylor expanded in cos2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 75.0

          \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphax \cdot alphax\right) \cdot u0, 0.5 \cdot \left(alphax \cdot alphax\right)\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   8. Applied rewrites 75.0%

      \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphax \cdot alphax\right) \cdot u0, 0.5 \cdot \left(alphax \cdot alphax\right)\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   9. Taylor expanded in alphax around 0

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

       1. lower-*.f32 N/A

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

       2. pow2 N/A

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

       3. lift-*.f32 N/A

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

       4. lower-+.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower-+.f32 N/A

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

       7. lower-*.f32 74.8

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

   11. Applied rewrites 74.8%

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

   if 5.00000016e-23 < sin2phi

   1. Initial program 58.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 78.2

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

   8. Applied rewrites 78.2%

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

Alternative 19: 75.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphax \cdot alphax\right) \cdot u0, alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphay \cdot alphay\right) \cdot u0, alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.000000156871975e-23)
   (/ (* u0 (fma 0.5 (* (* alphax alphax) u0) (* alphax alphax))) cos2phi)
   (/ (* u0 (fma 0.5 (* (* alphay alphay) u0) (* alphay alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.000000156871975e-23f) {
		tmp = (u0 * fmaf(0.5f, ((alphax * alphax) * u0), (alphax * alphax))) / cos2phi;
	} else {
		tmp = (u0 * fmaf(0.5f, ((alphay * alphay) * u0), (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000156871975e-23))
		tmp = Float32(Float32(u0 * fma(Float32(0.5), Float32(Float32(alphax * alphax) * u0), Float32(alphax * alphax))) / cos2phi);
	else
		tmp = Float32(Float32(u0 * fma(Float32(0.5), Float32(Float32(alphay * alphay) * u0), Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000016e-23

   1. Initial program 54.4%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in cos2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 73.4

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

   8. Applied rewrites 73.4%

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

   if 5.00000016e-23 < sin2phi

   1. Initial program 58.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 78.2

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

   8. Applied rewrites 78.2%

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

Alternative 20: 68.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.6030000627979972 \cdot 10^{-20}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(0.5, \left(alphax \cdot alphax\right) \cdot u0, alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 2.6030000627979972e-20)
   (/ (* u0 (fma 0.5 (* (* alphax alphax) u0) (* alphax alphax))) cos2phi)
   (* (/ (* alphay alphay) sin2phi) u0)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 2.6030000627979972e-20f) {
		tmp = (u0 * fmaf(0.5f, ((alphax * alphax) * u0), (alphax * alphax))) / cos2phi;
	} else {
		tmp = ((alphay * alphay) / sin2phi) * u0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(2.6030000627979972e-20))
		tmp = Float32(Float32(u0 * fma(Float32(0.5), Float32(Float32(alphax * alphax) * u0), Float32(alphax * alphax))) / cos2phi);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * u0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.60300006e-20

   1. Initial program 52.8%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in cos2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 67.9

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

   8. Applied rewrites 67.9%

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

   if 2.60300006e-20 < sin2phi

   1. Initial program 59.0%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   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 92.4%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 83.7

          \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   8. Applied rewrites 83.7%

      \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{{alphay}^{2}}{sin2phi} \cdot u0 \]
   10. Step-by-step derivation

       1. pow2 N/A

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

       2. lift-*.f32 70.1

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

   11. Applied rewrites 70.1%

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

Alternative 21: 66.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.000000156871975e-23)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (* (/ (* alphay alphay) sin2phi) u0)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.000000156871975e-23f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} else {
		tmp = ((alphay * alphay) / sin2phi) * u0;
	}
	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) :: tmp
    if (sin2phi <= 5.000000156871975e-23) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    else
        tmp = ((alphay * alphay) / sin2phi) * u0
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000156871975e-23))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(5.000000156871975e-23))
		tmp = u0 / (cos2phi / (alphax * alphax));
	else
		tmp = ((alphay * alphay) / sin2phi) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000016e-23

   1. Initial program 54.4%

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

   3. Taylor expanded in u0 around 0

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in alphax around 0

      \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]
   7. Step-by-step derivation

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 63.7

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

   8. Applied rewrites 63.7%

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

   if 5.00000016e-23 < sin2phi

   1. Initial program 58.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 92.7%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 81.4

          \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   8. Applied rewrites 81.4%

      \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{{alphay}^{2}}{sin2phi} \cdot u0 \]
   10. Step-by-step derivation

       1. pow2 N/A

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

       2. lift-*.f32 68.3

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

   11. Applied rewrites 68.3%

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

Alternative 22: 66.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{alphax \cdot alphax}{cos2phi} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.000000156871975e-23)
   (* (/ (* alphax alphax) cos2phi) u0)
   (* (/ (* alphay alphay) sin2phi) u0)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.000000156871975e-23f) {
		tmp = ((alphax * alphax) / cos2phi) * u0;
	} else {
		tmp = ((alphay * alphay) / sin2phi) * u0;
	}
	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) :: tmp
    if (sin2phi <= 5.000000156871975e-23) then
        tmp = ((alphax * alphax) / cos2phi) * u0
    else
        tmp = ((alphay * alphay) / sin2phi) * u0
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000156871975e-23))
		tmp = Float32(Float32(Float32(alphax * alphax) / cos2phi) * u0);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(5.000000156871975e-23))
		tmp = ((alphax * alphax) / cos2phi) * u0;
	else
		tmp = ((alphay * alphay) / sin2phi) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000016e-23

   1. Initial program 54.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 92.0%

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

   6. Taylor expanded in cos2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 75.0

          \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphax \cdot alphax\right) \cdot u0, 0.5 \cdot \left(alphax \cdot alphax\right)\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   8. Applied rewrites 75.0%

      \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphax \cdot alphax\right) \cdot u0, 0.5 \cdot \left(alphax \cdot alphax\right)\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{{alphax}^{2}}{cos2phi} \cdot u0 \]
   10. Step-by-step derivation

       1. pow2 N/A

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

       2. lift-*.f32 63.5

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

   11. Applied rewrites 63.5%

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

   if 5.00000016e-23 < sin2phi

   1. Initial program 58.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 92.7%

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

   6. Taylor expanded in sin2phi around inf

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. pow2 N/A

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

      9. lift-*.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 81.4

          \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   8. Applied rewrites 81.4%

      \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphay \cdot alphay\right) \cdot u0, 0.5 \cdot \left(alphay \cdot alphay\right)\right), alphay \cdot alphay\right)}{sin2phi} \cdot u0 \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{{alphay}^{2}}{sin2phi} \cdot u0 \]
   10. Step-by-step derivation

       1. pow2 N/A

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

       2. lift-*.f32 68.3

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

   11. Applied rewrites 68.3%

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

Alternative 23: 23.7% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \frac{alphax \cdot alphax}{cos2phi} \cdot u0 \end{array} \]

FPCore

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

C

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

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 = ((alphax * alphax) / cos2phi) * u0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
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 92.5%

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

6. Taylor expanded in cos2phi around inf

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

   1. lower-/.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift-*.f32 N/A

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

   10. pow2 N/A

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

   11. lift-*.f32 28.8

       \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphax \cdot alphax\right) \cdot u0, 0.5 \cdot \left(alphax \cdot alphax\right)\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

8. Applied rewrites 28.8%

   \[\leadsto \frac{\mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, \left(alphax \cdot alphax\right) \cdot u0, 0.5 \cdot \left(alphax \cdot alphax\right)\right), alphax \cdot alphax\right)}{cos2phi} \cdot u0 \]

9. Taylor expanded in u0 around 0

   \[\leadsto \frac{{alphax}^{2}}{cos2phi} \cdot u0 \]
10. Step-by-step derivation

    1. pow2 N/A

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

    2. lift-*.f32 25.2

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

11. Applied rewrites 25.2%

    \[\leadsto \frac{alphax \cdot alphax}{cos2phi} \cdot u0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025037 
(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)))))