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

Percentage Accurate: 60.8% → 98.2%

Time: 5.2s

Alternatives: 12

Speedup: 1.4×

Specification

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

Math

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

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: 60.8% 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: 98.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))

C

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

Julia

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

Derivation

1. Initial program 60.8%

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

2. Taylor expanded in sin2phi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. pow2 N/A

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

   11. lift-*.f32 60.3

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

4. Applied rewrites 60.3%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. *-lft-identity N/A

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

   4. metadata-eval N/A

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

   5. fp-cancel-sign-sub-inv N/A

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

   6. mul-1-neg N/A

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

   7. lower-log1p.f32 N/A

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

   8. lower-neg.f32 96.8

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

6. Applied rewrites 96.8%

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

7. Taylor expanded in alphax around inf

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

   1. associate-/l/ N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. pow2 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-/.f32 98.2

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

9. Applied rewrites 98.2%

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

Alternative 2: 96.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0))) (t_1 (/ cos2phi (* alphax alphax))))
   (if (<= t_0 -0.003000000026077032)
     (/ (- t_0) (+ t_1 (/ (/ sin2phi alphay) alphay)))
     (/ (fma (* 0.5 u0) u0 u0) (+ (/ sin2phi (* alphay alphay)) 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);
	float tmp;
	if (t_0 <= -0.003000000026077032f) {
		tmp = -t_0 / (t_1 + ((sin2phi / alphay) / alphay));
	} else {
		tmp = fmaf((0.5f * u0), u0, u0) / ((sin2phi / (alphay * alphay)) + t_1);
	}
	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(-0.003000000026077032))
		tmp = Float32(Float32(-t_0) / Float32(t_1 + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(fma(Float32(Float32(0.5) * u0), u0, u0) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + t_1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

      1. lift-*.f32 N/A

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

      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 91.9

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

   3. Applied rewrites 91.9%

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

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

   1. Initial program 49.5%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. 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 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 97.9

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

   6. Applied rewrites 97.9%

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

      1. flip-- 97.9

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

      2. metadata-eval 97.9

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

      3. diff-log 97.9

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

      4. lift-+.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. lift-*.f32 97.9

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

      10. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot u0, u0, u0\right)}{\mathsf{Rewrite=>}\left(lift-+.f32, \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

      11. lift-*.f32 N/A

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

      12. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot u0, u0, u0\right)}{\mathsf{Rewrite<=}\left(lift-+.f32, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

   8. Applied rewrites 97.9%

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

Alternative 3: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ t_1 := \log \left(1 - u0\right)\\ t_2 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;t\_1 \leq -0.003000000026077032:\\ \;\;\;\;\frac{-t\_1}{t\_2 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{t\_0 + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay)))
        (t_1 (log (- 1.0 u0)))
        (t_2 (/ cos2phi (* alphax alphax))))
   (if (<= t_1 -0.003000000026077032)
     (/ (- t_1) (+ t_2 t_0))
     (/ (fma (* 0.5 u0) u0 u0) (+ t_0 t_2)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float t_1 = logf((1.0f - u0));
	float t_2 = cos2phi / (alphax * alphax);
	float tmp;
	if (t_1 <= -0.003000000026077032f) {
		tmp = -t_1 / (t_2 + t_0);
	} else {
		tmp = fmaf((0.5f * u0), u0, u0) / (t_0 + t_2);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	t_1 = log(Float32(Float32(1.0) - u0))
	t_2 = Float32(cos2phi / Float32(alphax * alphax))
	tmp = Float32(0.0)
	if (t_1 <= Float32(-0.003000000026077032))
		tmp = Float32(Float32(-t_1) / Float32(t_2 + t_0));
	else
		tmp = Float32(fma(Float32(Float32(0.5) * u0), u0, u0) / Float32(t_0 + t_2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.9%

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

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

   1. Initial program 49.5%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. 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 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 97.9

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

   6. Applied rewrites 97.9%

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

      1. flip-- 97.9

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

      2. metadata-eval 97.9

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

      3. diff-log 97.9

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

      4. lift-+.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. lift-*.f32 97.9

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

      10. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot u0, u0, u0\right)}{\mathsf{Rewrite=>}\left(lift-+.f32, \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

      11. lift-*.f32 N/A

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

      12. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot u0, u0, u0\right)}{\mathsf{Rewrite<=}\left(lift-+.f32, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

   8. Applied rewrites 97.9%

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

Alternative 4: 92.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 30:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 30.0)
     (/ (fma (* 0.5 u0) u0 u0) (+ t_0 (/ cos2phi (* alphax alphax))))
     (/ (- (log1p (- u0))) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 30.0f) {
		tmp = fmaf((0.5f * u0), u0, u0) / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = -log1pf(-u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(30.0))
		tmp = Float32(fma(Float32(Float32(0.5) * u0), u0, u0) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 30

   1. Initial program 55.4%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. 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 87.0

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

   4. Applied rewrites 87.0%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 87.1

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

   6. Applied rewrites 87.1%

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

      1. flip-- 87.1

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

      2. metadata-eval 87.1

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

      3. diff-log 87.1

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

      4. lift-+.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. lift-*.f32 87.1

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

      10. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot u0, u0, u0\right)}{\mathsf{Rewrite=>}\left(lift-+.f32, \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)} \]

      11. lift-*.f32 N/A

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

      12. lift-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot u0, u0, u0\right)}{\mathsf{Rewrite<=}\left(lift-+.f32, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right)} \]

   8. Applied rewrites 87.1%

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

   if 30 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.4%

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

   2. Taylor expanded in sin2phi around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 65.4

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

   4. Applied rewrites 65.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 97.7

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

   6. Applied rewrites 97.7%

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

   7. Taylor expanded in alphax around inf

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{\color{blue}{{alphay}^{2}}}} \]
   8. Step-by-step derivation

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 96.9

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

   9. Applied rewrites 96.9%

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

Alternative 5: 92.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 30:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 30.0)
     (/ (* (fma 0.5 u0 1.0) u0) (+ (/ cos2phi (* alphax alphax)) t_0))
     (/ (- (log1p (- u0))) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 30.0f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = -log1pf(-u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(30.0))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 30

   1. Initial program 55.4%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. 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 87.0

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

   4. Applied rewrites 87.0%

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

   if 30 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.4%

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

   2. Taylor expanded in sin2phi around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 65.4

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

   4. Applied rewrites 65.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 97.7

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

   6. Applied rewrites 97.7%

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

   7. Taylor expanded in alphax around inf

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{\color{blue}{{alphay}^{2}}}} \]
   8. Step-by-step derivation

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 96.9

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

   9. Applied rewrites 96.9%

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

Alternative 6: 86.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 5:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 5.0)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/ (- (log1p (- u0))) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 5.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = -log1pf(-u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(5.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5

   1. Initial program 55.4%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 74.6%

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

   if 5 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.2%

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

   2. Taylor expanded in sin2phi around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 65.2

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

   4. Applied rewrites 65.2%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 97.7

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

   6. Applied rewrites 97.7%

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

   7. Taylor expanded in alphax around inf

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{\color{blue}{{alphay}^{2}}}} \]
   8. Step-by-step derivation

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 96.6

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

   9. Applied rewrites 96.6%

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

Alternative 7: 82.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u0\right)\\ \mathbf{if}\;t\_0 \leq -0.000295000005280599:\\ \;\;\;\;-\left(alphay \cdot alphay\right) \cdot \frac{t\_0}{sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0))))
   (if (<= t_0 -0.000295000005280599)
     (- (* (* alphay alphay) (/ t_0 sin2phi)))
     (/ 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 <= -0.000295000005280599f) {
		tmp = -((alphay * alphay) * (t_0 / sin2phi));
	} else {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	}
	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 = log((1.0e0 - u0))
    if (t_0 <= (-0.000295000005280599e0)) then
        tmp = -((alphay * alphay) * (t_0 / sin2phi))
    else
        tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.000295000005280599))
		tmp = Float32(-Float32(Float32(alphay * alphay) * Float32(t_0 / sin2phi)));
	else
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log((single(1.0) - u0));
	tmp = single(0.0);
	if (t_0 <= single(-0.000295000005280599))
		tmp = -((alphay * alphay) * (t_0 / sin2phi));
	else
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -2.95000005e-4

   1. Initial program 88.1%

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

   2. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right) \]

      2. lower-neg.f32 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 68.6

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

   4. Applied rewrites 68.6%

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

   if -2.95000005e-4 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 44.7%

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

   2. Taylor expanded in u0 around 0

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

   4. Applied rewrites 91.0%

      \[\leadsto \frac{\color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ t_1 := \mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\\ \mathbf{if}\;t\_0 \leq 1.5000000170217692 \cdot 10^{-18}:\\ \;\;\;\;\frac{t\_1}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))) (t_1 (* (fma 0.5 u0 1.0) u0)))
   (if (<= t_0 1.5000000170217692e-18)
     (/ t_1 (/ cos2phi (* alphax alphax)))
     (/ 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 = fmaf(0.5f, u0, 1.0f) * u0;
	float tmp;
	if (t_0 <= 1.5000000170217692e-18f) {
		tmp = t_1 / (cos2phi / (alphax * alphax));
	} else {
		tmp = t_1 / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	t_1 = Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0)
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.5000000170217692e-18))
		tmp = Float32(t_1 / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(t_1 / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.50000002e-18

   1. Initial program 54.5%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. 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 86.9

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

   4. Applied rewrites 86.9%

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

   5. Taylor expanded in alphax around 0

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

      1. pow2 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 65.1

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

   7. Applied rewrites 65.1%

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

   if 1.50000002e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.4%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. 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 87.2

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in alphax around inf

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

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 77.5

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

   7. Applied rewrites 77.5%

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

Alternative 9: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 1.5000000170217692 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 1.5000000170217692e-18)
     (/ (* (fma 0.5 u0 1.0) u0) (/ cos2phi (* alphax alphax)))
     (/ u0 t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 1.5000000170217692e-18f) {
		tmp = (fmaf(0.5f, u0, 1.0f) * u0) / (cos2phi / (alphax * alphax));
	} else {
		tmp = u0 / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.5000000170217692e-18))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(u0 / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.50000002e-18

   1. Initial program 54.5%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   3. 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 86.9

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

   4. Applied rewrites 86.9%

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

   5. Taylor expanded in alphax around 0

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

      1. pow2 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-/.f32 65.1

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

   7. Applied rewrites 65.1%

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

   if 1.50000002e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.4%

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

   2. Taylor expanded in alphax around inf

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

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 57.2

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

   4. Applied rewrites 57.2%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0}}{\frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. flip-- 68.1

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

      2. metadata-eval 68.1

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

      3. diff-log 68.1

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

   7. Applied rewrites 68.1%

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

Alternative 10: 65.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 1.5000000170217692 \cdot 10^{-18}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 1.5000000170217692e-18)
     (* alphax (* alphax (/ u0 cos2phi)))
     (/ u0 t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 1.5000000170217692e-18f) {
		tmp = alphax * (alphax * (u0 / cos2phi));
	} else {
		tmp = u0 / t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 1.5000000170217692e-18) then
        tmp = alphax * (alphax * (u0 / cos2phi))
    else
        tmp = u0 / t_0
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.5000000170217692e-18))
		tmp = Float32(alphax * Float32(alphax * Float32(u0 / cos2phi)));
	else
		tmp = Float32(u0 / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(1.5000000170217692e-18))
		tmp = alphax * (alphax * (u0 / cos2phi));
	else
		tmp = u0 / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.50000002e-18

   1. Initial program 54.5%

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

   2. Taylor expanded in alphax around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 43.5

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

   4. Applied rewrites 43.5%

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

   5. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

         \[\leadsto \frac{{alphax}^{2} \cdot u0}{cos2phi} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{{alphax}^{2} \cdot u0}{cos2phi} \]

      3. pow2 N/A

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

      4. lift-*.f32 57.4

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

   7. Applied rewrites 57.4%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. pow2 N/A

         \[\leadsto \frac{{alphax}^{2} \cdot u0}{cos2phi} \]

      5. associate-/l* N/A

         \[\leadsto {alphax}^{2} \cdot \frac{u0}{\color{blue}{cos2phi}} \]

      6. lower-*.f32 N/A

         \[\leadsto {alphax}^{2} \cdot \frac{u0}{\color{blue}{cos2phi}} \]

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. lower-/.f32 57.4

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

   9. Applied rewrites 57.4%

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

       1. lift-*.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. lift-/.f32 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f32 N/A

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

       6. lower-*.f32 N/A

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

       7. lift-/.f32 57.4

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

   11. Applied rewrites 57.4%

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

   if 1.50000002e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.4%

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

   2. Taylor expanded in alphax around inf

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

      1. pow2 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 57.2

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

   4. Applied rewrites 57.2%

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

   5. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0}}{\frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. flip-- 68.1

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

      2. metadata-eval 68.1

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

      3. diff-log 68.1

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

   7. Applied rewrites 68.1%

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

Alternative 11: 23.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi} \end{array} \]

FPCore

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

C

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

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) * u0) / cos2phi
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.8%

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

2. Taylor expanded in alphax around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-log.f32 N/A

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

   9. lift--.f32 21.8

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

4. Applied rewrites 21.8%

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

5. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

      \[\leadsto \frac{{alphax}^{2} \cdot u0}{cos2phi} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{{alphax}^{2} \cdot u0}{cos2phi} \]

   3. pow2 N/A

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

   4. lift-*.f32 23.4

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

7. Applied rewrites 23.4%

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

Alternative 12: 23.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right) \end{array} \]

FPCore

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

C

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

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 * (u0 / cos2phi))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.8%

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

2. Taylor expanded in alphax around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-log.f32 N/A

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

   9. lift--.f32 21.8

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

4. Applied rewrites 21.8%

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

5. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

      \[\leadsto \frac{{alphax}^{2} \cdot u0}{cos2phi} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{{alphax}^{2} \cdot u0}{cos2phi} \]

   3. pow2 N/A

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

   4. lift-*.f32 23.4

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

7. Applied rewrites 23.4%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. pow2 N/A

      \[\leadsto \frac{{alphax}^{2} \cdot u0}{cos2phi} \]

   5. associate-/l* N/A

      \[\leadsto {alphax}^{2} \cdot \frac{u0}{\color{blue}{cos2phi}} \]

   6. lower-*.f32 N/A

      \[\leadsto {alphax}^{2} \cdot \frac{u0}{\color{blue}{cos2phi}} \]

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. lower-/.f32 23.4

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

9. Applied rewrites 23.4%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. associate-*l* N/A

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

    5. lower-*.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lift-/.f32 23.4

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

11. Applied rewrites 23.4%

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

Reproduce

herbie shell --seed 2025133 
(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)))))