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

Percentage Accurate: 60.9% → 98.3%

Time: 6.2s

Alternatives: 13

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

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

MATLAB

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

Initial Program: 60.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (+ (/ (/ 1.0 (/ alphax cos2phi)) alphax) (/ sin2phi (* alphay alphay)))))

C

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

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(Float32(Float32(1.0) / Float32(alphax / cos2phi)) / alphax) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 60.9%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3

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

3. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-/.f32 N/A

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

   5. lift-/.f32 98.3

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

5. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

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

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

   4. lower-unsound-/.f32 98.2

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

7. Applied rewrites 98.2%

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

Alternative 2: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.9%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3

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

3. Applied rewrites 98.3%

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

Alternative 3: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.002420000033453107:\\ \;\;\;\;\frac{u0 \cdot \left(1 + 0.5 \cdot u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{\mathsf{fma}\left(\frac{cos2phi}{alphax \cdot alphax}, alphay, \frac{sin2phi}{alphay}\right)} \cdot \left(-\log \left(1 - u0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= u0 0.002420000033453107)
   (/
    (* u0 (+ 1.0 (* 0.5 u0)))
    (+ (/ (/ cos2phi alphax) alphax) (/ (/ sin2phi alphay) alphay)))
   (*
    (/ alphay (fma (/ cos2phi (* alphax alphax)) alphay (/ sin2phi alphay)))
    (- (log (- 1.0 u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (u0 <= 0.002420000033453107f) {
		tmp = (u0 * (1.0f + (0.5f * u0))) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = (alphay / fmaf((cos2phi / (alphax * alphax)), alphay, (sin2phi / alphay))) * -logf((1.0f - u0));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.002420000033453107))
		tmp = Float32(Float32(u0 * Float32(Float32(1.0) + Float32(Float32(0.5) * u0))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(alphay / fma(Float32(cos2phi / Float32(alphax * alphax)), alphay, Float32(sin2phi / alphay))) * Float32(-log(Float32(Float32(1.0) - u0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00242000003

   1. Initial program 60.9%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f32 N/A

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

      5. lift-/.f32 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 87.0

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

   8. Applied rewrites 87.0%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 87.0

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

   10. Applied rewrites 87.0%

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

   if 0.00242000003 < u0

   1. Initial program 60.9%

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

      1. lift-/.f32 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

   3. Applied rewrites 61.3%

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

Alternative 4: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.002899999963119626:\\ \;\;\;\;\frac{u0 \cdot \left(1 + 0.5 \cdot u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\frac{sin2phi}{\left(-alphay\right) \cdot alphay} - \frac{cos2phi}{alphax \cdot alphax}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= u0 0.002899999963119626)
   (/
    (* u0 (+ 1.0 (* 0.5 u0)))
    (+ (/ (/ cos2phi alphax) alphax) (/ (/ sin2phi alphay) alphay)))
   (/
    (log (- 1.0 u0))
    (- (/ sin2phi (* (- alphay) alphay)) (/ cos2phi (* alphax alphax))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (u0 <= 0.002899999963119626f) {
		tmp = (u0 * (1.0f + (0.5f * u0))) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = logf((1.0f - u0)) / ((sin2phi / (-alphay * alphay)) - (cos2phi / (alphax * alphax)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (u0 <= 0.002899999963119626e0) then
        tmp = (u0 * (1.0e0 + (0.5e0 * u0))) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) / alphay))
    else
        tmp = log((1.0e0 - u0)) / ((sin2phi / (-alphay * alphay)) - (cos2phi / (alphax * alphax)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.002899999963119626))
		tmp = Float32(Float32(u0 * Float32(Float32(1.0) + Float32(Float32(0.5) * u0))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(log(Float32(Float32(1.0) - u0)) / Float32(Float32(sin2phi / Float32(Float32(-alphay) * alphay)) - Float32(cos2phi / Float32(alphax * alphax))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (u0 <= single(0.002899999963119626))
		tmp = (u0 * (single(1.0) + (single(0.5) * u0))) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) / alphay));
	else
		tmp = log((single(1.0) - u0)) / ((sin2phi / (-alphay * alphay)) - (cos2phi / (alphax * alphax)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00289999996

   1. Initial program 60.9%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f32 N/A

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

      5. lift-/.f32 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 87.0

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

   8. Applied rewrites 87.0%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. lower-/.f32 87.0

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

   10. Applied rewrites 87.0%

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

   if 0.00289999996 < u0

   1. Initial program 60.9%

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

      1. lift-/.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. add-flip N/A

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

      8. sub-negate N/A

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

      9. lower--.f32 N/A

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

      10. lift-/.f32 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lower-/.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f32 N/A

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

      16. lower-neg.f32 60.9

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

   3. Applied rewrites 60.9%

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

Alternative 5: 96.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u0\right)\\ \mathbf{if}\;t\_0 \leq -0.002899999963119626:\\ \;\;\;\;\frac{t\_0}{\frac{sin2phi}{\left(-alphay\right) \cdot alphay} - \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u0 \cdot u0, 0.5, u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0))))
   (if (<= t_0 -0.002899999963119626)
     (/
      t_0
      (- (/ sin2phi (* (- alphay) alphay)) (/ cos2phi (* alphax alphax))))
     (/
      (fma (* u0 u0) 0.5 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 <= -0.002899999963119626f) {
		tmp = t_0 / ((sin2phi / (-alphay * alphay)) - (cos2phi / (alphax * alphax)));
	} else {
		tmp = fmaf((u0 * u0), 0.5f, u0) / (((cos2phi / alphax) / alphax) + (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.002899999963119626))
		tmp = Float32(t_0 / Float32(Float32(sin2phi / Float32(Float32(-alphay) * alphay)) - Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(fma(Float32(u0 * u0), Float32(0.5), u0) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.00289999996

   1. Initial program 60.9%

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

      1. lift-/.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. add-flip N/A

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

      8. sub-negate N/A

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

      9. lower--.f32 N/A

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

      10. lift-/.f32 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lower-/.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f32 N/A

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

      16. lower-neg.f32 60.9

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

   3. Applied rewrites 60.9%

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

   if -0.00289999996 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 60.9%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f32 N/A

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

      5. lift-/.f32 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 87.0

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

   8. Applied rewrites 87.0%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f32 N/A

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

      10. lower-unsound-*.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-unsound-*.f32 87.2

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

   10. Applied rewrites 87.2%

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

Alternative 6: 87.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.9%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3

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

3. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-/.f32 N/A

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

   5. lift-/.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 87.0

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

8. Applied rewrites 87.0%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-unsound-*.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-unsound-*.f32 87.2

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

10. Applied rewrites 87.2%

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

Alternative 7: 87.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.9%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3

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

3. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-/.f32 N/A

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

   5. lift-/.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 87.0

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

8. Applied rewrites 87.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 87.0

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 87.0

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

10. Applied rewrites 87.0%

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

Alternative 8: 87.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.9%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3

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

3. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-/.f32 N/A

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

   5. lift-/.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 87.0

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

8. Applied rewrites 87.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 87.0

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 87.0

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

   8. lower-fma.f32 N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-fma.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-fma.f32 N/A

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

10. Applied rewrites 87.1%

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

Alternative 9: 75.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.9%

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

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 75.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 75.6

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

6. Applied rewrites 75.6%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. add-to-fraction N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-fma.f32 75.8

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

8. Applied rewrites 75.8%

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

Alternative 10: 75.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 / (((cos2phi / alphax) / alphax) + (sin2phi / (alphay * alphay)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.9%

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

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 75.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-/.f32 N/A

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

   5. lift-/.f32 75.6

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

6. Applied rewrites 75.6%

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

Alternative 11: 75.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.9%

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

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 75.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 75.6

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

6. Applied rewrites 75.6%

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

Alternative 12: 75.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.9%

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

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 75.6%

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

Alternative 13: 23.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 / (cos2phi / (alphax * alphax))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.9%

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

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 75.6%

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

5. Taylor expanded in alphax around 0

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

   1. lower-/.f32 N/A

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

   2. lower-pow.f32 23.6

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

7. Applied rewrites 23.6%

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

   1. lift-pow.f32 N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 23.6

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

9. Applied rewrites 23.6%

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(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)))))