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

Percentage Accurate: 60.2% → 98.3%

Time: 11.7s

Alternatives: 23

Speedup: 5.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Initial Program: 60.2% 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, 1.0× 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 62.5%

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

   1. remove-double-neg N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. neg-log N/A

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

   6. neg-log N/A

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

   7. remove-double-div N/A

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

   8. lift--.f32 N/A

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

   9. *-lft-identity N/A

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

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

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

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

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

   12. *-lft-identity N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 98.5

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

   15. lift-+.f32 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f32 98.5

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

4. Applied rewrites 98.5%

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

Alternative 2: 94.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u0\right) \leq -0.02500000037252903:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{log1p}\left(-u0\right)}{-sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right) \cdot u0, u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (log (- 1.0 u0)) -0.02500000037252903)
   (/ (* (* alphay alphay) (log1p (- u0))) (- sin2phi))
   (/
    (fma (* (fma (fma 0.25 u0 0.3333333333333333) u0 0.5) u0) u0 u0)
    (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (logf((1.0f - u0)) <= -0.02500000037252903f) {
		tmp = ((alphay * alphay) * log1pf(-u0)) / -sin2phi;
	} else {
		tmp = fmaf((fmaf(fmaf(0.25f, u0, 0.3333333333333333f), u0, 0.5f) * u0), u0, u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u0)) <= Float32(-0.02500000037252903))
		tmp = Float32(Float32(Float32(alphay * alphay) * log1p(Float32(-u0))) / Float32(-sin2phi));
	else
		tmp = Float32(fma(Float32(fma(fma(Float32(0.25), u0, Float32(0.3333333333333333)), u0, Float32(0.5)) * u0), u0, u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.3%

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

   3. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 N/A

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

      13. lower-neg.f32 85.1

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

   5. Applied rewrites 85.1%

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

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

   1. Initial program 54.5%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 98.2

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

   5. Applied rewrites 98.2%

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

   7. Applied rewrites 98.6%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u0\right) \leq -0.02500000037252903:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \mathsf{log1p}\left(-u0\right)}{-sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5\right) \cdot u0, u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 300:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 300.0)
     (/ (fma (* 0.5 u0) u0 u0) (+ (/ cos2phi (* alphax alphax)) t_0))
     (/
      (*
       (* alphay alphay)
       (*
        (- (* (- (* (- (* -0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0)
        u0))
      (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 300.0f) {
		tmp = fmaf((0.5f * u0), u0, u0) / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = ((alphay * alphay) * (((((((-0.25f * u0) - 0.3333333333333333f) * u0) - 0.5f) * u0) - 1.0f) * u0)) / -sin2phi;
	}
	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(300.0))
		tmp = Float32(fma(Float32(Float32(0.5) * u0), u0, u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333)) * u0) - Float32(0.5)) * u0) - Float32(1.0)) * u0)) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 300

   1. Initial program 52.6%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 85.7

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

   5. Applied rewrites 85.7%

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

   7. Applied rewrites 85.9%

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

   if 300 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 70.4%

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

   3. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 N/A

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

      13. lower-neg.f32 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 92.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 300:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot u0, u0, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 300:\\ \;\;\;\;\mathsf{fma}\left(0.5, u0, 1\right) \cdot \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 300.0)
     (* (fma 0.5 u0 1.0) (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0)))
     (/
      (*
       (* alphay alphay)
       (*
        (- (* (- (* (- (* -0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0)
        u0))
      (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 300.0f) {
		tmp = fmaf(0.5f, u0, 1.0f) * (u0 / ((cos2phi / (alphax * alphax)) + t_0));
	} else {
		tmp = ((alphay * alphay) * (((((((-0.25f * u0) - 0.3333333333333333f) * u0) - 0.5f) * u0) - 1.0f) * u0)) / -sin2phi;
	}
	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(300.0))
		tmp = Float32(fma(Float32(0.5), u0, Float32(1.0)) * Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333)) * u0) - Float32(0.5)) * u0) - Float32(1.0)) * u0)) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 300

   1. Initial program 52.6%

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

      1. remove-double-neg N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. remove-double-div N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 98.6

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f32 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 85.7%

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

   9. Applied rewrites 85.8%

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

   if 300 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 70.4%

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

   3. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 N/A

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

      13. lower-neg.f32 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 92.6%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 300:\\ \;\;\;\;\mathsf{fma}\left(0.5, u0, 1\right) \cdot \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 300:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right)}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 300.0)
     (* (/ (fma 0.5 u0 1.0) (+ t_0 (/ cos2phi (* alphax alphax)))) u0)
     (/
      (*
       (* alphay alphay)
       (*
        (- (* (- (* (- (* -0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0)
        u0))
      (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 300.0f) {
		tmp = (fmaf(0.5f, u0, 1.0f) / (t_0 + (cos2phi / (alphax * alphax)))) * u0;
	} else {
		tmp = ((alphay * alphay) * (((((((-0.25f * u0) - 0.3333333333333333f) * u0) - 0.5f) * u0) - 1.0f) * u0)) / -sin2phi;
	}
	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(300.0))
		tmp = Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax)))) * u0);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333)) * u0) - Float32(0.5)) * u0) - Float32(1.0)) * u0)) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 300

   1. Initial program 52.6%

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

      1. remove-double-neg N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. remove-double-div N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 98.6

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f32 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 85.7%

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

   if 300 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 70.4%

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

   3. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 N/A

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

      13. lower-neg.f32 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 92.6%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 300:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, u0, 1\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.6000000730164174 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 2.6000000730164174e-7)
   (/
    u0
    (/
     (fma (* alphay alphay) (/ cos2phi (* alphax alphax)) sin2phi)
     (* alphay alphay)))
   (/
    (*
     (* alphay alphay)
     (* (- (* (- (* (- (* -0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0) u0))
    (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.6000000730164174e-7f) {
		tmp = u0 / (fmaf((alphay * alphay), (cos2phi / (alphax * alphax)), sin2phi) / (alphay * alphay));
	} else {
		tmp = ((alphay * alphay) * (((((((-0.25f * u0) - 0.3333333333333333f) * u0) - 0.5f) * u0) - 1.0f) * u0)) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.6000000730164174e-7))
		tmp = Float32(u0 / Float32(fma(Float32(alphay * alphay), Float32(cos2phi / Float32(alphax * alphax)), sin2phi) / Float32(alphay * alphay)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333)) * u0) - Float32(0.5)) * u0) - Float32(1.0)) * u0)) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.60000007e-7

   1. Initial program 53.5%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. lower-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

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

      9. lower-*.f32 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in alphay around 0

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

   8. Applied rewrites 74.8%

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

   if 2.60000007e-7 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.7%

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

   3. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 N/A

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

      13. lower-neg.f32 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 90.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.6000000730164174 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{\mathsf{fma}\left(alphay \cdot alphay, \frac{cos2phi}{alphax \cdot alphax}, sin2phi\right)}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 2.6000000730164174 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 2.6000000730164174e-7)
     (/ u0 (+ t_0 (/ cos2phi (* alphax alphax))))
     (/
      (*
       (* alphay alphay)
       (*
        (- (* (- (* (- (* -0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0)
        u0))
      (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 2.6000000730164174e-7f) {
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = ((alphay * alphay) * (((((((-0.25f * u0) - 0.3333333333333333f) * u0) - 0.5f) * u0) - 1.0f) * u0)) / -sin2phi;
	}
	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 <= 2.6000000730164174e-7) then
        tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)))
    else
        tmp = ((alphay * alphay) * ((((((((-0.25e0) * u0) - 0.3333333333333333e0) * u0) - 0.5e0) * u0) - 1.0e0) * u0)) / -sin2phi
    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(2.6000000730164174e-7))
		tmp = Float32(u0 / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333)) * u0) - Float32(0.5)) * u0) - Float32(1.0)) * u0)) / Float32(-sin2phi));
	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(2.6000000730164174e-7))
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	else
		tmp = ((alphay * alphay) * (((((((single(-0.25) * u0) - single(0.3333333333333333)) * u0) - single(0.5)) * u0) - single(1.0)) * u0)) / -sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.60000007e-7

   1. Initial program 53.5%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. lower-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

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

      9. lower-*.f32 74.7

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

   5. Applied rewrites 74.7%

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

   if 2.60000007e-7 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.7%

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

   3. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 N/A

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

      13. lower-neg.f32 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 90.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.6000000730164174 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 91.6

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

5. Applied rewrites 91.6%

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

7. Applied rewrites 91.8%

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

Alternative 9: 93.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 91.6

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

5. Applied rewrites 91.6%

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

7. Applied rewrites 91.8%

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

Alternative 10: 93.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 91.6

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

5. Applied rewrites 91.6%

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

Alternative 11: 83.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 2.6000000730164174 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(-0.3333333333333333 \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 2.6000000730164174e-7)
     (/ u0 (+ t_0 (/ cos2phi (* alphax alphax))))
     (/
      (*
       (* alphay alphay)
       (* (- (* (- (* -0.3333333333333333 u0) 0.5) u0) 1.0) u0))
      (- sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 2.6000000730164174e-7f) {
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = ((alphay * alphay) * (((((-0.3333333333333333f * u0) - 0.5f) * u0) - 1.0f) * u0)) / -sin2phi;
	}
	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 <= 2.6000000730164174e-7) then
        tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)))
    else
        tmp = ((alphay * alphay) * ((((((-0.3333333333333333e0) * u0) - 0.5e0) * u0) - 1.0e0) * u0)) / -sin2phi
    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(2.6000000730164174e-7))
		tmp = Float32(u0 / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(Float32(Float32(Float32(Float32(Float32(-0.3333333333333333) * u0) - Float32(0.5)) * u0) - Float32(1.0)) * u0)) / Float32(-sin2phi));
	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(2.6000000730164174e-7))
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	else
		tmp = ((alphay * alphay) * (((((single(-0.3333333333333333) * u0) - single(0.5)) * u0) - single(1.0)) * u0)) / -sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.60000007e-7

   1. Initial program 53.5%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. lower-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

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

      9. lower-*.f32 74.7

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

   5. Applied rewrites 74.7%

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

   if 2.60000007e-7 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.7%

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

   3. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 N/A

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

      13. lower-neg.f32 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 88.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.6000000730164174 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(-0.3333333333333333 \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 91.6

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

5. Applied rewrites 91.6%

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

7. Applied rewrites 91.8%

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

8. Taylor expanded in u0 around 0

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

10. Applied rewrites 89.9%

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

Alternative 13: 91.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 89.7

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

5. Applied rewrites 89.7%

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

Alternative 14: 77.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.000000067449534 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(0.5, u0, 1\right)}{cos2phi}\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(-0.3333333333333333 \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.000000067449534e-16)
   (* (* (* alphax alphax) (/ (fma 0.5 u0 1.0) cos2phi)) u0)
   (/
    (*
     (* alphay alphay)
     (* (- (* (- (* -0.3333333333333333 u0) 0.5) u0) 1.0) u0))
    (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.000000067449534e-16f) {
		tmp = ((alphax * alphax) * (fmaf(0.5f, u0, 1.0f) / cos2phi)) * u0;
	} else {
		tmp = ((alphay * alphay) * (((((-0.3333333333333333f * u0) - 0.5f) * u0) - 1.0f) * u0)) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.000000067449534e-16))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(fma(Float32(0.5), u0, Float32(1.0)) / cos2phi)) * u0);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(Float32(Float32(Float32(Float32(Float32(-0.3333333333333333) * u0) - Float32(0.5)) * u0) - Float32(1.0)) * u0)) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.00000007e-16

   1. Initial program 55.8%

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

      1. remove-double-neg N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. remove-double-div N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 98.8

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f32 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in alphax around 0

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

   10. Applied rewrites 62.0%

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

   if 4.00000007e-16 < sin2phi

   1. Initial program 64.7%

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

   3. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 N/A

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

      13. lower-neg.f32 94.0

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 85.3%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.000000067449534 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(0.5, u0, 1\right)}{cos2phi}\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(\left(\left(-0.3333333333333333 \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(0.5, u0, 1\right)}{cos2phi}\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-alphay, alphay, -0.5 \cdot \left(\left(alphay \cdot alphay\right) \cdot u0\right)\right) \cdot u0}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.9999998245167e-14)
   (* (* (* alphax alphax) (/ (fma 0.5 u0 1.0) cos2phi)) u0)
   (/
    (* (fma (- alphay) alphay (* -0.5 (* (* alphay alphay) u0))) u0)
    (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.9999998245167e-14f) {
		tmp = ((alphax * alphax) * (fmaf(0.5f, u0, 1.0f) / cos2phi)) * u0;
	} else {
		tmp = (fmaf(-alphay, alphay, (-0.5f * ((alphay * alphay) * u0))) * u0) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.9999998245167e-14))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(fma(Float32(0.5), u0, Float32(1.0)) / cos2phi)) * u0);
	else
		tmp = Float32(Float32(fma(Float32(-alphay), alphay, Float32(Float32(-0.5) * Float32(Float32(alphay * alphay) * u0))) * u0) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.99999982e-14

   1. Initial program 54.3%

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

      1. remove-double-neg N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. remove-double-div N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 98.7

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f32 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in alphax around 0

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

   10. Applied rewrites 58.8%

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

   if 9.99999982e-14 < sin2phi

   1. Initial program 65.9%

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

   3. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 N/A

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

      13. lower-neg.f32 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 84.6%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(0.5, u0, 1\right)}{cos2phi}\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-alphay, alphay, -0.5 \cdot \left(\left(alphay \cdot alphay\right) \cdot u0\right)\right) \cdot u0}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(0.5, u0, 1\right)}{cos2phi}\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(\left(alphay \cdot alphay\right) \cdot u0\right)\right)}{sin2phi} \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.9999998245167e-14)
   (* (* (* alphax alphax) (/ (fma 0.5 u0 1.0) cos2phi)) u0)
   (* (/ (fma alphay alphay (* 0.5 (* (* alphay alphay) u0))) sin2phi) u0)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.9999998245167e-14f) {
		tmp = ((alphax * alphax) * (fmaf(0.5f, u0, 1.0f) / cos2phi)) * u0;
	} else {
		tmp = (fmaf(alphay, alphay, (0.5f * ((alphay * alphay) * u0))) / sin2phi) * u0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.9999998245167e-14))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(fma(Float32(0.5), u0, Float32(1.0)) / cos2phi)) * u0);
	else
		tmp = Float32(Float32(fma(alphay, alphay, Float32(Float32(0.5) * Float32(Float32(alphay * alphay) * u0))) / sin2phi) * u0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.99999982e-14

   1. Initial program 54.3%

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

      1. remove-double-neg N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. remove-double-div N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 98.7

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f32 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in alphax around 0

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

   10. Applied rewrites 58.8%

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

   if 9.99999982e-14 < sin2phi

   1. Initial program 65.9%

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

   3. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 N/A

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

      13. lower-neg.f32 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 84.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(0.5, u0, 1\right)}{cos2phi}\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(alphay, alphay, 0.5 \cdot \left(\left(alphay \cdot alphay\right) \cdot u0\right)\right)}{sin2phi} \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 74.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(0.5, u0, 1\right)}{cos2phi}\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot u0 - 1\right) \cdot u0\right) \cdot \frac{\left(-alphay\right) \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.9999998245167e-14)
   (* (* (* alphax alphax) (/ (fma 0.5 u0 1.0) cos2phi)) u0)
   (* (* (- (* -0.5 u0) 1.0) u0) (/ (* (- alphay) alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.9999998245167e-14f) {
		tmp = ((alphax * alphax) * (fmaf(0.5f, u0, 1.0f) / cos2phi)) * u0;
	} else {
		tmp = (((-0.5f * u0) - 1.0f) * u0) * ((-alphay * alphay) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.9999998245167e-14))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(fma(Float32(0.5), u0, Float32(1.0)) / cos2phi)) * u0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-0.5) * u0) - Float32(1.0)) * u0) * Float32(Float32(Float32(-alphay) * alphay) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.99999982e-14

   1. Initial program 54.3%

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

      1. remove-double-neg N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. remove-double-div N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 98.7

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f32 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in alphax around 0

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

   10. Applied rewrites 58.8%

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

   if 9.99999982e-14 < sin2phi

   1. Initial program 65.9%

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

   3. Taylor expanded in alphax around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. lower-neg.f32 N/A

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

      13. lower-neg.f32 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 84.4%

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

   10. Applied rewrites 84.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(0.5, u0, 1\right)}{cos2phi}\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot u0 - 1\right) \cdot u0\right) \cdot \frac{\left(-alphay\right) \cdot alphay}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 74.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(0.5, u0, 1\right)}{cos2phi}\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.9999998245167e-14)
   (* (* (* alphax alphax) (/ (fma 0.5 u0 1.0) cos2phi)) u0)
   (* (* alphay alphay) (/ (* (fma 0.5 u0 1.0) u0) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.9999998245167e-14f) {
		tmp = ((alphax * alphax) * (fmaf(0.5f, u0, 1.0f) / cos2phi)) * u0;
	} else {
		tmp = (alphay * alphay) * ((fmaf(0.5f, u0, 1.0f) * u0) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.9999998245167e-14))
		tmp = Float32(Float32(Float32(alphax * alphax) * Float32(fma(Float32(0.5), u0, Float32(1.0)) / cos2phi)) * u0);
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.99999982e-14

   1. Initial program 54.3%

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

      1. remove-double-neg N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. remove-double-div N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 98.7

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f32 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in alphax around 0

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

   10. Applied rewrites 58.8%

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

   if 9.99999982e-14 < sin2phi

   1. Initial program 65.9%

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

      1. remove-double-neg N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. remove-double-div N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 98.4

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f32 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 85.7%

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

   8. Taylor expanded in alphax around inf

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

   10. Applied rewrites 84.4%

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

Alternative 19: 74.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\\ \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{t\_0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{t\_0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* (fma 0.5 u0 1.0) u0)))
   (if (<= sin2phi 9.9999998245167e-14)
     (* (* alphax alphax) (/ t_0 cos2phi))
     (* (* alphay alphay) (/ t_0 sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = fmaf(0.5f, u0, 1.0f) * u0;
	float tmp;
	if (sin2phi <= 9.9999998245167e-14f) {
		tmp = (alphax * alphax) * (t_0 / cos2phi);
	} else {
		tmp = (alphay * alphay) * (t_0 / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.9999998245167e-14))
		tmp = Float32(Float32(alphax * alphax) * Float32(t_0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(t_0 / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.99999982e-14

   1. Initial program 54.3%

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

      1. remove-double-neg N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. remove-double-div N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 98.7

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f32 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in alphax around 0

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

   10. Applied rewrites 58.7%

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

   if 9.99999982e-14 < sin2phi

   1. Initial program 65.9%

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

      1. remove-double-neg N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. remove-double-div N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 98.4

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f32 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 85.7%

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

   8. Taylor expanded in alphax around inf

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

   10. Applied rewrites 84.4%

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

Alternative 20: 68.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.9999998245167e-14)
   (* (* alphax alphax) (/ (* (fma 0.5 u0 1.0) u0) cos2phi))
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.9999998245167e-14f) {
		tmp = (alphax * alphax) * ((fmaf(0.5f, u0, 1.0f) * u0) / cos2phi);
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.9999998245167e-14))
		tmp = Float32(Float32(alphax * alphax) * Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * u0) / cos2phi));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.99999982e-14

   1. Initial program 54.3%

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

      1. remove-double-neg N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-log.f32 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. remove-double-div N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 98.7

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f32 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in alphax around 0

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

   10. Applied rewrites 58.7%

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

   if 9.99999982e-14 < sin2phi

   1. Initial program 65.9%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. lower-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

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

      9. lower-*.f32 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 74.2%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 66.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;alphax \cdot \frac{alphax \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.9999998245167e-14)
   (* alphax (/ (* alphax u0) cos2phi))
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.9999998245167e-14f) {
		tmp = alphax * ((alphax * u0) / cos2phi);
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
use fmin_fmax_functions
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 9.9999998245167e-14) then
        tmp = alphax * ((alphax * u0) / cos2phi)
    else
        tmp = ((alphay * alphay) * u0) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.9999998245167e-14))
		tmp = Float32(alphax * Float32(Float32(alphax * u0) / cos2phi));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(9.9999998245167e-14))
		tmp = alphax * ((alphax * u0) / cos2phi);
	else
		tmp = ((alphay * alphay) * u0) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.99999982e-14

   1. Initial program 54.3%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. lower-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

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

      9. lower-*.f32 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 52.2%

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

   10. Applied rewrites 52.3%

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

   if 9.99999982e-14 < sin2phi

   1. Initial program 65.9%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      3. lower-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      8. unpow2 N/A

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

      9. lower-*.f32 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 74.2%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;alphax \cdot \frac{alphax \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 23.7% accurate, 6.9× speedup

Math

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

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(alphax * Float32(Float32(alphax * u0) / cos2phi))
end

MATLAB

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

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

   9. lower-*.f32 75.1

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

5. Applied rewrites 75.1%

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

6. Taylor expanded in alphax around 0

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

8. Applied rewrites 21.4%

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

10. Applied rewrites 21.5%

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

11. Final simplification 21.5%

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

Alternative 23: 23.7% accurate, 6.9× 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 62.5%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}} + \frac{cos2phi}{{alphax}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   8. unpow2 N/A

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

   9. lower-*.f32 75.1

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

5. Applied rewrites 75.1%

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

6. Taylor expanded in alphax around 0

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

8. Applied rewrites 21.4%

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

10. Applied rewrites 21.4%

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

11. Final simplification 21.4%

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

Reproduce

herbie shell --seed 2024358 
(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)))))