Numeric.AD.Rank1.Halley:findZero from ad-4.2.4

Percentage Accurate: 82.3% → 94.8%

Time: 3.9s

Alternatives: 7

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))

C

double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}

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(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (((y * 2.0d0) * z) / (((z * 2.0d0) * z) - (y * t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}

Python

def code(x, y, z, t):
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(Float64(z * 2.0) * z) - Float64(y * t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))

C

double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}

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(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (((y * 2.0d0) * z) / (((z * 2.0d0) * z) - (y * t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}

Python

def code(x, y, z, t):
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(Float64(z * 2.0) * z) - Float64(y * t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-128}:\\ \;\;\;\;x - \frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \left(y + y\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+154}:\\ \;\;\;\;x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(z + z, z, \left(-y\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(t \cdot \left(\frac{y}{z} \cdot \frac{y}{z}\right), 0.5, y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.4e-128)
   (- x (* (/ z (fma (- t) y (* (* z z) 2.0))) (+ y y)))
   (if (<= z 2e-181)
     (fma (/ z t) 2.0 x)
     (if (<= z 8e+154)
       (- x (* (+ y y) (/ z (fma (+ z z) z (* (- y) t)))))
       (- x (/ (fma (* t (* (/ y z) (/ y z))) 0.5 y) z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.4e-128) {
		tmp = x - ((z / fma(-t, y, ((z * z) * 2.0))) * (y + y));
	} else if (z <= 2e-181) {
		tmp = fma((z / t), 2.0, x);
	} else if (z <= 8e+154) {
		tmp = x - ((y + y) * (z / fma((z + z), z, (-y * t))));
	} else {
		tmp = x - (fma((t * ((y / z) * (y / z))), 0.5, y) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.4e-128)
		tmp = Float64(x - Float64(Float64(z / fma(Float64(-t), y, Float64(Float64(z * z) * 2.0))) * Float64(y + y)));
	elseif (z <= 2e-181)
		tmp = fma(Float64(z / t), 2.0, x);
	elseif (z <= 8e+154)
		tmp = Float64(x - Float64(Float64(y + y) * Float64(z / fma(Float64(z + z), z, Float64(Float64(-y) * t)))));
	else
		tmp = Float64(x - Float64(fma(Float64(t * Float64(Float64(y / z) * Float64(y / z))), 0.5, y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.4e-128], N[(x - N[(N[(z / N[((-t) * y + N[(N[(z * z), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e-181], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision], If[LessEqual[z, 8e+154], N[(x - N[(N[(y + y), $MachinePrecision] * N[(z / N[(N[(z + z), $MachinePrecision] * z + N[((-y) * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(t * N[(N[(y / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.39999999999999975e-128

   1. Initial program 79.8%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\left(y \cdot 2\right)} \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\left(y \cdot 2\right) \cdot z}}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      4. lift--.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      5. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right) \cdot z} - y \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right)} \cdot z - y \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - \color{blue}{y \cdot t}} \]

      8. associate-/l* N/A

         \[\leadsto x - \color{blue}{\left(y \cdot 2\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(y \cdot 2\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      11. lower-*.f64 N/A

          \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      12. lower-/.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \color{blue}{\frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      13. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right)} \cdot z - y \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z} - y \cdot t} \]

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

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t}} \]

      16. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z} + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right)} \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      18. *-commutative N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(2 \cdot z\right)} \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      19. lower-fma.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(2 \cdot z, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)}} \]

      20. *-commutative N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)} \]

      21. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)} \]

      22. mul-1-neg N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-1 \cdot y\right)} \cdot t\right)} \]

      23. lower-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-1 \cdot y\right) \cdot t}\right)} \]

      24. mul-1-neg N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot t\right)} \]

      25. lower-neg.f64 95.8

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-y\right)} \cdot t\right)} \]

   4. Applied rewrites 95.8%

      \[\leadsto x - \color{blue}{\left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto x - \left(2 \cdot y\right) \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(-y\right) \cdot t\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z + \left(-y\right) \cdot t}} \]

      6. lift-neg.f64 N/A

         \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot t}} \]

      8. *-commutative N/A

         \[\leadsto x - \color{blue}{\frac{z}{\left(z \cdot 2\right) \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \cdot \left(2 \cdot y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{z}{\left(z \cdot 2\right) \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \cdot \left(2 \cdot y\right)} \]

   6. Applied rewrites 95.8%

      \[\leadsto x - \color{blue}{\frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \left(y \cdot 2\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x - \frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \color{blue}{\left(y \cdot 2\right)} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \color{blue}{\left(2 \cdot y\right)} \]

      3. count-2-rev N/A

         \[\leadsto x - \frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \color{blue}{\left(y + y\right)} \]

      4. lift-+.f64 95.8

         \[\leadsto x - \frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \color{blue}{\left(y + y\right)} \]

   8. Applied rewrites 95.8%

      \[\leadsto x - \frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \color{blue}{\left(y + y\right)} \]

   if -3.39999999999999975e-128 < z < 2.00000000000000009e-181

   1. Initial program 85.9%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - -2 \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x - \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{z}}{t} \]

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

         \[\leadsto x + \color{blue}{2 \cdot \frac{z}{t}} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \frac{z}{t} + \color{blue}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{z}{t} \cdot 2 + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \color{blue}{2}, x\right) \]

      6. lower-/.f64 100.0

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, 2, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, 2, x\right)} \]

   if 2.00000000000000009e-181 < z < 8.0000000000000003e154

   1. Initial program 82.4%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\left(y \cdot 2\right)} \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\left(y \cdot 2\right) \cdot z}}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      4. lift--.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      5. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right) \cdot z} - y \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right)} \cdot z - y \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - \color{blue}{y \cdot t}} \]

      8. associate-/l* N/A

         \[\leadsto x - \color{blue}{\left(y \cdot 2\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(y \cdot 2\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      11. lower-*.f64 N/A

          \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      12. lower-/.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \color{blue}{\frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      13. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right)} \cdot z - y \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z} - y \cdot t} \]

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

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t}} \]

      16. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z} + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right)} \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      18. *-commutative N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(2 \cdot z\right)} \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      19. lower-fma.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(2 \cdot z, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)}} \]

      20. *-commutative N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)} \]

      21. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)} \]

      22. mul-1-neg N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-1 \cdot y\right)} \cdot t\right)} \]

      23. lower-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-1 \cdot y\right) \cdot t}\right)} \]

      24. mul-1-neg N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot t\right)} \]

      25. lower-neg.f64 95.9

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-y\right)} \cdot t\right)} \]

   4. Applied rewrites 95.9%

      \[\leadsto x - \color{blue}{\left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]

      2. count-2-rev N/A

         \[\leadsto x - \color{blue}{\left(y + y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]

      3. lower-+.f64 95.9

         \[\leadsto x - \color{blue}{\left(y + y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]

   6. Applied rewrites 95.9%

      \[\leadsto x - \color{blue}{\left(y + y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(-y\right) \cdot t\right)} \]

      2. *-commutative N/A

         \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{2 \cdot z}, z, \left(-y\right) \cdot t\right)} \]

      3. count-2-rev N/A

         \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z + z}, z, \left(-y\right) \cdot t\right)} \]

      4. lower-+.f64 95.9

         \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z + z}, z, \left(-y\right) \cdot t\right)} \]

   8. Applied rewrites 95.9%

      \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z + z}, z, \left(-y\right) \cdot t\right)} \]

   if 8.0000000000000003e154 < z

   1. Initial program 39.8%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto x - \color{blue}{\frac{y + \frac{1}{2} \cdot \frac{t \cdot {y}^{2}}{{z}^{2}}}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{y + \frac{1}{2} \cdot \frac{t \cdot {y}^{2}}{{z}^{2}}}{\color{blue}{z}} \]

      2. +-commutative N/A

         \[\leadsto x - \frac{\frac{1}{2} \cdot \frac{t \cdot {y}^{2}}{{z}^{2}} + y}{z} \]

      3. *-commutative N/A

         \[\leadsto x - \frac{\frac{t \cdot {y}^{2}}{{z}^{2}} \cdot \frac{1}{2} + y}{z} \]

      4. lower-fma.f64 N/A

         \[\leadsto x - \frac{\mathsf{fma}\left(\frac{t \cdot {y}^{2}}{{z}^{2}}, \frac{1}{2}, y\right)}{z} \]

      5. associate-/l* N/A

         \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot \frac{{y}^{2}}{{z}^{2}}, \frac{1}{2}, y\right)}{z} \]

      6. unpow2 N/A

         \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot \frac{y \cdot y}{{z}^{2}}, \frac{1}{2}, y\right)}{z} \]

      7. unpow2 N/A

         \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot \frac{y \cdot y}{z \cdot z}, \frac{1}{2}, y\right)}{z} \]

      8. frac-times N/A

         \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot \left(\frac{y}{z} \cdot \frac{y}{z}\right), \frac{1}{2}, y\right)}{z} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot \left(\frac{y}{z} \cdot \frac{y}{z}\right), \frac{1}{2}, y\right)}{z} \]

      10. pow2 N/A

          \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot {\left(\frac{y}{z}\right)}^{2}, \frac{1}{2}, y\right)}{z} \]

      11. lower-pow.f64 N/A

          \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot {\left(\frac{y}{z}\right)}^{2}, \frac{1}{2}, y\right)}{z} \]

      12. lower-/.f64 99.9

          \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot {\left(\frac{y}{z}\right)}^{2}, 0.5, y\right)}{z} \]

   5. Applied rewrites 99.9%

      \[\leadsto x - \color{blue}{\frac{\mathsf{fma}\left(t \cdot {\left(\frac{y}{z}\right)}^{2}, 0.5, y\right)}{z}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot {\left(\frac{y}{z}\right)}^{2}, \frac{1}{2}, y\right)}{z} \]

      2. lift-pow.f64 N/A

         \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot {\left(\frac{y}{z}\right)}^{2}, \frac{1}{2}, y\right)}{z} \]

      3. unpow2 N/A

         \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot \left(\frac{y}{z} \cdot \frac{y}{z}\right), \frac{1}{2}, y\right)}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot \left(\frac{y}{z} \cdot \frac{y}{z}\right), \frac{1}{2}, y\right)}{z} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot \left(\frac{y}{z} \cdot \frac{y}{z}\right), \frac{1}{2}, y\right)}{z} \]

      6. lift-/.f64 99.9

         \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot \left(\frac{y}{z} \cdot \frac{y}{z}\right), 0.5, y\right)}{z} \]

   7. Applied rewrites 99.9%

      \[\leadsto x - \frac{\mathsf{fma}\left(t \cdot \left(\frac{y}{z} \cdot \frac{y}{z}\right), 0.5, y\right)}{z} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-128}:\\ \;\;\;\;x - \frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \left(y + y\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+153}:\\ \;\;\;\;x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(z + z, z, \left(-y\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.4e-128)
   (- x (* (/ z (fma (- t) y (* (* z z) 2.0))) (+ y y)))
   (if (<= z 2e-181)
     (fma (/ z t) 2.0 x)
     (if (<= z 1.6e+153)
       (- x (* (+ y y) (/ z (fma (+ z z) z (* (- y) t)))))
       (- x (/ y z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.4e-128) {
		tmp = x - ((z / fma(-t, y, ((z * z) * 2.0))) * (y + y));
	} else if (z <= 2e-181) {
		tmp = fma((z / t), 2.0, x);
	} else if (z <= 1.6e+153) {
		tmp = x - ((y + y) * (z / fma((z + z), z, (-y * t))));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.4e-128)
		tmp = Float64(x - Float64(Float64(z / fma(Float64(-t), y, Float64(Float64(z * z) * 2.0))) * Float64(y + y)));
	elseif (z <= 2e-181)
		tmp = fma(Float64(z / t), 2.0, x);
	elseif (z <= 1.6e+153)
		tmp = Float64(x - Float64(Float64(y + y) * Float64(z / fma(Float64(z + z), z, Float64(Float64(-y) * t)))));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.4e-128], N[(x - N[(N[(z / N[((-t) * y + N[(N[(z * z), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e-181], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision], If[LessEqual[z, 1.6e+153], N[(x - N[(N[(y + y), $MachinePrecision] * N[(z / N[(N[(z + z), $MachinePrecision] * z + N[((-y) * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.39999999999999975e-128

   1. Initial program 79.8%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\left(y \cdot 2\right)} \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\left(y \cdot 2\right) \cdot z}}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      4. lift--.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      5. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right) \cdot z} - y \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right)} \cdot z - y \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - \color{blue}{y \cdot t}} \]

      8. associate-/l* N/A

         \[\leadsto x - \color{blue}{\left(y \cdot 2\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(y \cdot 2\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      11. lower-*.f64 N/A

          \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      12. lower-/.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \color{blue}{\frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      13. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right)} \cdot z - y \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z} - y \cdot t} \]

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

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t}} \]

      16. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z} + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right)} \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      18. *-commutative N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(2 \cdot z\right)} \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      19. lower-fma.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(2 \cdot z, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)}} \]

      20. *-commutative N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)} \]

      21. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)} \]

      22. mul-1-neg N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-1 \cdot y\right)} \cdot t\right)} \]

      23. lower-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-1 \cdot y\right) \cdot t}\right)} \]

      24. mul-1-neg N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot t\right)} \]

      25. lower-neg.f64 95.8

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-y\right)} \cdot t\right)} \]

   4. Applied rewrites 95.8%

      \[\leadsto x - \color{blue}{\left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto x - \left(2 \cdot y\right) \cdot \color{blue}{\frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(-y\right) \cdot t\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z + \left(-y\right) \cdot t}} \]

      6. lift-neg.f64 N/A

         \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot t}} \]

      8. *-commutative N/A

         \[\leadsto x - \color{blue}{\frac{z}{\left(z \cdot 2\right) \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \cdot \left(2 \cdot y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{z}{\left(z \cdot 2\right) \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \cdot \left(2 \cdot y\right)} \]

   6. Applied rewrites 95.8%

      \[\leadsto x - \color{blue}{\frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \left(y \cdot 2\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x - \frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \color{blue}{\left(y \cdot 2\right)} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \color{blue}{\left(2 \cdot y\right)} \]

      3. count-2-rev N/A

         \[\leadsto x - \frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \color{blue}{\left(y + y\right)} \]

      4. lift-+.f64 95.8

         \[\leadsto x - \frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \color{blue}{\left(y + y\right)} \]

   8. Applied rewrites 95.8%

      \[\leadsto x - \frac{z}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \cdot \color{blue}{\left(y + y\right)} \]

   if -3.39999999999999975e-128 < z < 2.00000000000000009e-181

   1. Initial program 85.9%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - -2 \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x - \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{z}}{t} \]

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

         \[\leadsto x + \color{blue}{2 \cdot \frac{z}{t}} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \frac{z}{t} + \color{blue}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{z}{t} \cdot 2 + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \color{blue}{2}, x\right) \]

      6. lower-/.f64 100.0

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, 2, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, 2, x\right)} \]

   if 2.00000000000000009e-181 < z < 1.6000000000000001e153

   1. Initial program 84.7%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\left(y \cdot 2\right)} \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\left(y \cdot 2\right) \cdot z}}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      4. lift--.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      5. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right) \cdot z} - y \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right)} \cdot z - y \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - \color{blue}{y \cdot t}} \]

      8. associate-/l* N/A

         \[\leadsto x - \color{blue}{\left(y \cdot 2\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(y \cdot 2\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      11. lower-*.f64 N/A

          \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      12. lower-/.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \color{blue}{\frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      13. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right)} \cdot z - y \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z} - y \cdot t} \]

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

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t}} \]

      16. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z} + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right)} \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      18. *-commutative N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(2 \cdot z\right)} \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      19. lower-fma.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(2 \cdot z, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)}} \]

      20. *-commutative N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)} \]

      21. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)} \]

      22. mul-1-neg N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-1 \cdot y\right)} \cdot t\right)} \]

      23. lower-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-1 \cdot y\right) \cdot t}\right)} \]

      24. mul-1-neg N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot t\right)} \]

      25. lower-neg.f64 95.8

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-y\right)} \cdot t\right)} \]

   4. Applied rewrites 95.8%

      \[\leadsto x - \color{blue}{\left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]

      2. count-2-rev N/A

         \[\leadsto x - \color{blue}{\left(y + y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]

      3. lower-+.f64 95.8

         \[\leadsto x - \color{blue}{\left(y + y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]

   6. Applied rewrites 95.8%

      \[\leadsto x - \color{blue}{\left(y + y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(-y\right) \cdot t\right)} \]

      2. *-commutative N/A

         \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{2 \cdot z}, z, \left(-y\right) \cdot t\right)} \]

      3. count-2-rev N/A

         \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z + z}, z, \left(-y\right) \cdot t\right)} \]

      4. lower-+.f64 95.8

         \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z + z}, z, \left(-y\right) \cdot t\right)} \]

   8. Applied rewrites 95.8%

      \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z + z}, z, \left(-y\right) \cdot t\right)} \]

   if 1.6000000000000001e153 < z

   1. Initial program 37.4%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 99.3

         \[\leadsto x - \frac{y}{\color{blue}{z}} \]

   5. Applied rewrites 99.3%

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(z + z, z, \left(-y\right) \cdot t\right)}\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{-128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (* (+ y y) (/ z (fma (+ z z) z (* (- y) t)))))))
   (if (<= z -3.4e-128)
     t_1
     (if (<= z 2e-181)
       (fma (/ z t) 2.0 x)
       (if (<= z 1.6e+153) t_1 (- x (/ y z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((y + y) * (z / fma((z + z), z, (-y * t))));
	double tmp;
	if (z <= -3.4e-128) {
		tmp = t_1;
	} else if (z <= 2e-181) {
		tmp = fma((z / t), 2.0, x);
	} else if (z <= 1.6e+153) {
		tmp = t_1;
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(y + y) * Float64(z / fma(Float64(z + z), z, Float64(Float64(-y) * t)))))
	tmp = 0.0
	if (z <= -3.4e-128)
		tmp = t_1;
	elseif (z <= 2e-181)
		tmp = fma(Float64(z / t), 2.0, x);
	elseif (z <= 1.6e+153)
		tmp = t_1;
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[(y + y), $MachinePrecision] * N[(z / N[(N[(z + z), $MachinePrecision] * z + N[((-y) * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.4e-128], t$95$1, If[LessEqual[z, 2e-181], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision], If[LessEqual[z, 1.6e+153], t$95$1, N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.39999999999999975e-128 or 2.00000000000000009e-181 < z < 1.6000000000000001e153

   1. Initial program 81.9%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\left(y \cdot 2\right)} \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\left(y \cdot 2\right) \cdot z}}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      4. lift--.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      5. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right) \cdot z} - y \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(z \cdot 2\right)} \cdot z - y \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - \color{blue}{y \cdot t}} \]

      8. associate-/l* N/A

         \[\leadsto x - \color{blue}{\left(y \cdot 2\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(y \cdot 2\right) \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      10. *-commutative N/A

          \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      11. lower-*.f64 N/A

          \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

      12. lower-/.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \color{blue}{\frac{z}{\left(z \cdot 2\right) \cdot z - y \cdot t}} \]

      13. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right)} \cdot z - y \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z} - y \cdot t} \]

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

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t}} \]

      16. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right) \cdot z} + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      17. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(z \cdot 2\right)} \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      18. *-commutative N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\left(2 \cdot z\right)} \cdot z + \left(\mathsf{neg}\left(y\right)\right) \cdot t} \]

      19. lower-fma.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(2 \cdot z, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)}} \]

      20. *-commutative N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)} \]

      21. lift-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(\mathsf{neg}\left(y\right)\right) \cdot t\right)} \]

      22. mul-1-neg N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-1 \cdot y\right)} \cdot t\right)} \]

      23. lower-*.f64 N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-1 \cdot y\right) \cdot t}\right)} \]

      24. mul-1-neg N/A

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot t\right)} \]

      25. lower-neg.f64 95.8

          \[\leadsto x - \left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \color{blue}{\left(-y\right)} \cdot t\right)} \]

   4. Applied rewrites 95.8%

      \[\leadsto x - \color{blue}{\left(2 \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\left(2 \cdot y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]

      2. count-2-rev N/A

         \[\leadsto x - \color{blue}{\left(y + y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]

      3. lower-+.f64 95.8

         \[\leadsto x - \color{blue}{\left(y + y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]

   6. Applied rewrites 95.8%

      \[\leadsto x - \color{blue}{\left(y + y\right)} \cdot \frac{z}{\mathsf{fma}\left(z \cdot 2, z, \left(-y\right) \cdot t\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z \cdot 2}, z, \left(-y\right) \cdot t\right)} \]

      2. *-commutative N/A

         \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{2 \cdot z}, z, \left(-y\right) \cdot t\right)} \]

      3. count-2-rev N/A

         \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z + z}, z, \left(-y\right) \cdot t\right)} \]

      4. lower-+.f64 95.8

         \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z + z}, z, \left(-y\right) \cdot t\right)} \]

   8. Applied rewrites 95.8%

      \[\leadsto x - \left(y + y\right) \cdot \frac{z}{\mathsf{fma}\left(\color{blue}{z + z}, z, \left(-y\right) \cdot t\right)} \]

   if -3.39999999999999975e-128 < z < 2.00000000000000009e-181

   1. Initial program 85.9%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - -2 \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x - \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{z}}{t} \]

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

         \[\leadsto x + \color{blue}{2 \cdot \frac{z}{t}} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \frac{z}{t} + \color{blue}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{z}{t} \cdot 2 + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \color{blue}{2}, x\right) \]

      6. lower-/.f64 100.0

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, 2, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, 2, x\right)} \]

   if 1.6000000000000001e153 < z

   1. Initial program 37.4%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 99.3

         \[\leadsto x - \frac{y}{\color{blue}{z}} \]

   5. Applied rewrites 99.3%

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-8} \lor \neg \left(z \leq 9.5 \cdot 10^{+55}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.35e-8) (not (<= z 9.5e+55)))
   (- x (/ y z))
   (fma (/ z t) 2.0 x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.35e-8) || !(z <= 9.5e+55)) {
		tmp = x - (y / z);
	} else {
		tmp = fma((z / t), 2.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.35e-8) || !(z <= 9.5e+55))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = fma(Float64(z / t), 2.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.35e-8], N[Not[LessEqual[z, 9.5e+55]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35000000000000001e-8 or 9.49999999999999989e55 < z

   1. Initial program 66.3%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 94.3

         \[\leadsto x - \frac{y}{\color{blue}{z}} \]

   5. Applied rewrites 94.3%

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]

   if -1.35000000000000001e-8 < z < 9.49999999999999989e55

   1. Initial program 88.0%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - -2 \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x - \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{z}}{t} \]

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

         \[\leadsto x + \color{blue}{2 \cdot \frac{z}{t}} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \frac{z}{t} + \color{blue}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{z}{t} \cdot 2 + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \color{blue}{2}, x\right) \]

      6. lower-/.f64 88.8

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, 2, x\right) \]

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, 2, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-8} \lor \neg \left(z \leq 9.5 \cdot 10^{+55}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.18 \lor \neg \left(z \leq 360000000\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -0.18) (not (<= z 360000000.0))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.18) || !(z <= 360000000.0)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	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(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-0.18d0)) .or. (.not. (z <= 360000000.0d0))) then
        tmp = x - (y / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.18) || !(z <= 360000000.0)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -0.18) or not (z <= 360000000.0):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -0.18) || !(z <= 360000000.0))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -0.18) || ~((z <= 360000000.0)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -0.18], N[Not[LessEqual[z, 360000000.0]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -0.17999999999999999 or 3.6e8 < z

   1. Initial program 66.8%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 90.4

         \[\leadsto x - \frac{y}{\color{blue}{z}} \]

   5. Applied rewrites 90.4%

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]

   if -0.17999999999999999 < z < 3.6e8

   1. Initial program 88.9%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 75.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.18 \lor \neg \left(z \leq 360000000\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-183} \lor \neg \left(x \leq 2.2 \cdot 10^{-175}\right):\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.9e-183) (not (<= x 2.2e-175))) x (/ (- y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.9e-183) || !(x <= 2.2e-175)) {
		tmp = x;
	} else {
		tmp = -y / z;
	}
	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(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2.9d-183)) .or. (.not. (x <= 2.2d-175))) then
        tmp = x
    else
        tmp = -y / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.9e-183) || !(x <= 2.2e-175)) {
		tmp = x;
	} else {
		tmp = -y / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.9e-183) or not (x <= 2.2e-175):
		tmp = x
	else:
		tmp = -y / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.9e-183) || !(x <= 2.2e-175))
		tmp = x;
	else
		tmp = Float64(Float64(-y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.9e-183) || ~((x <= 2.2e-175)))
		tmp = x;
	else
		tmp = -y / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.9e-183], N[Not[LessEqual[x, 2.2e-175]], $MachinePrecision]], x, N[((-y) / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e-183 or 2.2e-175 < x

   1. Initial program 82.3%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 86.9%

      \[\leadsto \color{blue}{x} \]

   if -2.9e-183 < x < 2.2e-175

   1. Initial program 57.6%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{y \cdot z}{2 \cdot {z}^{2} - t \cdot y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot \left(y \cdot z\right)}{\color{blue}{2 \cdot {z}^{2} - t \cdot y}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-2 \cdot \left(y \cdot z\right)}{\color{blue}{2 \cdot {z}^{2} - t \cdot y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-2 \cdot \left(y \cdot z\right)}{\color{blue}{2 \cdot {z}^{2}} - t \cdot y} \]

      4. *-commutative N/A

         \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{2 \cdot \color{blue}{{z}^{2}} - t \cdot y} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{2 \cdot \color{blue}{{z}^{2}} - t \cdot y} \]

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

         \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{2 \cdot {z}^{2} + \color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot y}} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{2 \cdot {z}^{2} + \left(-1 \cdot t\right) \cdot y} \]

      8. associate-*r* N/A

         \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{2 \cdot {z}^{2} + -1 \cdot \color{blue}{\left(t \cdot y\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{-1 \cdot \left(t \cdot y\right) + \color{blue}{2 \cdot {z}^{2}}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{\left(-1 \cdot t\right) \cdot y + \color{blue}{2} \cdot {z}^{2}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{\left(\mathsf{neg}\left(t\right)\right) \cdot y + 2 \cdot {z}^{2}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{\mathsf{fma}\left(\mathsf{neg}\left(t\right), \color{blue}{y}, 2 \cdot {z}^{2}\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{\mathsf{fma}\left(-t, y, 2 \cdot {z}^{2}\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{\mathsf{fma}\left(-t, y, {z}^{2} \cdot 2\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{\mathsf{fma}\left(-t, y, {z}^{2} \cdot 2\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \]

      17. lower-*.f64 43.3

          \[\leadsto \frac{-2 \cdot \left(z \cdot y\right)}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)} \]

   5. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\frac{-2 \cdot \left(z \cdot y\right)}{\mathsf{fma}\left(-t, y, \left(z \cdot z\right) \cdot 2\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{y}{z}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot y}{z} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(y\right)}{z} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(y\right)}{z} \]

      4. lift-neg.f64 53.7

         \[\leadsto \frac{-y}{z} \]

   8. Applied rewrites 53.7%

      \[\leadsto \frac{-y}{\color{blue}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-183} \lor \neg \left(x \leq 2.2 \cdot 10^{-175}\right):\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 43.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 x)

C

double code(double x, double y, double z, double t) {
	return x;
}

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(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x
end function

Java

public static double code(double x, double y, double z, double t) {
	return x;
}

Python

def code(x, y, z, t):
	return x

Julia

function code(x, y, z, t)
	return x
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 77.1%

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Applied rewrites 72.1%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z)))))

C

double code(double x, double y, double z, double t) {
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)));
}

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(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (1.0d0 / ((z / y) - ((t / 2.0d0) / z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)));
}

Python

def code(x, y, z, t):
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(1.0 / Float64(Float64(z / y) - Float64(Float64(t / 2.0) / z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (1.0 / ((z / y) - ((t / 2.0) / z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(1.0 / N[(N[(z / y), $MachinePrecision] - N[(N[(t / 2.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025057 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (/ 1 (- (/ z y) (/ (/ t 2) z)))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))