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

Percentage Accurate: 81.7% → 94.4%

Time: 4.4s

Alternatives: 5

Speedup: 1.7×

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: 81.7% 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: 94.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \leq 10^{+298}:\\ \;\;\;\;x - \left(y + y\right) \cdot \frac{z}{\left(z + z\right) \cdot z - t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))) 1e+298)
   (- x (* (+ y y) (/ z (- (* (+ z z) z) (* t y)))))
   (- x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)))) <= 1e+298) {
		tmp = x - ((y + y) * (z / (((z + z) * z) - (t * y))));
	} else {
		tmp = x - (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 - (((y * 2.0d0) * z) / (((z * 2.0d0) * z) - (y * t)))) <= 1d+298) then
        tmp = x - ((y + y) * (z / (((z + z) * z) - (t * y))))
    else
        tmp = x - (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 - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)))) <= 1e+298) {
		tmp = x - ((y + y) * (z / (((z + z) * z) - (t * y))));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)))) <= 1e+298:
		tmp = x - ((y + y) * (z / (((z + z) * z) - (t * y))))
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x - Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(Float64(z * 2.0) * z) - Float64(y * t)))) <= 1e+298)
		tmp = Float64(x - Float64(Float64(y + y) * Float64(z / Float64(Float64(Float64(z + z) * z) - Float64(t * y)))));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)))) <= 1e+298)
		tmp = x - ((y + y) * (z / (((z + z) * z) - (t * y))));
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[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], 1e+298], N[(x - N[(N[(y + y), $MachinePrecision] * N[(z / N[(N[(N[(z + z), $MachinePrecision] * z), $MachinePrecision] - N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))) < 9.9999999999999996e297

   1. Initial program 95.2%

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

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

      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

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

      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. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 N/A

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

      22. lower-*.f64 96.7

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

   3. Applied rewrites 96.7%

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

   if 9.9999999999999996e297 < (-.f64 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t))))

   1. Initial program 4.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 81.7

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

   4. Applied rewrites 81.7%

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

Alternative 2: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \leq 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot y}{\left(z \cdot z\right) \cdot 2 - t \cdot y}, -2, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))) 1e+298)
   (fma (/ (* z y) (- (* (* z z) 2.0) (* t y))) -2.0 x)
   (- x (/ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)))) <= 1e+298) {
		tmp = fma(((z * y) / (((z * z) * 2.0) - (t * y))), -2.0, x);
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x - Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(Float64(z * 2.0) * z) - Float64(y * t)))) <= 1e+298)
		tmp = fma(Float64(Float64(z * y) / Float64(Float64(Float64(z * z) * 2.0) - Float64(t * y))), -2.0, x);
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[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], 1e+298], N[(N[(N[(z * y), $MachinePrecision] / N[(N[(N[(z * z), $MachinePrecision] * 2.0), $MachinePrecision] - N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + x), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))) < 9.9999999999999996e297

   1. Initial program 95.2%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - -2 \cdot \frac{z}{t}} \]
   3. 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 66.0

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(1 + 2 \cdot \frac{z}{t \cdot x}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 62.0

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

   7. Applied rewrites 62.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \left(2 \cdot \frac{z}{t \cdot x}\right) \cdot x \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 11.4

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

   10. Applied rewrites 11.4%

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

   11. Taylor expanded in x around 0

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

       1. metadata-eval N/A

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

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

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. pow2 N/A

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

       13. lift-*.f64 N/A

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

       14. lower-*.f64 95.2

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

   13. Applied rewrites 95.2%

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

   if 9.9999999999999996e297 < (-.f64 x (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t))))

   1. Initial program 4.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 81.7

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

   4. Applied rewrites 81.7%

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

Alternative 3: 88.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -2.95 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -2.95e-22) t_1 (if (<= z 2.3e+16) (fma (/ z t) 2.0 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -2.95e-22) {
		tmp = t_1;
	} else if (z <= 2.3e+16) {
		tmp = fma((z / t), 2.0, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -2.95e-22)
		tmp = t_1;
	elseif (z <= 2.3e+16)
		tmp = fma(Float64(z / t), 2.0, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.95e-22], t$95$1, If[LessEqual[z, 2.3e+16], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.95000000000000004e-22 or 2.3e16 < z

   1. Initial program 72.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 88.8

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

   4. Applied rewrites 88.8%

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

   if -2.95000000000000004e-22 < z < 2.3e16

   1. Initial program 91.8%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - -2 \cdot \frac{z}{t}} \]
   3. 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 89.0

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

   4. Applied rewrites 89.0%

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

Alternative 4: 81.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -3.9 \cdot 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -3.9e-25) t_1 (if (<= z 2.3e+16) x t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -3.9e-25) {
		tmp = t_1;
	} else if (z <= 2.3e+16) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x - (y / z)
    if (z <= (-3.9d-25)) then
        tmp = t_1
    else if (z <= 2.3d+16) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -3.9e-25) {
		tmp = t_1;
	} else if (z <= 2.3e+16) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -3.9e-25:
		tmp = t_1
	elif z <= 2.3e+16:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -3.9e-25)
		tmp = t_1;
	elseif (z <= 2.3e+16)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / z);
	tmp = 0.0;
	if (z <= -3.9e-25)
		tmp = t_1;
	elseif (z <= 2.3e+16)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.9e-25], t$95$1, If[LessEqual[z, 2.3e+16], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.9e-25 or 2.3e16 < z

   1. Initial program 72.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 88.6

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

   4. Applied rewrites 88.6%

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

   if -3.9e-25 < z < 2.3e16

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 74.7%

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

Alternative 5: 74.3% accurate, 24.4× 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 81.7%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 74.3%

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

Reproduce

herbie shell --seed 2025117 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64
  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))