fabs fraction 1

Percentage Accurate: 92.2% → 99.8%

Time: 9.7s

Alternatives: 11

Speedup: 1.2×

• 79.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

FPCore

(FPCore (x y z)
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

C

double code(double x, double y, double z) {
	return fabs((((x + 4.0) / y) - ((x / y) * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((((x + 4.0d0) / y) - ((x / y) * z)))
end function

Java

public static double code(double x, double y, double z) {
	return Math.abs((((x + 4.0) / y) - ((x / y) * z)));
}

Python

def code(x, y, z):
	return math.fabs((((x + 4.0) / y) - ((x / y) * z)))

Julia

function code(x, y, z)
	return abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = abs((((x + 4.0) / y) - ((x / y) * z)));
end

Wolfram

code[x_, y_, z_] := N[Abs[N[(N[(N[(x + 4), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 92.2% accurate, 1.0× speedup

Math

\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

FPCore

(FPCore (x y z)
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

C

double code(double x, double y, double z) {
	return fabs((((x + 4.0) / y) - ((x / y) * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((((x + 4.0d0) / y) - ((x / y) * z)))
end function

Java

public static double code(double x, double y, double z) {
	return Math.abs((((x + 4.0) / y) - ((x / y) * z)));
}

Python

def code(x, y, z):
	return math.fabs((((x + 4.0) / y) - ((x / y) * z)))

Julia

function code(x, y, z)
	return abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = abs((((x + 4.0) / y) - ((x / y) * z)));
end

Wolfram

code[x_, y_, z_] := N[Abs[N[(N[(N[(x + 4), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{if}\;t\_0 \leq \frac{7339195571168229}{36695977855841144185773134324833391052745039826692497979801421430190766017415756929120296849762010984873984}:\\ \;\;\;\;\left|\frac{4 - z \cdot x}{y} + \frac{x}{y}\right|\\ \mathbf{elif}\;t\_0 \leq 3999999999999999831443609401385136528614207512386057135410071092916132616022289914504946148287614500592330451563947454856809828080168025678726105499863496711154494173799977940229033050646983792107070529102452199587840315845272602181673858644271966678327153142116021922822752784275414552936448:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(z \cdot x - 4\right) - x}{y}\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (- (/ (+ x 4) y) (* (/ x y) z)))))
  (if (<=
       t_0
       7339195571168229/36695977855841144185773134324833391052745039826692497979801421430190766017415756929120296849762010984873984)
    (fabs (+ (/ (- 4 (* z x)) y) (/ x y)))
    (if (<=
         t_0
         3999999999999999831443609401385136528614207512386057135410071092916132616022289914504946148287614500592330451563947454856809828080168025678726105499863496711154494173799977940229033050646983792107070529102452199587840315845272602181673858644271966678327153142116021922822752784275414552936448)
      t_0
      (fabs (/ (- (- (* z x) 4) x) y))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((((x + 4.0) / y) - ((x / y) * z)));
	double tmp;
	if (t_0 <= 2e-91) {
		tmp = fabs((((4.0 - (z * x)) / y) + (x / y)));
	} else if (t_0 <= 4e+291) {
		tmp = t_0;
	} else {
		tmp = fabs(((((z * x) - 4.0) - x) / y));
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((((x + 4.0d0) / y) - ((x / y) * z)))
    if (t_0 <= 2d-91) then
        tmp = abs((((4.0d0 - (z * x)) / y) + (x / y)))
    else if (t_0 <= 4d+291) then
        tmp = t_0
    else
        tmp = abs(((((z * x) - 4.0d0) - x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((((x + 4.0) / y) - ((x / y) * z)));
	double tmp;
	if (t_0 <= 2e-91) {
		tmp = Math.abs((((4.0 - (z * x)) / y) + (x / y)));
	} else if (t_0 <= 4e+291) {
		tmp = t_0;
	} else {
		tmp = Math.abs(((((z * x) - 4.0) - x) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((((x + 4.0) / y) - ((x / y) * z)))
	tmp = 0
	if t_0 <= 2e-91:
		tmp = math.fabs((((4.0 - (z * x)) / y) + (x / y)))
	elif t_0 <= 4e+291:
		tmp = t_0
	else:
		tmp = math.fabs(((((z * x) - 4.0) - x) / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z)))
	tmp = 0.0
	if (t_0 <= 2e-91)
		tmp = abs(Float64(Float64(Float64(4.0 - Float64(z * x)) / y) + Float64(x / y)));
	elseif (t_0 <= 4e+291)
		tmp = t_0;
	else
		tmp = abs(Float64(Float64(Float64(Float64(z * x) - 4.0) - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((((x + 4.0) / y) - ((x / y) * z)));
	tmp = 0.0;
	if (t_0 <= 2e-91)
		tmp = abs((((4.0 - (z * x)) / y) + (x / y)));
	elseif (t_0 <= 4e+291)
		tmp = t_0;
	else
		tmp = abs(((((z * x) - 4.0) - x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(N[(x + 4), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 7339195571168229/36695977855841144185773134324833391052745039826692497979801421430190766017415756929120296849762010984873984], N[Abs[N[(N[(N[(4 - N[(z * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 3999999999999999831443609401385136528614207512386057135410071092916132616022289914504946148287614500592330451563947454856809828080168025678726105499863496711154494173799977940229033050646983792107070529102452199587840315845272602181673858644271966678327153142116021922822752784275414552936448], t$95$0, N[Abs[N[(N[(N[(N[(z * x), $MachinePrecision] - 4), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 2e-91

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x}{y} \cdot z}\right| \]

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

         \[\leadsto \left|\color{blue}{\frac{x + 4}{y} + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot z}\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x + 4}{y}} + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot z\right| \]

      5. lift-+.f64 N/A

         \[\leadsto \left|\frac{\color{blue}{x + 4}}{y} + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot z\right| \]

      6. div-add N/A

         \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + \frac{4}{y}\right)} + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot z\right| \]

      7. lift-/.f64 N/A

         \[\leadsto \left|\left(\color{blue}{\frac{x}{y}} + \frac{4}{y}\right) + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot z\right| \]

      8. associate-+l+ N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y} + \left(\frac{4}{y} + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot z\right)}\right| \]

      9. +-commutative N/A

         \[\leadsto \left|\color{blue}{\left(\frac{4}{y} + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot z\right) + \frac{x}{y}}\right| \]

      10. lower-+.f64 N/A

          \[\leadsto \left|\color{blue}{\left(\frac{4}{y} + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right) \cdot z\right) + \frac{x}{y}}\right| \]

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

          \[\leadsto \left|\color{blue}{\left(\frac{4}{y} - \frac{x}{y} \cdot z\right)} + \frac{x}{y}\right| \]

      12. lift-*.f64 N/A

          \[\leadsto \left|\left(\frac{4}{y} - \color{blue}{\frac{x}{y} \cdot z}\right) + \frac{x}{y}\right| \]

      13. lift-*.f64 N/A

          \[\leadsto \left|\left(\frac{4}{y} - \color{blue}{\frac{x}{y} \cdot z}\right) + \frac{x}{y}\right| \]

      14. lift-/.f64 N/A

          \[\leadsto \left|\left(\frac{4}{y} - \color{blue}{\frac{x}{y}} \cdot z\right) + \frac{x}{y}\right| \]

      15. associate-*l/ N/A

          \[\leadsto \left|\left(\frac{4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right) + \frac{x}{y}\right| \]

      16. sub-div N/A

          \[\leadsto \left|\color{blue}{\frac{4 - x \cdot z}{y}} + \frac{x}{y}\right| \]

      17. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{4 - x \cdot z}{y}} + \frac{x}{y}\right| \]

      18. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{4 - x \cdot z}}{y} + \frac{x}{y}\right| \]

      19. *-commutative N/A

          \[\leadsto \left|\frac{4 - \color{blue}{z \cdot x}}{y} + \frac{x}{y}\right| \]

      20. lower-*.f64 92.8%

          \[\leadsto \left|\frac{4 - \color{blue}{z \cdot x}}{y} + \frac{x}{y}\right| \]

   3. Applied rewrites 92.8%

      \[\leadsto \left|\color{blue}{\frac{4 - z \cdot x}{y} + \frac{x}{y}}\right| \]

   if 2e-91 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 3.9999999999999998e291

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   if 3.9999999999999998e291 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      2. lift--.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]

      3. fabs-sub N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      4. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} - \frac{x + 4}{y}\right| \]

      6. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y}} \cdot z - \frac{x + 4}{y}\right| \]

      7. associate-*l/ N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{x \cdot z}{y} - \color{blue}{\frac{x + 4}{y}}\right| \]

      9. sub-div N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      10. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      11. lift-+.f64 N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(x + 4\right)}}{y}\right| \]

      12. +-commutative N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(4 + x\right)}}{y}\right| \]

      13. associate--r+ N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      14. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      15. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right)} - x}{y}\right| \]

      16. *-commutative N/A

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

      17. lower-*.f64 96.4%

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

   3. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\left|\frac{\left(z \cdot x - 4\right) - x}{y}\right|} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{134\_z0z1z2z3z4}\left(\left(\frac{1}{y}\right), 4, 1, z, x\right)\right|\\ \mathbf{if}\;z \leq -4999999999999999965699095179735106473829597184:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 699999999999999962599239418951134825273498349517073521493650464426374089253398435122312280289223336609586768632125540303233709625062010957121507716759552:\\ \;\;\;\;\left|\frac{x \cdot \left(z - 1\right) - 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (134-z0z1z2z3z4 (/ 1 y) 4 1 z x))))
  (if (<= z -4999999999999999965699095179735106473829597184)
    t_0
    (if (<=
         z
         699999999999999962599239418951134825273498349517073521493650464426374089253398435122312280289223336609586768632125540303233709625062010957121507716759552)
      (fabs (/ (- (* x (- z 1)) 4) y))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if z < -5e45 or 6.9999999999999996e152 < z

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   3. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{134\_z0z1z2z3z4}\left(\left(\frac{1}{y}\right), \left(x - -4\right), 1, z, x\right)}\right| \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\mathsf{134\_z0z1z2z3z4}\left(\left(\frac{1}{y}\right), \color{blue}{4}, 1, z, x\right)\right| \]
   5. Step-by-step derivation

   6. Applied rewrites 76.9%

      \[\leadsto \left|\mathsf{134\_z0z1z2z3z4}\left(\left(\frac{1}{y}\right), \color{blue}{4}, 1, z, x\right)\right| \]

   if -5e45 < z < 6.9999999999999996e152

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      2. lift--.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]

      3. fabs-sub N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      4. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} - \frac{x + 4}{y}\right| \]

      6. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y}} \cdot z - \frac{x + 4}{y}\right| \]

      7. associate-*l/ N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{x \cdot z}{y} - \color{blue}{\frac{x + 4}{y}}\right| \]

      9. sub-div N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      10. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      11. lift-+.f64 N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(x + 4\right)}}{y}\right| \]

      12. +-commutative N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(4 + x\right)}}{y}\right| \]

      13. associate--r+ N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      14. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      15. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right)} - x}{y}\right| \]

      16. *-commutative N/A

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

      17. lower-*.f64 96.4%

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

   3. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\left|\frac{\left(z \cdot x - 4\right) - x}{y}\right|} \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\frac{\color{blue}{x \cdot \left(z - 1\right) - 4}}{y}\right| \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left|\frac{x \cdot \left(z - 1\right) - \color{blue}{4}}{y}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{x \cdot \left(z - 1\right) - 4}{y}\right| \]

      3. lower--.f64 96.4%

         \[\leadsto \left|\frac{x \cdot \left(z - 1\right) - 4}{y}\right| \]

   6. Applied rewrites 96.4%

      \[\leadsto \left|\frac{\color{blue}{x \cdot \left(z - 1\right) - 4}}{y}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)\right|\\ \mathbf{if}\;x \leq -100000000000000004764729344:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 499999999999999988524756632622668314223421359962075003064997987365996726090394955651630647240755773440:\\ \;\;\;\;\left|\frac{x \cdot \left(z - 1\right) - 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (* x (- (/ 1 y) (/ z y))))))
  (if (<= x -100000000000000004764729344)
    t_0
    (if (<=
         x
         499999999999999988524756632622668314223421359962075003064997987365996726090394955651630647240755773440)
      (fabs (/ (- (* x (- z 1)) 4) y))
      t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x * ((1.0 / y) - (z / y))));
	double tmp;
	if (x <= -1e+26) {
		tmp = t_0;
	} else if (x <= 5e+101) {
		tmp = fabs((((x * (z - 1.0)) - 4.0) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x * ((1.0d0 / y) - (z / y))))
    if (x <= (-1d+26)) then
        tmp = t_0
    else if (x <= 5d+101) then
        tmp = abs((((x * (z - 1.0d0)) - 4.0d0) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x * ((1.0 / y) - (z / y))));
	double tmp;
	if (x <= -1e+26) {
		tmp = t_0;
	} else if (x <= 5e+101) {
		tmp = Math.abs((((x * (z - 1.0)) - 4.0) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x * ((1.0 / y) - (z / y))))
	tmp = 0
	if x <= -1e+26:
		tmp = t_0
	elif x <= 5e+101:
		tmp = math.fabs((((x * (z - 1.0)) - 4.0) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x * Float64(Float64(1.0 / y) - Float64(z / y))))
	tmp = 0.0
	if (x <= -1e+26)
		tmp = t_0;
	elseif (x <= 5e+101)
		tmp = abs(Float64(Float64(Float64(x * Float64(z - 1.0)) - 4.0) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x * ((1.0 / y) - (z / y))));
	tmp = 0.0;
	if (x <= -1e+26)
		tmp = t_0;
	elseif (x <= 5e+101)
		tmp = abs((((x * (z - 1.0)) - 4.0) / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x * N[(N[(1 / y), $MachinePrecision] - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -100000000000000004764729344], t$95$0, If[LessEqual[x, 499999999999999988524756632622668314223421359962075003064997987365996726090394955651630647240755773440], N[Abs[N[(N[(N[(x * N[(z - 1), $MachinePrecision]), $MachinePrecision] - 4), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1e26 or 4.9999999999999999e101 < x

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   2. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      2. lower--.f64 N/A

         \[\leadsto \left|x \cdot \left(\frac{1}{y} - \color{blue}{\frac{z}{y}}\right)\right| \]

      3. lower-/.f64 N/A

         \[\leadsto \left|x \cdot \left(\frac{1}{y} - \frac{\color{blue}{z}}{y}\right)\right| \]

      4. lower-/.f64 61.5%

         \[\leadsto \left|x \cdot \left(\frac{1}{y} - \frac{z}{\color{blue}{y}}\right)\right| \]

   4. Applied rewrites 61.5%

      \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

   if -1e26 < x < 4.9999999999999999e101

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      2. lift--.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]

      3. fabs-sub N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      4. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} - \frac{x + 4}{y}\right| \]

      6. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y}} \cdot z - \frac{x + 4}{y}\right| \]

      7. associate-*l/ N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{x \cdot z}{y} - \color{blue}{\frac{x + 4}{y}}\right| \]

      9. sub-div N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      10. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      11. lift-+.f64 N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(x + 4\right)}}{y}\right| \]

      12. +-commutative N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(4 + x\right)}}{y}\right| \]

      13. associate--r+ N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      14. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      15. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right)} - x}{y}\right| \]

      16. *-commutative N/A

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

      17. lower-*.f64 96.4%

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

   3. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\left|\frac{\left(z \cdot x - 4\right) - x}{y}\right|} \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\frac{\color{blue}{x \cdot \left(z - 1\right) - 4}}{y}\right| \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left|\frac{x \cdot \left(z - 1\right) - \color{blue}{4}}{y}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{x \cdot \left(z - 1\right) - 4}{y}\right| \]

      3. lower--.f64 96.4%

         \[\leadsto \left|\frac{x \cdot \left(z - 1\right) - 4}{y}\right| \]

   6. Applied rewrites 96.4%

      \[\leadsto \left|\frac{\color{blue}{x \cdot \left(z - 1\right) - 4}}{y}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\left|\mathsf{134\_z0z1z2z3z4}\left(\left(\frac{1}{y}\right), \left(x - -4\right), 1, z, x\right)\right| \]

FPCore

(FPCore (x y z)
  :precision binary64
  (fabs (134-z0z1z2z3z4 (/ 1 y) (- x -4) 1 z x)))

Derivation

1. Initial program 92.2%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\color{blue}{\mathsf{134\_z0z1z2z3z4}\left(\left(\frac{1}{y}\right), \left(x - -4\right), 1, z, x\right)}\right| \]
4. Add Preprocessing

Alternative 5: 96.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -5000000000000000282377055102604207074203131909915291873502825820777282819837890985946098807947299914898840846737681810482829903223034619386525800728016398897098919701520311599092821190412956384597997941526508766362009243481475645440:\\ \;\;\;\;\left|\left(-x\right) \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(z - 1\right) - 4}{y}\right|\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<=
     z
     -5000000000000000282377055102604207074203131909915291873502825820777282819837890985946098807947299914898840846737681810482829903223034619386525800728016398897098919701520311599092821190412956384597997941526508766362009243481475645440)
  (fabs (* (- x) (/ z y)))
  (fabs (/ (- (* x (- z 1)) 4) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e+231) {
		tmp = fabs((-x * (z / y)));
	} else {
		tmp = fabs((((x * (z - 1.0)) - 4.0) / y));
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-5d+231)) then
        tmp = abs((-x * (z / y)))
    else
        tmp = abs((((x * (z - 1.0d0)) - 4.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e+231) {
		tmp = Math.abs((-x * (z / y)));
	} else {
		tmp = Math.abs((((x * (z - 1.0)) - 4.0) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5e+231:
		tmp = math.fabs((-x * (z / y)))
	else:
		tmp = math.fabs((((x * (z - 1.0)) - 4.0) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5e+231)
		tmp = abs(Float64(Float64(-x) * Float64(z / y)));
	else
		tmp = abs(Float64(Float64(Float64(x * Float64(z - 1.0)) - 4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5e+231)
		tmp = abs((-x * (z / y)));
	else
		tmp = abs((((x * (z - 1.0)) - 4.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5000000000000000282377055102604207074203131909915291873502825820777282819837890985946098807947299914898840846737681810482829903223034619386525800728016398897098919701520311599092821190412956384597997941526508766362009243481475645440], N[Abs[N[((-x) * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x * N[(z - 1), $MachinePrecision]), $MachinePrecision] - 4), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.0000000000000003e231

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   2. Taylor expanded in z around inf

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{x \cdot z}{y}}\right| \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|-1 \cdot \color{blue}{\frac{x \cdot z}{y}}\right| \]

      2. lower-/.f64 N/A

         \[\leadsto \left|-1 \cdot \frac{x \cdot z}{\color{blue}{y}}\right| \]

      3. lower-*.f64 37.1%

         \[\leadsto \left|-1 \cdot \frac{x \cdot z}{y}\right| \]

   4. Applied rewrites 37.1%

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{x \cdot z}{y}}\right| \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|-1 \cdot \color{blue}{\frac{x \cdot z}{y}}\right| \]

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

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

         \[\leadsto \left|\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{z}{y}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{z}{y}}\right| \]

      11. lower-neg.f64 39.0%

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

   6. Applied rewrites 39.0%

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

   if -5.0000000000000003e231 < z

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      2. lift--.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]

      3. fabs-sub N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      4. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} - \frac{x + 4}{y}\right| \]

      6. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y}} \cdot z - \frac{x + 4}{y}\right| \]

      7. associate-*l/ N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{x \cdot z}{y} - \color{blue}{\frac{x + 4}{y}}\right| \]

      9. sub-div N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      10. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      11. lift-+.f64 N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(x + 4\right)}}{y}\right| \]

      12. +-commutative N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(4 + x\right)}}{y}\right| \]

      13. associate--r+ N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      14. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      15. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right)} - x}{y}\right| \]

      16. *-commutative N/A

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

      17. lower-*.f64 96.4%

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

   3. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\left|\frac{\left(z \cdot x - 4\right) - x}{y}\right|} \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\frac{\color{blue}{x \cdot \left(z - 1\right) - 4}}{y}\right| \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left|\frac{x \cdot \left(z - 1\right) - \color{blue}{4}}{y}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{x \cdot \left(z - 1\right) - 4}{y}\right| \]

      3. lower--.f64 96.4%

         \[\leadsto \left|\frac{x \cdot \left(z - 1\right) - 4}{y}\right| \]

   6. Applied rewrites 96.4%

      \[\leadsto \left|\frac{\color{blue}{x \cdot \left(z - 1\right) - 4}}{y}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{x \cdot \left(z - 1\right)}{y}\right|\\ \mathbf{if}\;x \leq -212:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq \frac{11}{2}:\\ \;\;\;\;\left|\frac{x \cdot z - 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (/ (* x (- z 1)) y))))
  (if (<= x -212)
    t_0
    (if (<= x 11/2) (fabs (/ (- (* x z) 4) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs(((x * (z - 1.0)) / y));
	double tmp;
	if (x <= -212.0) {
		tmp = t_0;
	} else if (x <= 5.5) {
		tmp = fabs((((x * z) - 4.0) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((x * (z - 1.0d0)) / y))
    if (x <= (-212.0d0)) then
        tmp = t_0
    else if (x <= 5.5d0) then
        tmp = abs((((x * z) - 4.0d0) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs(((x * (z - 1.0)) / y));
	double tmp;
	if (x <= -212.0) {
		tmp = t_0;
	} else if (x <= 5.5) {
		tmp = Math.abs((((x * z) - 4.0) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs(((x * (z - 1.0)) / y))
	tmp = 0
	if x <= -212.0:
		tmp = t_0
	elif x <= 5.5:
		tmp = math.fabs((((x * z) - 4.0) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(x * Float64(z - 1.0)) / y))
	tmp = 0.0
	if (x <= -212.0)
		tmp = t_0;
	elseif (x <= 5.5)
		tmp = abs(Float64(Float64(Float64(x * z) - 4.0) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs(((x * (z - 1.0)) / y));
	tmp = 0.0;
	if (x <= -212.0)
		tmp = t_0;
	elseif (x <= 5.5)
		tmp = abs((((x * z) - 4.0) / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x * N[(z - 1), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -212], t$95$0, If[LessEqual[x, 11/2], N[Abs[N[(N[(N[(x * z), $MachinePrecision] - 4), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -212 or 5.5 < x

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      2. lift--.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]

      3. fabs-sub N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      4. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} - \frac{x + 4}{y}\right| \]

      6. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y}} \cdot z - \frac{x + 4}{y}\right| \]

      7. associate-*l/ N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{x \cdot z}{y} - \color{blue}{\frac{x + 4}{y}}\right| \]

      9. sub-div N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      10. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      11. lift-+.f64 N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(x + 4\right)}}{y}\right| \]

      12. +-commutative N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(4 + x\right)}}{y}\right| \]

      13. associate--r+ N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      14. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      15. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right)} - x}{y}\right| \]

      16. *-commutative N/A

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

      17. lower-*.f64 96.4%

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

   3. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\left|\frac{\left(z \cdot x - 4\right) - x}{y}\right|} \]

   4. Taylor expanded in x around inf

      \[\leadsto \left|\frac{\color{blue}{x \cdot \left(z - 1\right)}}{y}\right| \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\frac{x \cdot \color{blue}{\left(z - 1\right)}}{y}\right| \]

      2. lower--.f64 59.6%

         \[\leadsto \left|\frac{x \cdot \left(z - \color{blue}{1}\right)}{y}\right| \]

   6. Applied rewrites 59.6%

      \[\leadsto \left|\frac{\color{blue}{x \cdot \left(z - 1\right)}}{y}\right| \]

   if -212 < x < 5.5

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      2. lift--.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]

      3. fabs-sub N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      4. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} - \frac{x + 4}{y}\right| \]

      6. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y}} \cdot z - \frac{x + 4}{y}\right| \]

      7. associate-*l/ N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{x \cdot z}{y} - \color{blue}{\frac{x + 4}{y}}\right| \]

      9. sub-div N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      10. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      11. lift-+.f64 N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(x + 4\right)}}{y}\right| \]

      12. +-commutative N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(4 + x\right)}}{y}\right| \]

      13. associate--r+ N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      14. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      15. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right)} - x}{y}\right| \]

      16. *-commutative N/A

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

      17. lower-*.f64 96.4%

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

   3. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\left|\frac{\left(z \cdot x - 4\right) - x}{y}\right|} \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\frac{\color{blue}{x \cdot \left(z - 1\right) - 4}}{y}\right| \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left|\frac{x \cdot \left(z - 1\right) - \color{blue}{4}}{y}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{x \cdot \left(z - 1\right) - 4}{y}\right| \]

      3. lower--.f64 96.4%

         \[\leadsto \left|\frac{x \cdot \left(z - 1\right) - 4}{y}\right| \]

   6. Applied rewrites 96.4%

      \[\leadsto \left|\frac{\color{blue}{x \cdot \left(z - 1\right) - 4}}{y}\right| \]

   7. Taylor expanded in z around inf

      \[\leadsto \left|\frac{x \cdot z - 4}{y}\right| \]
   8. Step-by-step derivation

      1. lower-*.f64 73.5%

         \[\leadsto \left|\frac{x \cdot z - 4}{y}\right| \]

   9. Applied rewrites 73.5%

      \[\leadsto \left|\frac{x \cdot z - 4}{y}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 85.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left|\left(-x\right) \cdot \frac{z}{y}\right|\\ \mathbf{if}\;z \leq -28999999999999998464120220974988138453039504620863880001971619371869304592430947864765457768030427676672:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1349999999999999906355496506828360125520427059234156687607193436946432:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (* (- x) (/ z y)))))
  (if (<=
       z
       -28999999999999998464120220974988138453039504620863880001971619371869304592430947864765457768030427676672)
    t_0
    (if (<=
         z
         1349999999999999906355496506828360125520427059234156687607193436946432)
      (fabs (/ (- -4 x) y))
      t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((-x * (z / y)));
	double tmp;
	if (z <= -2.9e+103) {
		tmp = t_0;
	} else if (z <= 1.35e+69) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((-x * (z / y)))
    if (z <= (-2.9d+103)) then
        tmp = t_0
    else if (z <= 1.35d+69) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((-x * (z / y)));
	double tmp;
	if (z <= -2.9e+103) {
		tmp = t_0;
	} else if (z <= 1.35e+69) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((-x * (z / y)))
	tmp = 0
	if z <= -2.9e+103:
		tmp = t_0
	elif z <= 1.35e+69:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(-x) * Float64(z / y)))
	tmp = 0.0
	if (z <= -2.9e+103)
		tmp = t_0;
	elseif (z <= 1.35e+69)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((-x * (z / y)));
	tmp = 0.0;
	if (z <= -2.9e+103)
		tmp = t_0;
	elseif (z <= 1.35e+69)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[((-x) * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -28999999999999998464120220974988138453039504620863880001971619371869304592430947864765457768030427676672], t$95$0, If[LessEqual[z, 1349999999999999906355496506828360125520427059234156687607193436946432], N[Abs[N[(N[(-4 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8999999999999998e103 or 1.3499999999999999e69 < z

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   2. Taylor expanded in z around inf

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{x \cdot z}{y}}\right| \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|-1 \cdot \color{blue}{\frac{x \cdot z}{y}}\right| \]

      2. lower-/.f64 N/A

         \[\leadsto \left|-1 \cdot \frac{x \cdot z}{\color{blue}{y}}\right| \]

      3. lower-*.f64 37.1%

         \[\leadsto \left|-1 \cdot \frac{x \cdot z}{y}\right| \]

   4. Applied rewrites 37.1%

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{x \cdot z}{y}}\right| \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|-1 \cdot \color{blue}{\frac{x \cdot z}{y}}\right| \]

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

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

         \[\leadsto \left|\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{z}{y}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{z}{y}}\right| \]

      11. lower-neg.f64 39.0%

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

   6. Applied rewrites 39.0%

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

   if -2.8999999999999998e103 < z < 1.3499999999999999e69

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      2. lift--.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]

      3. fabs-sub N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      4. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} - \frac{x + 4}{y}\right| \]

      6. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y}} \cdot z - \frac{x + 4}{y}\right| \]

      7. associate-*l/ N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{x \cdot z}{y} - \color{blue}{\frac{x + 4}{y}}\right| \]

      9. sub-div N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      10. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      11. lift-+.f64 N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(x + 4\right)}}{y}\right| \]

      12. +-commutative N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(4 + x\right)}}{y}\right| \]

      13. associate--r+ N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      14. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      15. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right)} - x}{y}\right| \]

      16. *-commutative N/A

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

      17. lower-*.f64 96.4%

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

   3. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\left|\frac{\left(z \cdot x - 4\right) - x}{y}\right|} \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\frac{\color{blue}{-4} - x}{y}\right| \]
   5. Step-by-step derivation

   6. Applied rewrites 70.4%

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

Alternative 8: 84.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{x \cdot z}{y}\right|\\ \mathbf{if}\;z \leq -28999999999999998464120220974988138453039504620863880001971619371869304592430947864765457768030427676672:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1349999999999999906355496506828360125520427059234156687607193436946432:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (/ (* x z) y))))
  (if (<=
       z
       -28999999999999998464120220974988138453039504620863880001971619371869304592430947864765457768030427676672)
    t_0
    (if (<=
         z
         1349999999999999906355496506828360125520427059234156687607193436946432)
      (fabs (/ (- -4 x) y))
      t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs(((x * z) / y));
	double tmp;
	if (z <= -2.9e+103) {
		tmp = t_0;
	} else if (z <= 1.35e+69) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((x * z) / y))
    if (z <= (-2.9d+103)) then
        tmp = t_0
    else if (z <= 1.35d+69) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs(((x * z) / y));
	double tmp;
	if (z <= -2.9e+103) {
		tmp = t_0;
	} else if (z <= 1.35e+69) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs(((x * z) / y))
	tmp = 0
	if z <= -2.9e+103:
		tmp = t_0
	elif z <= 1.35e+69:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(x * z) / y))
	tmp = 0.0
	if (z <= -2.9e+103)
		tmp = t_0;
	elseif (z <= 1.35e+69)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs(((x * z) / y));
	tmp = 0.0;
	if (z <= -2.9e+103)
		tmp = t_0;
	elseif (z <= 1.35e+69)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -28999999999999998464120220974988138453039504620863880001971619371869304592430947864765457768030427676672], t$95$0, If[LessEqual[z, 1349999999999999906355496506828360125520427059234156687607193436946432], N[Abs[N[(N[(-4 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8999999999999998e103 or 1.3499999999999999e69 < z

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      2. lift--.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]

      3. fabs-sub N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      4. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} - \frac{x + 4}{y}\right| \]

      6. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y}} \cdot z - \frac{x + 4}{y}\right| \]

      7. associate-*l/ N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{x \cdot z}{y} - \color{blue}{\frac{x + 4}{y}}\right| \]

      9. sub-div N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      10. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      11. lift-+.f64 N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(x + 4\right)}}{y}\right| \]

      12. +-commutative N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(4 + x\right)}}{y}\right| \]

      13. associate--r+ N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      14. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      15. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right)} - x}{y}\right| \]

      16. *-commutative N/A

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

      17. lower-*.f64 96.4%

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

   3. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\left|\frac{\left(z \cdot x - 4\right) - x}{y}\right|} \]

   4. Taylor expanded in x around inf

      \[\leadsto \left|\frac{\color{blue}{x \cdot \left(z - 1\right)}}{y}\right| \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\frac{x \cdot \color{blue}{\left(z - 1\right)}}{y}\right| \]

      2. lower--.f64 59.6%

         \[\leadsto \left|\frac{x \cdot \left(z - \color{blue}{1}\right)}{y}\right| \]

   6. Applied rewrites 59.6%

      \[\leadsto \left|\frac{\color{blue}{x \cdot \left(z - 1\right)}}{y}\right| \]

   7. Taylor expanded in z around inf

      \[\leadsto \left|\frac{x \cdot \color{blue}{z}}{y}\right| \]
   8. Step-by-step derivation

      1. lower-*.f64 37.1%

         \[\leadsto \left|\frac{x \cdot z}{y}\right| \]

   9. Applied rewrites 37.1%

      \[\leadsto \left|\frac{x \cdot \color{blue}{z}}{y}\right| \]

   if -2.8999999999999998e103 < z < 1.3499999999999999e69

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      2. lift--.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]

      3. fabs-sub N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      4. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} - \frac{x + 4}{y}\right| \]

      6. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y}} \cdot z - \frac{x + 4}{y}\right| \]

      7. associate-*l/ N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{x \cdot z}{y} - \color{blue}{\frac{x + 4}{y}}\right| \]

      9. sub-div N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      10. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      11. lift-+.f64 N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(x + 4\right)}}{y}\right| \]

      12. +-commutative N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(4 + x\right)}}{y}\right| \]

      13. associate--r+ N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      14. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      15. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right)} - x}{y}\right| \]

      16. *-commutative N/A

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

      17. lower-*.f64 96.4%

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

   3. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\left|\frac{\left(z \cdot x - 4\right) - x}{y}\right|} \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\frac{\color{blue}{-4} - x}{y}\right| \]
   5. Step-by-step derivation

   6. Applied rewrites 70.4%

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

Alternative 9: 70.4% accurate, 2.1× speedup

Math

\[\left|\frac{-4 - x}{y}\right| \]

FPCore

(FPCore (x y z)
  :precision binary64
  (fabs (/ (- -4 x) y)))

C

double code(double x, double y, double z) {
	return fabs(((-4.0 - x) / y));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((((-4.0d0) - x) / y))
end function

Java

public static double code(double x, double y, double z) {
	return Math.abs(((-4.0 - x) / y));
}

Python

def code(x, y, z):
	return math.fabs(((-4.0 - x) / y))

Julia

function code(x, y, z)
	return abs(Float64(Float64(-4.0 - x) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = abs(((-4.0 - x) / y));
end

Wolfram

code[x_, y_, z_] := N[Abs[N[(N[(-4 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.2%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
2. Step-by-step derivation

   1. lift-fabs.f64 N/A

      \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

   2. lift--.f64 N/A

      \[\leadsto \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]

   3. fabs-sub N/A

      \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

   4. lower-fabs.f64 N/A

      \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

   5. lift-*.f64 N/A

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} - \frac{x + 4}{y}\right| \]

   6. lift-/.f64 N/A

      \[\leadsto \left|\color{blue}{\frac{x}{y}} \cdot z - \frac{x + 4}{y}\right| \]

   7. associate-*l/ N/A

      \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

   8. lift-/.f64 N/A

      \[\leadsto \left|\frac{x \cdot z}{y} - \color{blue}{\frac{x + 4}{y}}\right| \]

   9. sub-div N/A

      \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

   10. lower-/.f64 N/A

       \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

   11. lift-+.f64 N/A

       \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(x + 4\right)}}{y}\right| \]

   12. +-commutative N/A

       \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(4 + x\right)}}{y}\right| \]

   13. associate--r+ N/A

       \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

   14. lower--.f64 N/A

       \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

   15. lower--.f64 N/A

       \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right)} - x}{y}\right| \]

   16. *-commutative N/A

       \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

   17. lower-*.f64 96.4%

       \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

3. Applied rewrites 96.4%

   \[\leadsto \color{blue}{\left|\frac{\left(z \cdot x - 4\right) - x}{y}\right|} \]

4. Taylor expanded in x around 0

   \[\leadsto \left|\frac{\color{blue}{-4} - x}{y}\right| \]
5. Step-by-step derivation

6. Applied rewrites 70.4%

   \[\leadsto \left|\frac{\color{blue}{-4} - x}{y}\right| \]
7. Add Preprocessing

Alternative 10: 69.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -245:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq \frac{11}{2}:\\ \;\;\;\;\left|\frac{-4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (fabs (/ x y))))
  (if (<= x -245) t_0 (if (<= x 11/2) (fabs (/ -4 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double tmp;
	if (x <= -245.0) {
		tmp = t_0;
	} else if (x <= 5.5) {
		tmp = fabs((-4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y))
    if (x <= (-245.0d0)) then
        tmp = t_0
    else if (x <= 5.5d0) then
        tmp = abs(((-4.0d0) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if (x <= -245.0) {
		tmp = t_0;
	} else if (x <= 5.5) {
		tmp = Math.abs((-4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	tmp = 0
	if x <= -245.0:
		tmp = t_0
	elif x <= 5.5:
		tmp = math.fabs((-4.0 / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -245.0)
		tmp = t_0;
	elseif (x <= 5.5)
		tmp = abs(Float64(-4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	tmp = 0.0;
	if (x <= -245.0)
		tmp = t_0;
	elseif (x <= 5.5)
		tmp = abs((-4.0 / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -245], t$95$0, If[LessEqual[x, 11/2], N[Abs[N[(-4 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -245 or 5.5 < x

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

   2. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

      2. lower--.f64 N/A

         \[\leadsto \left|x \cdot \left(\frac{1}{y} - \color{blue}{\frac{z}{y}}\right)\right| \]

      3. lower-/.f64 N/A

         \[\leadsto \left|x \cdot \left(\frac{1}{y} - \frac{\color{blue}{z}}{y}\right)\right| \]

      4. lower-/.f64 61.5%

         \[\leadsto \left|x \cdot \left(\frac{1}{y} - \frac{z}{\color{blue}{y}}\right)\right| \]

   4. Applied rewrites 61.5%

      \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

   5. Taylor expanded in z around 0

      \[\leadsto \left|\frac{x}{\color{blue}{y}}\right| \]
   6. Step-by-step derivation

      1. lower-/.f64 34.1%

         \[\leadsto \left|\frac{x}{y}\right| \]

   7. Applied rewrites 34.1%

      \[\leadsto \left|\frac{x}{\color{blue}{y}}\right| \]

   if -245 < x < 5.5

   1. Initial program 92.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]

      2. lift--.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]

      3. fabs-sub N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      4. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot z} - \frac{x + 4}{y}\right| \]

      6. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{x}{y}} \cdot z - \frac{x + 4}{y}\right| \]

      7. associate-*l/ N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      8. lift-/.f64 N/A

         \[\leadsto \left|\frac{x \cdot z}{y} - \color{blue}{\frac{x + 4}{y}}\right| \]

      9. sub-div N/A

         \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      10. lower-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{x \cdot z - \left(x + 4\right)}{y}}\right| \]

      11. lift-+.f64 N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(x + 4\right)}}{y}\right| \]

      12. +-commutative N/A

          \[\leadsto \left|\frac{x \cdot z - \color{blue}{\left(4 + x\right)}}{y}\right| \]

      13. associate--r+ N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      14. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right) - x}}{y}\right| \]

      15. lower--.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(x \cdot z - 4\right)} - x}{y}\right| \]

      16. *-commutative N/A

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

      17. lower-*.f64 96.4%

          \[\leadsto \left|\frac{\left(\color{blue}{z \cdot x} - 4\right) - x}{y}\right| \]

   3. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\left|\frac{\left(z \cdot x - 4\right) - x}{y}\right|} \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{\frac{-4}{y}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 40.6%

         \[\leadsto \left|\frac{-4}{\color{blue}{y}}\right| \]

   6. Applied rewrites 40.6%

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

Alternative 11: 34.1% accurate, 2.6× speedup

Math

\[\left|\frac{x}{y}\right| \]

FPCore

(FPCore (x y z)
  :precision binary64
  (fabs (/ x y)))

C

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

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

Java

public static double code(double x, double y, double z) {
	return Math.abs((x / y));
}

Python

def code(x, y, z):
	return math.fabs((x / y))

Julia

function code(x, y, z)
	return abs(Float64(x / y))
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.2%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

2. Taylor expanded in x around inf

   \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

   2. lower--.f64 N/A

      \[\leadsto \left|x \cdot \left(\frac{1}{y} - \color{blue}{\frac{z}{y}}\right)\right| \]

   3. lower-/.f64 N/A

      \[\leadsto \left|x \cdot \left(\frac{1}{y} - \frac{\color{blue}{z}}{y}\right)\right| \]

   4. lower-/.f64 61.5%

      \[\leadsto \left|x \cdot \left(\frac{1}{y} - \frac{z}{\color{blue}{y}}\right)\right| \]

4. Applied rewrites 61.5%

   \[\leadsto \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]

5. Taylor expanded in z around 0

   \[\leadsto \left|\frac{x}{\color{blue}{y}}\right| \]
6. Step-by-step derivation

   1. lower-/.f64 34.1%

      \[\leadsto \left|\frac{x}{y}\right| \]

7. Applied rewrites 34.1%

   \[\leadsto \left|\frac{x}{\color{blue}{y}}\right| \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))