fabs fraction 1

Percentage Accurate: 91.6% → 99.3%

Time: 3.7s

Alternatives: 8

Speedup: 1.1×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) 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 + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 91.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) 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 + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+108} \lor \neg \left(x \leq 1.5 \cdot 10^{+77}\right):\\ \;\;\;\;\left|\frac{1 - z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.4e+108) (not (<= x 1.5e+77)))
   (fabs (* (/ (- 1.0 z) y) x))
   (fabs (/ (fma z x (- -4.0 x)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.4e+108) || !(x <= 1.5e+77)) {
		tmp = fabs((((1.0 - z) / y) * x));
	} else {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.4e+108) || !(x <= 1.5e+77))
		tmp = abs(Float64(Float64(Float64(1.0 - z) / y) * x));
	else
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.4e+108], N[Not[LessEqual[x, 1.5e+77]], $MachinePrecision]], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.40000000000000019e108 or 1.4999999999999999e77 < x

   1. Initial program 81.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -2.40000000000000019e108 < x < 1.4999999999999999e77

   1. Initial program 97.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. neg-fabs N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. div-sub N/A

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

      11. distribute-neg-frac N/A

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

      12. mul-1-neg N/A

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

      13. distribute-lft-out-- N/A

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

      14. lower-fabs.f64 N/A

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

      15. distribute-lft-out-- N/A

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

      16. mul-1-neg N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+108} \lor \neg \left(x \leq 1.5 \cdot 10^{+77}\right):\\ \;\;\;\;\left|\frac{1 - z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-68}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{1 - z}{y}, x, \frac{4}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.5e-68)
   (fabs (/ (fma z x (- -4.0 x)) y))
   (fabs (fma (/ (- 1.0 z) y) x (/ 4.0 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e-68) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y));
	} else {
		tmp = fabs(fma(((1.0 - z) / y), x, (4.0 / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.5e-68)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y));
	else
		tmp = abs(fma(Float64(Float64(1.0 - z) / y), x, Float64(4.0 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.5e-68], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision] * x + N[(4.0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.50000000000000026e-68

   1. Initial program 90.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. neg-fabs N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. div-sub N/A

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

      11. distribute-neg-frac N/A

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

      12. mul-1-neg N/A

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

      13. distribute-lft-out-- N/A

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

      14. lower-fabs.f64 N/A

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

      15. distribute-lft-out-- N/A

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

      16. mul-1-neg N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 94.4%

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

   if 8.50000000000000026e-68 < y

   1. Initial program 94.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. 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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. div-add N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. +-commutative N/A

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

      11. associate--l+ N/A

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

      12. +-commutative N/A

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

   4. Applied rewrites 99.8%

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

Alternative 3: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -420000 \lor \neg \left(x \leq 3.55\right):\\ \;\;\;\;\left|\frac{1 - z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -420000.0) (not (<= x 3.55)))
   (fabs (* (/ (- 1.0 z) y) x))
   (fabs (/ (fma z x -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -420000.0) || !(x <= 3.55)) {
		tmp = fabs((((1.0 - z) / y) * x));
	} else {
		tmp = fabs((fma(z, x, -4.0) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -420000.0) || !(x <= 3.55))
		tmp = abs(Float64(Float64(Float64(1.0 - z) / y) * x));
	else
		tmp = abs(Float64(fma(z, x, -4.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -420000.0], N[Not[LessEqual[x, 3.55]], $MachinePrecision]], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * x + -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e5 or 3.5499999999999998 < x

   1. Initial program 86.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -4.2e5 < x < 3.5499999999999998

   1. Initial program 96.3%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. neg-fabs N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. div-sub N/A

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

      11. distribute-neg-frac N/A

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

      12. mul-1-neg N/A

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

      13. distribute-lft-out-- N/A

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

      14. lower-fabs.f64 N/A

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

      15. distribute-lft-out-- N/A

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

      16. mul-1-neg N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -420000 \lor \neg \left(x \leq 3.55\right):\\ \;\;\;\;\left|\frac{1 - z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -260000000 \lor \neg \left(z \leq 8.5 \cdot 10^{+17}\right):\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -260000000.0) (not (<= z 8.5e+17)))
   (fabs (/ (fma z x -4.0) y))
   (fabs (/ (- x -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -260000000.0) || !(z <= 8.5e+17)) {
		tmp = fabs((fma(z, x, -4.0) / y));
	} else {
		tmp = fabs(((x - -4.0) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -260000000.0) || !(z <= 8.5e+17))
		tmp = abs(Float64(fma(z, x, -4.0) / y));
	else
		tmp = abs(Float64(Float64(x - -4.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -260000000.0], N[Not[LessEqual[z, 8.5e+17]], $MachinePrecision]], N[Abs[N[(N[(z * x + -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.6e8 or 8.5e17 < z

   1. Initial program 89.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. neg-fabs N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. div-sub N/A

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

      11. distribute-neg-frac N/A

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

      12. mul-1-neg N/A

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

      13. distribute-lft-out-- N/A

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

      14. lower-fabs.f64 N/A

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

      15. distribute-lft-out-- N/A

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

      16. mul-1-neg N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 88.5%

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

   if -2.6e8 < z < 8.5e17

   1. Initial program 93.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

         \[\leadsto \left|\frac{4 \cdot 1}{y} + \frac{\color{blue}{x}}{y}\right| \]

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

         \[\leadsto \left|\frac{x + 4 \cdot 1}{y}\right| \]

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

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

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

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

      9. metadata-eval N/A

         \[\leadsto \left|\frac{x - \left(\mathsf{neg}\left(4\right)\right)}{y}\right| \]

      10. lower--.f64 N/A

          \[\leadsto \left|\frac{x - \left(\mathsf{neg}\left(4\right)\right)}{y}\right| \]

      11. metadata-eval 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -260000000 \lor \neg \left(z \leq 8.5 \cdot 10^{+17}\right):\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+93} \lor \neg \left(z \leq 2 \cdot 10^{+62}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.55e+93) (not (<= z 2e+62)))
   (fabs (* (/ x y) z))
   (fabs (/ (- x -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.55e+93) || !(z <= 2e+62)) {
		tmp = fabs(((x / y) * z));
	} else {
		tmp = fabs(((x - -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 <= (-1.55d+93)) .or. (.not. (z <= 2d+62))) then
        tmp = abs(((x / y) * z))
    else
        tmp = abs(((x - (-4.0d0)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.55e+93) || !(z <= 2e+62)) {
		tmp = Math.abs(((x / y) * z));
	} else {
		tmp = Math.abs(((x - -4.0) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.55e+93) or not (z <= 2e+62):
		tmp = math.fabs(((x / y) * z))
	else:
		tmp = math.fabs(((x - -4.0) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.55e+93) || !(z <= 2e+62))
		tmp = abs(Float64(Float64(x / y) * z));
	else
		tmp = abs(Float64(Float64(x - -4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.55e+93) || ~((z <= 2e+62)))
		tmp = abs(((x / y) * z));
	else
		tmp = abs(((x - -4.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.55e+93], N[Not[LessEqual[z, 2e+62]], $MachinePrecision]], N[Abs[N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5500000000000001e93 or 2.00000000000000007e62 < z

   1. Initial program 87.7%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. neg-fabs N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. div-sub N/A

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

      11. distribute-neg-frac N/A

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

      12. mul-1-neg N/A

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

      13. distribute-lft-out-- N/A

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

      14. lower-fabs.f64 N/A

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

      15. distribute-lft-out-- N/A

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

      16. mul-1-neg N/A

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

      17. lower-/.f64 N/A

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 78.1

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

   7. Applied rewrites 78.1%

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

   if -1.5500000000000001e93 < z < 2.00000000000000007e62

   1. Initial program 94.2%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

         \[\leadsto \left|\frac{4 \cdot 1}{y} + \frac{\color{blue}{x}}{y}\right| \]

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

         \[\leadsto \left|\frac{x + 4 \cdot 1}{y}\right| \]

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

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

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

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

      9. metadata-eval N/A

         \[\leadsto \left|\frac{x - \left(\mathsf{neg}\left(4\right)\right)}{y}\right| \]

      10. lower--.f64 N/A

          \[\leadsto \left|\frac{x - \left(\mathsf{neg}\left(4\right)\right)}{y}\right| \]

      11. metadata-eval 94.6

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

   5. Applied rewrites 94.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+93} \lor \neg \left(z \leq 2 \cdot 10^{+62}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = fabs((x / y));
	} else {
		tmp = fabs((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 ((x <= (-1.55d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = Math.abs((x / y));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.55) or not (x <= 4.0):
		tmp = math.fabs((x / y))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 4.0))
		tmp = abs(Float64(x / y));
	else
		tmp = abs(Float64(4.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 4.0)))
		tmp = abs((x / y));
	else
		tmp = abs((4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 4 < x

   1. Initial program 86.3%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

         \[\leadsto \left|\frac{4 \cdot 1}{y} + \frac{\color{blue}{x}}{y}\right| \]

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

         \[\leadsto \left|\frac{x + 4 \cdot 1}{y}\right| \]

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

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

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

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

      9. metadata-eval N/A

         \[\leadsto \left|\frac{x - \left(\mathsf{neg}\left(4\right)\right)}{y}\right| \]

      10. lower--.f64 N/A

          \[\leadsto \left|\frac{x - \left(\mathsf{neg}\left(4\right)\right)}{y}\right| \]

      11. metadata-eval 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 58.0%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 79.7

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

   5. Applied rewrites 79.7%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left|\frac{x - -4}{y}\right| \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fabs (/ (- x -4.0) y)))

C

double code(double x, double y, double z) {
	return fabs(((x - -4.0) / 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 - (-4.0d0)) / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in z around 0

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

   1. associate-*r/ N/A

      \[\leadsto \left|\frac{4 \cdot 1}{y} + \frac{\color{blue}{x}}{y}\right| \]

   2. metadata-eval N/A

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

   3. div-add N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

      \[\leadsto \left|\frac{x + 4 \cdot 1}{y}\right| \]

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

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

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

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

   9. metadata-eval N/A

      \[\leadsto \left|\frac{x - \left(\mathsf{neg}\left(4\right)\right)}{y}\right| \]

   10. lower--.f64 N/A

       \[\leadsto \left|\frac{x - \left(\mathsf{neg}\left(4\right)\right)}{y}\right| \]

   11. metadata-eval 70.7

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

5. Applied rewrites 70.7%

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

Alternative 8: 40.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left|\frac{4}{y}\right| \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fabs (/ 4.0 y)))

C

double code(double x, double y, double z) {
	return fabs((4.0 / 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 / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 45.3

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

5. Applied rewrites 45.3%

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

Reproduce

herbie shell --seed 2025037 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))