fabs fraction 1

Percentage Accurate: 91.6% → 98.5%

Time: 3.1s

Alternatives: 9

Speedup: 1.2×

• 21.3% 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: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+15} \lor \neg \left(x \leq 0.31\right):\\ \;\;\;\;\left|\frac{1 - z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.1e+15) (not (<= x 0.31)))
   (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 <= -1.1e+15) || !(x <= 0.31)) {
		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 <= -1.1e+15) || !(x <= 0.31))
		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, -1.1e+15], N[Not[LessEqual[x, 0.31]], $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 < -1.1e15 or 0.309999999999999998 < x

   1. Initial program 85.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   if -1.1e15 < x < 0.309999999999999998

   1. Initial program 95.0%

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

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

   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 99.2%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+15} \lor \neg \left(x \leq 0.31\right):\\ \;\;\;\;\left|\frac{1 - z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+42} \lor \neg \left(x \leq 0.31\right):\\ \;\;\;\;\left|\frac{x - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5e+42) (not (<= x 0.31)))
   (fabs (/ (- x (* x z)) y))
   (fabs (/ (fma z x -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5e+42) || !(x <= 0.31)) {
		tmp = fabs(((x - (x * z)) / y));
	} else {
		tmp = fabs((fma(z, x, -4.0) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5e+42) || !(x <= 0.31))
		tmp = abs(Float64(Float64(x - Float64(x * z)) / y));
	else
		tmp = abs(Float64(fma(z, x, -4.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5e+42], N[Not[LessEqual[x, 0.31]], $MachinePrecision]], N[Abs[N[(N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * x + -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000007e42 or 0.309999999999999998 < x

   1. Initial program 85.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. associate-*r/ N/A

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

       3. div-add-rev N/A

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

       4. associate-*r* N/A

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

       5. mul-1-neg N/A

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

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

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

       7. lower-/.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lift-*.f64 90.0

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

   11. Applied rewrites 90.0%

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

   if -5.00000000000000007e42 < x < 0.309999999999999998

   1. Initial program 95.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 99.2%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+42} \lor \neg \left(x \leq 0.31\right):\\ \;\;\;\;\left|\frac{x - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -18 \lor \neg \left(z \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -18.0) (not (<= z 3.7e-6)))
   (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 <= -18.0) || !(z <= 3.7e-6)) {
		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 <= -18.0) || !(z <= 3.7e-6))
		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, -18.0], N[Not[LessEqual[z, 3.7e-6]], $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 < -18 or 3.7000000000000002e-6 < z

   1. Initial program 87.9%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 90.5%

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

   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 89.1%

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

   if -18 < z < 3.7000000000000002e-6

   1. Initial program 92.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -18 \lor \neg \left(z \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+118} \lor \neg \left(z \leq 5.2 \cdot 10^{+82}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.95e+118) (not (<= z 5.2e+82)))
   (fabs (* x (/ z y)))
   (fabs (/ (- x -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.95e+118) || !(z <= 5.2e+82)) {
		tmp = fabs((x * (z / y)));
	} 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 <= (-1.95d+118)) .or. (.not. (z <= 5.2d+82))) then
        tmp = abs((x * (z / y)))
    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 <= -1.95e+118) || !(z <= 5.2e+82)) {
		tmp = Math.abs((x * (z / y)));
	} else {
		tmp = Math.abs(((x - -4.0) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.95e+118) or not (z <= 5.2e+82):
		tmp = math.fabs((x * (z / y)))
	else:
		tmp = math.fabs(((x - -4.0) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.95e+118) || !(z <= 5.2e+82))
		tmp = abs(Float64(x * Float64(z / y)));
	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 <= -1.95e+118) || ~((z <= 5.2e+82)))
		tmp = abs((x * (z / y)));
	else
		tmp = abs(((x - -4.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.95e+118], N[Not[LessEqual[z, 5.2e+82]], $MachinePrecision]], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.95e118 or 5.1999999999999997e82 < z

   1. Initial program 84.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 87.3%

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

   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. lower-*.f64 N/A

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

      3. lift-/.f64 84.1

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

   7. Applied rewrites 84.1%

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

      1. lift-*.f64 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. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 81.8

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

   9. Applied rewrites 81.8%

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

   if -1.95e118 < z < 5.1999999999999997e82

   1. Initial program 93.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 91.1

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

   5. Applied rewrites 91.1%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+118} \lor \neg \left(z \leq 5.2 \cdot 10^{+82}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+118}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+82}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.95e+118)
   (fabs (* x (/ z y)))
   (if (<= z 5.2e+82) (fabs (/ (- x -4.0) y)) (fabs (* (/ x y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.95e+118) {
		tmp = fabs((x * (z / y)));
	} else if (z <= 5.2e+82) {
		tmp = fabs(((x - -4.0) / y));
	} else {
		tmp = fabs(((x / y) * z));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.95d+118)) then
        tmp = abs((x * (z / y)))
    else if (z <= 5.2d+82) then
        tmp = abs(((x - (-4.0d0)) / y))
    else
        tmp = abs(((x / y) * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.95e+118) {
		tmp = Math.abs((x * (z / y)));
	} else if (z <= 5.2e+82) {
		tmp = Math.abs(((x - -4.0) / y));
	} else {
		tmp = Math.abs(((x / y) * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.95e+118:
		tmp = math.fabs((x * (z / y)))
	elif z <= 5.2e+82:
		tmp = math.fabs(((x - -4.0) / y))
	else:
		tmp = math.fabs(((x / y) * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.95e+118)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (z <= 5.2e+82)
		tmp = abs(Float64(Float64(x - -4.0) / y));
	else
		tmp = abs(Float64(Float64(x / y) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.95e+118)
		tmp = abs((x * (z / y)));
	elseif (z <= 5.2e+82)
		tmp = abs(((x - -4.0) / y));
	else
		tmp = abs(((x / y) * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.95e+118], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 5.2e+82], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.95e118

   1. Initial program 91.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 90.7%

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

   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. lower-*.f64 N/A

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

      3. lift-/.f64 78.7

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

   7. Applied rewrites 78.7%

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

      1. lift-*.f64 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. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.0

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

   9. Applied rewrites 80.0%

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

   if -1.95e118 < z < 5.1999999999999997e82

   1. Initial program 93.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 91.1

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

   5. Applied rewrites 91.1%

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

   if 5.1999999999999997e82 < z

   1. Initial program 81.1%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 85.2%

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

   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. lower-*.f64 N/A

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

      3. lift-/.f64 87.4

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

   7. Applied rewrites 87.4%

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

Alternative 6: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+112}:\\ \;\;\;\;\left|\frac{1 - z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.1e+112)
   (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 <= -2.1e+112) {
		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 <= -2.1e+112)
		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[LessEqual[x, -2.1e+112], 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 < -2.0999999999999999e112

   1. Initial program 78.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 -2.0999999999999999e112 < x

   1. Initial program 92.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 98.6%

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

Alternative 7: 69.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10.5) || !(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 <= (-10.5d0)) .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 <= -10.5) || !(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 <= -10.5) 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 <= -10.5) || !(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 <= -10.5) || ~((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, -10.5], 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 < -10.5 or 4 < x

   1. Initial program 85.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 65.7

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

   5. Applied rewrites 65.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 64.7%

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

   if -10.5 < x < 4

   1. Initial program 95.0%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.5% 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 90.4%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 69.4

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

5. Applied rewrites 69.4%

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

Alternative 9: 39.7% 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 90.4%

   \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
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 38.7

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

5. Applied rewrites 38.7%

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

Reproduce

herbie shell --seed 2025060 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))