fabs fraction 1

Percentage Accurate: 92.2% → 99.9%

Time: 2.8s

Alternatives: 8

Speedup: 1.0×

• 20.0% 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: 92.2% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{-1}{y\_m} \cdot z, \frac{x - -4}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4e-14)
   (fabs (/ (fma z x (- -4.0 x)) y_m))
   (fabs (fma x (* (/ -1.0 y_m) z) (/ (- x -4.0) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-14) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs(fma(x, ((-1.0 / y_m) * z), ((x - -4.0) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 4e-14)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(fma(x, Float64(Float64(-1.0 / y_m) * z), Float64(Float64(x - -4.0) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 4e-14], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(-1.0 / y$95$m), $MachinePrecision] * z), $MachinePrecision] + N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4e-14

   1. Initial program 92.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 95.7%

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

   if 4e-14 < y

   1. Initial program 92.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mult-flip N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. frac-2neg-rev N/A

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

      13. lower-/.f64 93.5

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 93.5

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

   3. Applied rewrites 93.5%

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

Alternative 2: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{z \cdot x - 4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y_m) (- 1.0 z)))))
   (if (<= x -1.45)
     t_0
     (if (<= x 5.9e-13) (fabs (/ (- (* z x) 4.0) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((x / y_m) * (1.0 - z)));
	double tmp;
	if (x <= -1.45) {
		tmp = t_0;
	} else if (x <= 5.9e-13) {
		tmp = fabs((((z * x) - 4.0) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m =     private
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_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((x / y_m) * (1.0d0 - z)))
    if (x <= (-1.45d0)) then
        tmp = t_0
    else if (x <= 5.9d-13) then
        tmp = abs((((z * x) - 4.0d0) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(((x / y_m) * (1.0 - z)));
	double tmp;
	if (x <= -1.45) {
		tmp = t_0;
	} else if (x <= 5.9e-13) {
		tmp = Math.abs((((z * x) - 4.0) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((x / y_m) * (1.0 - z)))
	tmp = 0
	if x <= -1.45:
		tmp = t_0
	elif x <= 5.9e-13:
		tmp = math.fabs((((z * x) - 4.0) / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)))
	tmp = 0.0
	if (x <= -1.45)
		tmp = t_0;
	elseif (x <= 5.9e-13)
		tmp = abs(Float64(Float64(Float64(z * x) - 4.0) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(((x / y_m) * (1.0 - z)));
	tmp = 0.0;
	if (x <= -1.45)
		tmp = t_0;
	elseif (x <= 5.9e-13)
		tmp = abs((((z * x) - 4.0) / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.45], t$95$0, If[LessEqual[x, 5.9e-13], N[Abs[N[(N[(N[(z * x), $MachinePrecision] - 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996 or 5.9000000000000001e-13 < x

   1. Initial program 92.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 60.9

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

   4. Applied rewrites 60.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 60.9

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

   6. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 61.2

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

   8. Applied rewrites 61.2%

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

   if -1.44999999999999996 < x < 5.9000000000000001e-13

   1. Initial program 92.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 79.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 74.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.0

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

   6. Applied rewrites 74.0%

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

Alternative 3: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+116}:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* (/ x y_m) z))))
   (if (<= t_0 -2e+116) (fabs t_0) (fabs (/ (fma z x (- -4.0 x)) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double tmp;
	if (t_0 <= -2e+116) {
		tmp = fabs(t_0);
	} else {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(x / y_m) * z))
	tmp = 0.0
	if (t_0 <= -2e+116)
		tmp = abs(t_0);
	else
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+116], N[Abs[t$95$0], $MachinePrecision], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -2.00000000000000003e116

   1. Initial program 92.2%

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

   if -2.00000000000000003e116 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 92.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 95.7%

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

Alternative 4: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+86}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1e+86)
   (fabs (* (/ x y_m) (- 1.0 z)))
   (fabs (/ (fma z x (- -4.0 x)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1e+86) {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1e+86)
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	else
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y_m));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1e+86], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e86

   1. Initial program 92.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 60.9

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

   4. Applied rewrites 60.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 60.9

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

   6. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 61.2

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

   8. Applied rewrites 61.2%

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

   if -1e86 < x

   1. Initial program 92.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 95.7%

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

Alternative 5: 86.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\\ t_1 := \left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-173}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y_m) (* (/ x y_m) z)))
        (t_1 (fabs (* (/ x y_m) (- 1.0 z)))))
   (if (<= t_0 -1e-173)
     t_1
     (if (<= t_0 5e+290) (fabs (/ (- -4.0 x) y_m)) t_1))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double t_1 = fabs(((x / y_m) * (1.0 - z)));
	double tmp;
	if (t_0 <= -1e-173) {
		tmp = t_1;
	} else if (t_0 <= 5e+290) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

y_m =     private
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_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + 4.0d0) / y_m) - ((x / y_m) * z)
    t_1 = abs(((x / y_m) * (1.0d0 - z)))
    if (t_0 <= (-1d-173)) then
        tmp = t_1
    else if (t_0 <= 5d+290) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	double t_1 = Math.abs(((x / y_m) * (1.0 - z)));
	double tmp;
	if (t_0 <= -1e-173) {
		tmp = t_1;
	} else if (t_0 <= 5e+290) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z)
	t_1 = math.fabs(((x / y_m) * (1.0 - z)))
	tmp = 0
	if t_0 <= -1e-173:
		tmp = t_1
	elif t_0 <= 5e+290:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(x / y_m) * z))
	t_1 = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)))
	tmp = 0.0
	if (t_0 <= -1e-173)
		tmp = t_1;
	elseif (t_0 <= 5e+290)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = ((x + 4.0) / y_m) - ((x / y_m) * z);
	t_1 = abs(((x / y_m) * (1.0 - z)));
	tmp = 0.0;
	if (t_0 <= -1e-173)
		tmp = t_1;
	elseif (t_0 <= 5e+290)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-173], t$95$1, If[LessEqual[t$95$0, 5e+290], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < -1e-173 or 4.9999999999999998e290 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))

   1. Initial program 92.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 60.9

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

   4. Applied rewrites 60.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 60.9

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

   6. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 61.2

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

   8. Applied rewrites 61.2%

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

   if -1e-173 < (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)) < 4.9999999999999998e290

   1. Initial program 92.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.0

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

   6. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. distribute-rgt-out-- N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-neg-out N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. metadata-eval N/A

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

      14. *-rgt-identity 70.0

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

   8. Applied rewrites 70.0%

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

Alternative 6: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x \cdot z}{y\_m}\right|\\ \mathbf{if}\;z \leq -5 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 122000000000:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (* x z) y_m))))
   (if (<= z -5e+154)
     t_0
     (if (<= z 122000000000.0) (fabs (/ (- -4.0 x) y_m)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs(((x * z) / y_m));
	double tmp;
	if (z <= -5e+154) {
		tmp = t_0;
	} else if (z <= 122000000000.0) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m =     private
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_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(((x * z) / y_m))
    if (z <= (-5d+154)) then
        tmp = t_0
    else if (z <= 122000000000.0d0) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs(((x * z) / y_m));
	double tmp;
	if (z <= -5e+154) {
		tmp = t_0;
	} else if (z <= 122000000000.0) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs(((x * z) / y_m))
	tmp = 0
	if z <= -5e+154:
		tmp = t_0
	elif z <= 122000000000.0:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(x * z) / y_m))
	tmp = 0.0
	if (z <= -5e+154)
		tmp = t_0;
	elseif (z <= 122000000000.0)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs(((x * z) / y_m));
	tmp = 0.0;
	if (z <= -5e+154)
		tmp = t_0;
	elseif (z <= 122000000000.0)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -5e+154], t$95$0, If[LessEqual[z, 122000000000.0], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.00000000000000004e154 or 1.22e11 < z

   1. Initial program 92.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in z around inf

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

      1. lower-*.f64 37.2

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

   6. Applied rewrites 37.2%

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

   if -5.00000000000000004e154 < z < 1.22e11

   1. Initial program 92.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift--.f64 N/A

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

      3. fabs-sub N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 70.0

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

   6. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. distribute-rgt-out-- N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-neg-out N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. metadata-eval N/A

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

      14. *-rgt-identity 70.0

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

   8. Applied rewrites 70.0%

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

Alternative 7: 70.0% accurate, 2.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (fabs (/ (- -4.0 x) y_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.2%

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

   1. lift-fabs.f64 N/A

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

   2. lift--.f64 N/A

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

   3. fabs-sub N/A

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

   4. lower-fabs.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lift-/.f64 N/A

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

   9. sub-div N/A

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

   10. lower-/.f64 N/A

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

3. Applied rewrites 95.7%

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

4. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 70.0

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

6. Applied rewrites 70.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. add-flip N/A

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

   4. distribute-rgt-out-- N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-neg-out N/A

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

   7. *-commutative N/A

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

   8. mul-1-neg N/A

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

   9. lower--.f64 N/A

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

   10. mul-1-neg N/A

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

   11. *-commutative N/A

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

   12. distribute-rgt-neg-out N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity 70.0

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

8. Applied rewrites 70.0%

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

Alternative 8: 40.9% accurate, 3.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (fabs (/ 4.0 y_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.2%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 40.9

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

4. Applied rewrites 40.9%

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

Reproduce

herbie shell --seed 2025156 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))