fabs fraction 1

Percentage Accurate: 91.9% → 99.9%

Time: 3.1s

Alternatives: 10

Speedup: 1.2×

• 20.2% 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: 91.9% 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, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 0.0001:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - \frac{z}{y\_m} \cdot x\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 0.0001)
   (fabs (/ (fma z x (- -4.0 x)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (* (/ z y_m) x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 0.0001) {
		tmp = fabs((fma(z, x, (-4.0 - x)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - ((z / y_m) * x)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 0.0001)
		tmp = abs(Float64(fma(z, x, Float64(-4.0 - x)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(z / y_m) * x)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 0.0001], N[Abs[N[(N[(z * x + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(N[(z / y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000005e-4

   1. Initial program 87.3%

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

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

   3. Applied rewrites 99.9%

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

   if 1.00000000000000005e-4 < y

   1. Initial program 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.8

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

   3. Applied rewrites 99.8%

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

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x + 4}{y\_m} - \frac{x}{y\_m} \cdot z\right|\\ \mathbf{if}\;t\_0 \leq 10^{+291}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ (+ x 4.0) y_m) (* (/ x y_m) z)))))
   (if (<= t_0 1e+291) 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 = fabs((((x + 4.0) / y_m) - ((x / y_m) * z)));
	double tmp;
	if (t_0 <= 1e+291) {
		tmp = 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 = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(Float64(x / y_m) * z)))
	tmp = 0.0
	if (t_0 <= 1e+291)
		tmp = 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[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(N[(x / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 1e+291], t$95$0, 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 (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 9.9999999999999996e290

   1. Initial program 97.6%

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

   if 9.9999999999999996e290 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 72.7%

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

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

   3. Applied rewrites 99.1%

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

Alternative 3: 94.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y\_m}\right|\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+216}:\\ \;\;\;\;\left|\frac{z}{y\_m} \cdot x\right|\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.6:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ (fma z x -4.0) y_m))))
   (if (<= z -2.7e+216)
     (fabs (* (/ z y_m) x))
     (if (<= z -1.0) t_0 (if (<= z 2.6) (fabs (/ (- 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((fma(z, x, -4.0) / y_m));
	double tmp;
	if (z <= -2.7e+216) {
		tmp = fabs(((z / y_m) * x));
	} else if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 2.6) {
		tmp = fabs(((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(fma(z, x, -4.0) / y_m))
	tmp = 0.0
	if (z <= -2.7e+216)
		tmp = abs(Float64(Float64(z / y_m) * x));
	elseif (z <= -1.0)
		tmp = t_0;
	elseif (z <= 2.6)
		tmp = abs(Float64(Float64(x - -4.0) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(z * x + -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -2.7e+216], N[Abs[N[(N[(z / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 2.6], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.7000000000000001e216

   1. Initial program 91.3%

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

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

   3. Applied rewrites 86.5%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--r+ N/A

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

      4. lower--.f64 N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 85.7

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

   6. Applied rewrites 85.7%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 83.4

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

   9. Applied rewrites 83.4%

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

   if -2.7000000000000001e216 < z < -1 or 2.60000000000000009 < z

   1. Initial program 90.0%

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

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

   3. Applied rewrites 92.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 91.7%

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

   if -1 < z < 2.60000000000000009

   1. Initial program 93.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + \frac{x}{y}}\right| \]
   3. 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 99.2

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

   4. Applied rewrites 99.2%

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

Alternative 4: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{1 - z}{y\_m} \cdot x\right|\\ \mathbf{if}\;x \leq -1.65:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4\right)}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ (- 1.0 z) y_m) x))))
   (if (<= x -1.65) t_0 (if (<= x 2.7e-6) (fabs (/ (fma 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((((1.0 - z) / y_m) * x));
	double tmp;
	if (x <= -1.65) {
		tmp = t_0;
	} else if (x <= 2.7e-6) {
		tmp = fabs((fma(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(Float64(1.0 - z) / y_m) * x))
	tmp = 0.0
	if (x <= -1.65)
		tmp = t_0;
	elseif (x <= 2.7e-6)
		tmp = abs(Float64(fma(z, x, -4.0) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.65], t$95$0, If[LessEqual[x, 2.7e-6], N[Abs[N[(N[(z * x + -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6499999999999999 or 2.69999999999999998e-6 < x

   1. Initial program 87.6%

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

   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. *-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 98.0

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

   4. Applied rewrites 98.0%

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

   if -1.6499999999999999 < x < 2.69999999999999998e-6

   1. Initial program 96.3%

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

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.3%

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

Alternative 5: 86.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m} \cdot z\right|\\ \mathbf{if}\;z \leq -500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7500000000:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ x y_m) z))))
   (if (<= z -500000000000.0)
     t_0
     (if (<= z 7500000000.0) (fabs (/ (- 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) * z));
	double tmp;
	if (z <= -500000000000.0) {
		tmp = t_0;
	} else if (z <= 7500000000.0) {
		tmp = fabs(((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) * z))
    if (z <= (-500000000000.0d0)) then
        tmp = t_0
    else if (z <= 7500000000.0d0) then
        tmp = abs(((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) * z));
	double tmp;
	if (z <= -500000000000.0) {
		tmp = t_0;
	} else if (z <= 7500000000.0) {
		tmp = Math.abs(((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) * z))
	tmp = 0
	if z <= -500000000000.0:
		tmp = t_0
	elif z <= 7500000000.0:
		tmp = math.fabs(((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) * z))
	tmp = 0.0
	if (z <= -500000000000.0)
		tmp = t_0;
	elseif (z <= 7500000000.0)
		tmp = abs(Float64(Float64(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) * z));
	tmp = 0.0;
	if (z <= -500000000000.0)
		tmp = t_0;
	elseif (z <= 7500000000.0)
		tmp = abs(((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] * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -500000000000.0], t$95$0, If[LessEqual[z, 7500000000.0], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5e11 or 7.5e9 < z

   1. Initial program 90.0%

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

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

   3. Applied rewrites 91.5%

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

   4. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 73.3

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

   6. Applied rewrites 73.3%

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

   if -5e11 < z < 7.5e9

   1. Initial program 93.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + \frac{x}{y}}\right| \]
   3. 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 98.1

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

   4. Applied rewrites 98.1%

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

Alternative 6: 86.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{z}{y\_m} \cdot x\right|\\ \mathbf{if}\;z \leq -500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7500000000:\\ \;\;\;\;\left|\frac{x - -4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* (/ z y_m) x))))
   (if (<= z -500000000000.0)
     t_0
     (if (<= z 7500000000.0) (fabs (/ (- 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(((z / y_m) * x));
	double tmp;
	if (z <= -500000000000.0) {
		tmp = t_0;
	} else if (z <= 7500000000.0) {
		tmp = fabs(((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(((z / y_m) * x))
    if (z <= (-500000000000.0d0)) then
        tmp = t_0
    else if (z <= 7500000000.0d0) then
        tmp = abs(((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(((z / y_m) * x));
	double tmp;
	if (z <= -500000000000.0) {
		tmp = t_0;
	} else if (z <= 7500000000.0) {
		tmp = Math.abs(((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(((z / y_m) * x))
	tmp = 0
	if z <= -500000000000.0:
		tmp = t_0
	elif z <= 7500000000.0:
		tmp = math.fabs(((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(z / y_m) * x))
	tmp = 0.0
	if (z <= -500000000000.0)
		tmp = t_0;
	elseif (z <= 7500000000.0)
		tmp = abs(Float64(Float64(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(((z / y_m) * x));
	tmp = 0.0;
	if (z <= -500000000000.0)
		tmp = t_0;
	elseif (z <= 7500000000.0)
		tmp = abs(((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[(z / y$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -500000000000.0], t$95$0, If[LessEqual[z, 7500000000.0], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5e11 or 7.5e9 < z

   1. Initial program 90.0%

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

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

   3. Applied rewrites 91.5%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--r+ N/A

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

      4. lower--.f64 N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 86.6

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

   6. Applied rewrites 86.6%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 72.8

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

   9. Applied rewrites 72.8%

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

   if -5e11 < z < 7.5e9

   1. Initial program 93.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + \frac{x}{y}}\right| \]
   3. 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 98.1

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

   4. Applied rewrites 98.1%

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

Alternative 7: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+50}:\\ \;\;\;\;\left|\frac{1 - z}{y\_m} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(z, x, -4 - x\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -5.6e+50)
   (fabs (* (/ (- 1.0 z) y_m) x))
   (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 <= -5.6e+50) {
		tmp = fabs((((1.0 - z) / y_m) * x));
	} 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 <= -5.6e+50)
		tmp = abs(Float64(Float64(Float64(1.0 - z) / y_m) * x));
	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, -5.6e+50], N[Abs[N[(N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $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 < -5.5999999999999996e50

   1. Initial program 85.2%

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

   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. *-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

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

   4. Applied rewrites 99.8%

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

   if -5.5999999999999996e50 < x

   1. Initial program 93.7%

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

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

   3. Applied rewrites 97.2%

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

Alternative 8: 68.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -1.55) t_0 (if (<= x 4.0) (fabs (/ 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));
	double tmp;
	if (x <= -1.55) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = fabs((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))
    if (x <= (-1.55d0)) then
        tmp = t_0
    else if (x <= 4.0d0) then
        tmp = abs((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));
	double tmp;
	if (x <= -1.55) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = Math.abs((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))
	tmp = 0
	if x <= -1.55:
		tmp = t_0
	elif x <= 4.0:
		tmp = math.fabs((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(x / y_m))
	tmp = 0.0
	if (x <= -1.55)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs(Float64(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));
	tmp = 0.0;
	if (x <= -1.55)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs((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[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.55], t$95$0, If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 4 < x

   1. Initial program 87.5%

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

   2. Taylor expanded in z around 0

      \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + \frac{x}{y}}\right| \]
   3. 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 63.3

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 62.1%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 75.4

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

   4. Applied rewrites 75.4%

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

Alternative 9: 69.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in z around 0

   \[\leadsto \left|\color{blue}{4 \cdot \frac{1}{y} + \frac{x}{y}}\right| \]
3. 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 69.8

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

4. Applied rewrites 69.8%

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

Alternative 10: 40.3% accurate, 2.6× 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 91.9%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 40.3

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

4. Applied rewrites 40.3%

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

Reproduce

herbie shell --seed 2025107 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))