fabs fraction 1

Percentage Accurate: 91.6% → 99.4%

Time: 9.3s

Alternatives: 11

Speedup: 1.0×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y_m}, \frac{-4 - x}{y_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 5e-85)
   (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5e-85) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 5e-85)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(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[y$95$m, 5e-85], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000002e-85

   1. Initial program 90.5%

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

   3. Taylor expanded in y around 0 96.8%

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

   if 5.0000000000000002e-85 < y

   1. Initial program 95.3%

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

   3. Simplified 99.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{4 + x}{y_m} - z \cdot \frac{x}{y_m}\right|\\ \mathbf{if}\;t_0 \leq 10^{+69}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{elif}\;t_0 \leq 7 \cdot 10^{+298}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ (+ 4.0 x) y_m) (* z (/ x y_m))))))
   (if (<= t_0 1e+69)
     (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
     (if (<= t_0 7e+298) t_0 (fabs (/ (+ z -1.0) (/ y_m x)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((((4.0 + x) / y_m) - (z * (x / y_m))));
	double tmp;
	if (t_0 <= 1e+69) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else if (t_0 <= 7e+298) {
		tmp = t_0;
	} else {
		tmp = fabs(((z + -1.0) / (y_m / x)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    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((((4.0d0 + x) / y_m) - (z * (x / y_m))))
    if (t_0 <= 1d+69) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else if (t_0 <= 7d+298) then
        tmp = t_0
    else
        tmp = abs(((z + (-1.0d0)) / (y_m / x)))
    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((((4.0 + x) / y_m) - (z * (x / y_m))));
	double tmp;
	if (t_0 <= 1e+69) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else if (t_0 <= 7e+298) {
		tmp = t_0;
	} else {
		tmp = Math.abs(((z + -1.0) / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((((4.0 + x) / y_m) - (z * (x / y_m))))
	tmp = 0
	if t_0 <= 1e+69:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	elif t_0 <= 7e+298:
		tmp = t_0
	else:
		tmp = math.fabs(((z + -1.0) / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(Float64(4.0 + x) / y_m) - Float64(z * Float64(x / y_m))))
	tmp = 0.0
	if (t_0 <= 1e+69)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	elseif (t_0 <= 7e+298)
		tmp = t_0;
	else
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((((4.0 + x) / y_m) - (z * (x / y_m))));
	tmp = 0.0;
	if (t_0 <= 1e+69)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	elseif (t_0 <= 7e+298)
		tmp = t_0;
	else
		tmp = abs(((z + -1.0) / (y_m / x)));
	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[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 1e+69], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 7e+298], t$95$0, N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 1.0000000000000001e69

   1. Initial program 93.2%

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

   3. Taylor expanded in y around 0 99.9%

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

   if 1.0000000000000001e69 < (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 7.00000000000000018e298

   1. Initial program 99.9%

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

   if 7.00000000000000018e298 < (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 73.3%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in x around inf 100.0%

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

   6. Taylor expanded in y around 0 96.2%

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

      1. sub-neg 96.2%

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

      2. metadata-eval 96.2%

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

      3. *-commutative 96.2%

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

      4. associate-/l* 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \left|\color{blue}{\frac{z + -1}{\frac{y}{x}}}\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{4 + x}{y} - z \cdot \frac{x}{y}\right| \leq 10^{+69}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{elif}\;\left|\frac{4 + x}{y} - z \cdot \frac{x}{y}\right| \leq 7 \cdot 10^{+298}:\\ \;\;\;\;\left|\frac{4 + x}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -126000 \lor \neg \left(x \leq 4.7 \cdot 10^{-16}\right):\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y_m} - \frac{1}{y_m}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -126000.0) (not (<= x 4.7e-16)))
   (fabs (* x (- (/ z y_m) (/ 1.0 y_m))))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -126000.0) || !(x <= 4.7e-16)) {
		tmp = fabs((x * ((z / y_m) - (1.0 / y_m))));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-126000.0d0)) .or. (.not. (x <= 4.7d-16))) then
        tmp = abs((x * ((z / y_m) - (1.0d0 / y_m))))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -126000.0) || !(x <= 4.7e-16)) {
		tmp = Math.abs((x * ((z / y_m) - (1.0 / y_m))));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -126000.0) or not (x <= 4.7e-16):
		tmp = math.fabs((x * ((z / y_m) - (1.0 / y_m))))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -126000.0) || !(x <= 4.7e-16))
		tmp = abs(Float64(x * Float64(Float64(z / y_m) - Float64(1.0 / y_m))));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -126000.0) || ~((x <= 4.7e-16)))
		tmp = abs((x * ((z / y_m) - (1.0 / y_m))));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -126000.0], N[Not[LessEqual[x, 4.7e-16]], $MachinePrecision]], N[Abs[N[(x * N[(N[(z / y$95$m), $MachinePrecision] - N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -126000 or 4.70000000000000044e-16 < x

   1. Initial program 90.1%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in x around inf 98.5%

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

   if -126000 < x < 4.70000000000000044e-16

   1. Initial program 93.7%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in z around 0 76.2%

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

      1. associate-*r/ 76.2%

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

      2. distribute-lft-in 76.2%

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

      3. metadata-eval 76.2%

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

      4. neg-mul-1 76.2%

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

      5. sub-neg 76.2%

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

   7. Simplified 76.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -126000 \lor \neg \left(x \leq 4.7 \cdot 10^{-16}\right):\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} - \frac{1}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+107}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y_m} - \frac{1}{y_m}\right)\right|\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+86}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.35e+107)
   (fabs (* x (- (/ z y_m) (/ 1.0 y_m))))
   (if (<= x 1.3e+86)
     (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
     (fabs (/ (+ z -1.0) (/ y_m x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.35e+107) {
		tmp = fabs((x * ((z / y_m) - (1.0 / y_m))));
	} else if (x <= 1.3e+86) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs(((z + -1.0) / (y_m / x)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.35d+107)) then
        tmp = abs((x * ((z / y_m) - (1.0d0 / y_m))))
    else if (x <= 1.3d+86) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else
        tmp = abs(((z + (-1.0d0)) / (y_m / x)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.35e+107) {
		tmp = Math.abs((x * ((z / y_m) - (1.0 / y_m))));
	} else if (x <= 1.3e+86) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = Math.abs(((z + -1.0) / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -1.35e+107:
		tmp = math.fabs((x * ((z / y_m) - (1.0 / y_m))))
	elif x <= 1.3e+86:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	else:
		tmp = math.fabs(((z + -1.0) / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1.35e+107)
		tmp = abs(Float64(x * Float64(Float64(z / y_m) - Float64(1.0 / y_m))));
	elseif (x <= 1.3e+86)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -1.35e+107)
		tmp = abs((x * ((z / y_m) - (1.0 / y_m))));
	elseif (x <= 1.3e+86)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	else
		tmp = abs(((z + -1.0) / (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1.35e+107], N[Abs[N[(x * N[(N[(z / y$95$m), $MachinePrecision] - N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3e+86], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001e107

   1. Initial program 84.4%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in x around inf 99.7%

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

   if -1.3500000000000001e107 < x < 1.2999999999999999e86

   1. Initial program 95.1%

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

   3. Taylor expanded in y around 0 99.9%

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

   if 1.2999999999999999e86 < x

   1. Initial program 87.1%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in x around inf 99.8%

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

   6. Taylor expanded in y around 0 88.3%

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

      1. sub-neg 88.3%

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

      2. metadata-eval 88.3%

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

      3. *-commutative 88.3%

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

      4. associate-/l* 99.8%

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

   8. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\frac{z + -1}{\frac{y}{x}}}\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+107}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} - \frac{1}{y}\right)\right|\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+86}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -5.9 \cdot 10^{+147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-23}:\\ \;\;\;\;\left|x \cdot \frac{z}{y_m}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -5.9e+147)
     t_0
     (if (<= x -4e-23)
       (fabs (* x (/ z y_m)))
       (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 <= -5.9e+147) {
		tmp = t_0;
	} else if (x <= -4e-23) {
		tmp = fabs((x * (z / y_m)));
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    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 <= (-5.9d+147)) then
        tmp = t_0
    else if (x <= (-4d-23)) then
        tmp = abs((x * (z / y_m)))
    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 <= -5.9e+147) {
		tmp = t_0;
	} else if (x <= -4e-23) {
		tmp = Math.abs((x * (z / y_m)));
	} 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 <= -5.9e+147:
		tmp = t_0
	elif x <= -4e-23:
		tmp = math.fabs((x * (z / y_m)))
	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 <= -5.9e+147)
		tmp = t_0;
	elseif (x <= -4e-23)
		tmp = abs(Float64(x * Float64(z / y_m)));
	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 <= -5.9e+147)
		tmp = t_0;
	elseif (x <= -4e-23)
		tmp = abs((x * (z / y_m)));
	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, -5.9e+147], t$95$0, If[LessEqual[x, -4e-23], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.9000000000000001e147 or 4 < x

   1. Initial program 87.8%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in x around inf 98.7%

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

   6. Taylor expanded in z around 0 75.2%

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

      1. neg-mul-1 75.2%

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

      2. distribute-neg-frac 75.2%

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

   8. Simplified 75.2%

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

   if -5.9000000000000001e147 < x < -3.99999999999999984e-23

   1. Initial program 97.1%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 89.7%

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

   6. Taylor expanded in z around inf 52.5%

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

      1. associate-*r/ 62.7%

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

   8. Simplified 62.7%

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

   if -3.99999999999999984e-23 < x < 4

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0 76.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{+147}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-23}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -8.4 \cdot 10^{+147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.85 \cdot 10^{-13}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -8.4e+147)
     t_0
     (if (<= x -3.85e-13)
       (fabs (* z (/ x y_m)))
       (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 <= -8.4e+147) {
		tmp = t_0;
	} else if (x <= -3.85e-13) {
		tmp = fabs((z * (x / y_m)));
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    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 <= (-8.4d+147)) then
        tmp = t_0
    else if (x <= (-3.85d-13)) then
        tmp = abs((z * (x / y_m)))
    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 <= -8.4e+147) {
		tmp = t_0;
	} else if (x <= -3.85e-13) {
		tmp = Math.abs((z * (x / y_m)));
	} 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 <= -8.4e+147:
		tmp = t_0
	elif x <= -3.85e-13:
		tmp = math.fabs((z * (x / y_m)))
	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 <= -8.4e+147)
		tmp = t_0;
	elseif (x <= -3.85e-13)
		tmp = abs(Float64(z * Float64(x / y_m)));
	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 <= -8.4e+147)
		tmp = t_0;
	elseif (x <= -3.85e-13)
		tmp = abs((z * (x / y_m)));
	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, -8.4e+147], t$95$0, If[LessEqual[x, -3.85e-13], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.40000000000000024e147 or 4 < x

   1. Initial program 87.8%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in x around inf 98.7%

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

   6. Taylor expanded in z around 0 75.2%

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

      1. neg-mul-1 75.2%

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

      2. distribute-neg-frac 75.2%

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

   8. Simplified 75.2%

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

   if -8.40000000000000024e147 < x < -3.8499999999999998e-13

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 92.4%

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

   6. Taylor expanded in z around inf 52.4%

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

      1. associate-*l/ 66.9%

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

      2. *-commutative 66.9%

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

   8. Simplified 66.9%

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

   if -3.8499999999999998e-13 < x < 4

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0 75.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{+147}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3.85 \cdot 10^{-13}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -270000 \lor \neg \left(x \leq 2.8 \cdot 10^{-18}\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y_m}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -270000.0) (not (<= x 2.8e-18)))
   (fabs (/ (+ z -1.0) (/ y_m x)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -270000.0) || !(x <= 2.8e-18)) {
		tmp = fabs(((z + -1.0) / (y_m / x)));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-270000.0d0)) .or. (.not. (x <= 2.8d-18))) then
        tmp = abs(((z + (-1.0d0)) / (y_m / x)))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -270000.0) || !(x <= 2.8e-18)) {
		tmp = Math.abs(((z + -1.0) / (y_m / x)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -270000.0) or not (x <= 2.8e-18):
		tmp = math.fabs(((z + -1.0) / (y_m / x)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -270000.0) || !(x <= 2.8e-18))
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y_m / x)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -270000.0) || ~((x <= 2.8e-18)))
		tmp = abs(((z + -1.0) / (y_m / x)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -270000.0], N[Not[LessEqual[x, 2.8e-18]], $MachinePrecision]], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7e5 or 2.80000000000000012e-18 < x

   1. Initial program 90.1%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in x around inf 98.5%

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

   6. Taylor expanded in y around 0 90.4%

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

      1. sub-neg 90.4%

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

      2. metadata-eval 90.4%

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

      3. *-commutative 90.4%

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

      4. associate-/l* 98.4%

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

   8. Simplified 98.4%

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

   if -2.7e5 < x < 2.80000000000000012e-18

   1. Initial program 93.7%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in z around 0 76.2%

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

      1. associate-*r/ 76.2%

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

      2. distribute-lft-in 76.2%

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

      3. metadata-eval 76.2%

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

      4. neg-mul-1 76.2%

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

      5. sub-neg 76.2%

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

   7. Simplified 76.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -270000 \lor \neg \left(x \leq 2.8 \cdot 10^{-18}\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.6 \cdot 10^{-49}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y_m} - \frac{x}{\frac{y_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1.6e-49)
   (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
   (fabs (- (/ (+ 4.0 x) y_m) (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.6e-49) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs((((4.0 + x) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1.6d-49) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else
        tmp = abs((((4.0d0 + x) / y_m) - (x / (y_m / z))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.6e-49) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((4.0 + x) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 1.6e-49:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((4.0 + x) / y_m) - (x / (y_m / z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.6e-49)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) / y_m) - Float64(x / Float64(y_m / z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.6e-49)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	else
		tmp = abs((((4.0 + x) / y_m) - (x / (y_m / z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 1.6e-49], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.60000000000000001e-49

   1. Initial program 90.8%

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

   3. Taylor expanded in y around 0 96.9%

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

   if 1.60000000000000001e-49 < y

   1. Initial program 95.0%

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

      1. associate-/r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-49}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{z}}\right|\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+63}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -6e+15)
   (fabs (/ x (/ y_m z)))
   (if (<= z 1.2e+63) (fabs (/ (- -4.0 x) y_m)) (fabs (* x (/ z y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -6e+15) {
		tmp = fabs((x / (y_m / z)));
	} else if (z <= 1.2e+63) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((x * (z / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-6d+15)) then
        tmp = abs((x / (y_m / z)))
    else if (z <= 1.2d+63) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((x * (z / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -6e+15) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (z <= 1.2e+63) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x * (z / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -6e+15:
		tmp = math.fabs((x / (y_m / z)))
	elif z <= 1.2e+63:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x * (z / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -6e+15)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (z <= 1.2e+63)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x * Float64(z / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -6e+15)
		tmp = abs((x / (y_m / z)));
	elseif (z <= 1.2e+63)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x * (z / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -6e+15], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.2e+63], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6e15

   1. Initial program 91.3%

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

   3. Taylor expanded in z around inf 74.5%

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

      1. mul-1-neg 74.5%

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

      2. associate-*l/ 72.0%

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

      3. distribute-rgt-neg-out 72.0%

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

   5. Simplified 72.0%

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

      1. add-sqr-sqrt 71.7%

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

      2. sqrt-unprod 58.3%

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

      3. sqr-neg 58.3%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 72.0%

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

      6. associate-/r/ 74.7%

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

   7. Applied egg-rr 74.7%

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

   if -6e15 < z < 1.2e63

   1. Initial program 95.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 97.0%

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

      1. associate-*r/ 97.0%

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

      2. distribute-lft-in 97.0%

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

      3. metadata-eval 97.0%

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

      4. neg-mul-1 97.0%

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

      5. sub-neg 97.0%

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

   7. Simplified 97.0%

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

   if 1.2e63 < z

   1. Initial program 80.3%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in x around inf 69.6%

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

   6. Taylor expanded in z around inf 55.7%

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

      1. associate-*r/ 69.6%

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

   8. Simplified 69.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+63}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.1% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y_m)) (fabs (/ 4.0 y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = fabs((x / y_m));
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-10.5d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y_m))
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = Math.abs((x / y_m));
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -10.5) or not (x <= 4.0):
		tmp = math.fabs((x / y_m))
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -10.5) || !(x <= 4.0))
		tmp = abs(Float64(x / y_m));
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -10.5) || ~((x <= 4.0)))
		tmp = abs((x / y_m));
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -10.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 90.1%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in x around inf 98.0%

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

   6. Taylor expanded in z around 0 67.7%

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

      1. neg-mul-1 67.7%

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

      2. distribute-neg-frac 67.7%

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

   8. Simplified 67.7%

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

   if -10.5 < x < 4

   1. Initial program 93.7%

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

   3. Taylor expanded in x around 0 74.9%

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

4. Final simplification 71.4%

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

Alternative 11: 39.8% accurate, 1.1× 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 = abs(y)
real(8) function code(x, y_m, z)
    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 92.0%

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

3. Taylor expanded in x around 0 41.4%

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

4. Final simplification 41.4%

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

Reproduce

herbie shell --seed 2024024 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))