fabs fraction 1

Percentage Accurate: 91.6% → 99.8%

Time: 7.9s

Alternatives: 11

Speedup: 1.0×

• 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

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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 3 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y_m}, \frac{-4 - x}{y_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 3e+34)
   (fabs (/ (fma x z (- -4.0 x)) 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 <= 3e+34) {
		tmp = fabs((fma(x, z, (-4.0 - x)) / 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 <= 3e+34)
		tmp = abs(Float64(fma(x, z, Float64(-4.0 - x)) / 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, 3e+34], N[Abs[N[(N[(x * z + N[(-4.0 - x), $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 < 3.00000000000000018e34

   1. Initial program 89.3%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in x around -inf 94.6%

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

   7. Simplified 96.5%

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

   if 3.00000000000000018e34 < y

   1. Initial program 94.7%

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

   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 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, z, -4 - x\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + 4}{y_m}\\ t_1 := z \cdot \frac{x}{y_m}\\ \mathbf{if}\;t_0 - t_1 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\left|t_1 - t_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y_m)) (t_1 (* z (/ x y_m))))
   (if (<= (- t_0 t_1) 5e+292)
     (fabs (- t_1 t_0))
     (fabs (/ (- (+ x 4.0) (* x z)) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = (x + 4.0) / y_m;
	double t_1 = z * (x / y_m);
	double tmp;
	if ((t_0 - t_1) <= 5e+292) {
		tmp = fabs((t_1 - t_0));
	} else {
		tmp = fabs((((x + 4.0) - (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y_m
    t_1 = z * (x / y_m)
    if ((t_0 - t_1) <= 5d+292) then
        tmp = abs((t_1 - t_0))
    else
        tmp = abs((((x + 4.0d0) - (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 t_0 = (x + 4.0) / y_m;
	double t_1 = z * (x / y_m);
	double tmp;
	if ((t_0 - t_1) <= 5e+292) {
		tmp = Math.abs((t_1 - t_0));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = (x + 4.0) / y_m
	t_1 = z * (x / y_m)
	tmp = 0
	if (t_0 - t_1) <= 5e+292:
		tmp = math.fabs((t_1 - t_0))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = Float64(Float64(x + 4.0) / y_m)
	t_1 = Float64(z * Float64(x / y_m))
	tmp = 0.0
	if (Float64(t_0 - t_1) <= 5e+292)
		tmp = abs(Float64(t_1 - t_0));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = (x + 4.0) / y_m;
	t_1 = z * (x / y_m);
	tmp = 0.0;
	if ((t_0 - t_1) <= 5e+292)
		tmp = abs((t_1 - t_0));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - t$95$1), $MachinePrecision], 5e+292], N[Abs[N[(t$95$1 - t$95$0), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < 4.9999999999999996e292

   1. Initial program 98.2%

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

   if 4.9999999999999996e292 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))

   1. Initial program 52.4%

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

   3. Taylor expanded in y around 0 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\left|z \cdot \frac{x}{y} - \frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y_m}\right|\\ t_1 := \left|x \cdot \frac{z}{y_m}\right|\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{+127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-111}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 185000000 \lor \neg \left(x \leq 1.46 \cdot 10^{+164}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* x (/ z y_m)))))
   (if (<= x -1.95e+127)
     t_0
     (if (<= x -1.1e-7)
       t_1
       (if (<= x 4.2e-111)
         (fabs (/ 4.0 y_m))
         (if (or (<= x 185000000.0) (not (<= x 1.46e+164))) t_1 t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((x * (z / y_m)));
	double tmp;
	if (x <= -1.95e+127) {
		tmp = t_0;
	} else if (x <= -1.1e-7) {
		tmp = t_1;
	} else if (x <= 4.2e-111) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 185000000.0) || !(x <= 1.46e+164)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y_m))
    t_1 = abs((x * (z / y_m)))
    if (x <= (-1.95d+127)) then
        tmp = t_0
    else if (x <= (-1.1d-7)) then
        tmp = t_1
    else if (x <= 4.2d-111) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 185000000.0d0) .or. (.not. (x <= 1.46d+164))) then
        tmp = t_1
    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 t_1 = Math.abs((x * (z / y_m)));
	double tmp;
	if (x <= -1.95e+127) {
		tmp = t_0;
	} else if (x <= -1.1e-7) {
		tmp = t_1;
	} else if (x <= 4.2e-111) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 185000000.0) || !(x <= 1.46e+164)) {
		tmp = t_1;
	} 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))
	t_1 = math.fabs((x * (z / y_m)))
	tmp = 0
	if x <= -1.95e+127:
		tmp = t_0
	elif x <= -1.1e-7:
		tmp = t_1
	elif x <= 4.2e-111:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 185000000.0) or not (x <= 1.46e+164):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(x * Float64(z / y_m)))
	tmp = 0.0
	if (x <= -1.95e+127)
		tmp = t_0;
	elseif (x <= -1.1e-7)
		tmp = t_1;
	elseif (x <= 4.2e-111)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 185000000.0) || !(x <= 1.46e+164))
		tmp = t_1;
	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));
	t_1 = abs((x * (z / y_m)));
	tmp = 0.0;
	if (x <= -1.95e+127)
		tmp = t_0;
	elseif (x <= -1.1e-7)
		tmp = t_1;
	elseif (x <= 4.2e-111)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 185000000.0) || ~((x <= 1.46e+164)))
		tmp = t_1;
	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]}, Block[{t$95$1 = N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.95e+127], t$95$0, If[LessEqual[x, -1.1e-7], t$95$1, If[LessEqual[x, 4.2e-111], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 185000000.0], N[Not[LessEqual[x, 1.46e+164]], $MachinePrecision]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.94999999999999991e127 or 1.85e8 < x < 1.45999999999999994e164

   1. Initial program 81.9%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in x around inf 98.4%

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

   6. Taylor expanded in z around 0 77.1%

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

      1. associate-*r/ 77.1%

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

      2. neg-mul-1 77.1%

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

   8. Simplified 77.1%

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

   if -1.94999999999999991e127 < x < -1.1000000000000001e-7 or 4.1999999999999997e-111 < x < 1.85e8 or 1.45999999999999994e164 < x

   1. Initial program 90.8%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in x around -inf 98.8%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in z around inf 61.7%

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

      1. associate-*r/ 69.9%

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

   10. Simplified 69.9%

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

   if -1.1000000000000001e-7 < x < 4.1999999999999997e-111

   1. Initial program 96.8%

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

   3. Taylor expanded in x around 0 81.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-111}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 185000000 \lor \neg \left(x \leq 1.46 \cdot 10^{+164}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y_m}\right|\\ t_1 := \left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-111}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 180000000 \lor \neg \left(x \leq 3.6 \cdot 10^{+166}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* z (/ x y_m)))))
   (if (<= x -6.5e+127)
     t_0
     (if (<= x -1.85e-7)
       t_1
       (if (<= x 4.2e-111)
         (fabs (/ 4.0 y_m))
         (if (or (<= x 180000000.0) (not (<= x 3.6e+166))) t_1 t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((z * (x / y_m)));
	double tmp;
	if (x <= -6.5e+127) {
		tmp = t_0;
	} else if (x <= -1.85e-7) {
		tmp = t_1;
	} else if (x <= 4.2e-111) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 180000000.0) || !(x <= 3.6e+166)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y_m))
    t_1 = abs((z * (x / y_m)))
    if (x <= (-6.5d+127)) then
        tmp = t_0
    else if (x <= (-1.85d-7)) then
        tmp = t_1
    else if (x <= 4.2d-111) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 180000000.0d0) .or. (.not. (x <= 3.6d+166))) then
        tmp = t_1
    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 t_1 = Math.abs((z * (x / y_m)));
	double tmp;
	if (x <= -6.5e+127) {
		tmp = t_0;
	} else if (x <= -1.85e-7) {
		tmp = t_1;
	} else if (x <= 4.2e-111) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 180000000.0) || !(x <= 3.6e+166)) {
		tmp = t_1;
	} 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))
	t_1 = math.fabs((z * (x / y_m)))
	tmp = 0
	if x <= -6.5e+127:
		tmp = t_0
	elif x <= -1.85e-7:
		tmp = t_1
	elif x <= 4.2e-111:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 180000000.0) or not (x <= 3.6e+166):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -6.5e+127)
		tmp = t_0;
	elseif (x <= -1.85e-7)
		tmp = t_1;
	elseif (x <= 4.2e-111)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 180000000.0) || !(x <= 3.6e+166))
		tmp = t_1;
	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));
	t_1 = abs((z * (x / y_m)));
	tmp = 0.0;
	if (x <= -6.5e+127)
		tmp = t_0;
	elseif (x <= -1.85e-7)
		tmp = t_1;
	elseif (x <= 4.2e-111)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 180000000.0) || ~((x <= 3.6e+166)))
		tmp = t_1;
	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]}, Block[{t$95$1 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.5e+127], t$95$0, If[LessEqual[x, -1.85e-7], t$95$1, If[LessEqual[x, 4.2e-111], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 180000000.0], N[Not[LessEqual[x, 3.6e+166]], $MachinePrecision]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5e127 or 1.8e8 < x < 3.5999999999999997e166

   1. Initial program 81.9%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in x around inf 98.4%

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

   6. Taylor expanded in z around 0 77.1%

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

      1. associate-*r/ 77.1%

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

      2. neg-mul-1 77.1%

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

   8. Simplified 77.1%

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

   if -6.5e127 < x < -1.85000000000000002e-7 or 4.1999999999999997e-111 < x < 1.8e8 or 3.5999999999999997e166 < x

   1. Initial program 90.8%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in x around -inf 98.8%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in z around inf 61.7%

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

      1. associate-*l/ 75.1%

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

   10. Applied egg-rr 75.1%

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

   if -1.85000000000000002e-7 < x < 4.1999999999999997e-111

   1. Initial program 96.8%

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

   3. Taylor expanded in x around 0 81.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-7}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-111}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 180000000 \lor \neg \left(x \leq 3.6 \cdot 10^{+166}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y_m}\right|\\ t_1 := \left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -7 \cdot 10^{+127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-112}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 5000000:\\ \;\;\;\;\left|\frac{x \cdot z}{y_m}\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+166}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* z (/ x y_m)))))
   (if (<= x -7e+127)
     t_0
     (if (<= x -1.85e-7)
       t_1
       (if (<= x 3.8e-112)
         (fabs (/ 4.0 y_m))
         (if (<= x 5000000.0)
           (fabs (/ (* x z) y_m))
           (if (<= x 6.5e+166) t_0 t_1)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((z * (x / y_m)));
	double tmp;
	if (x <= -7e+127) {
		tmp = t_0;
	} else if (x <= -1.85e-7) {
		tmp = t_1;
	} else if (x <= 3.8e-112) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 5000000.0) {
		tmp = fabs(((x * z) / y_m));
	} else if (x <= 6.5e+166) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y_m))
    t_1 = abs((z * (x / y_m)))
    if (x <= (-7d+127)) then
        tmp = t_0
    else if (x <= (-1.85d-7)) then
        tmp = t_1
    else if (x <= 3.8d-112) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 5000000.0d0) then
        tmp = abs(((x * z) / y_m))
    else if (x <= 6.5d+166) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double t_1 = Math.abs((z * (x / y_m)));
	double tmp;
	if (x <= -7e+127) {
		tmp = t_0;
	} else if (x <= -1.85e-7) {
		tmp = t_1;
	} else if (x <= 3.8e-112) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 5000000.0) {
		tmp = Math.abs(((x * z) / y_m));
	} else if (x <= 6.5e+166) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	t_1 = math.fabs((z * (x / y_m)))
	tmp = 0
	if x <= -7e+127:
		tmp = t_0
	elif x <= -1.85e-7:
		tmp = t_1
	elif x <= 3.8e-112:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 5000000.0:
		tmp = math.fabs(((x * z) / y_m))
	elif x <= 6.5e+166:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -7e+127)
		tmp = t_0;
	elseif (x <= -1.85e-7)
		tmp = t_1;
	elseif (x <= 3.8e-112)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 5000000.0)
		tmp = abs(Float64(Float64(x * z) / y_m));
	elseif (x <= 6.5e+166)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	t_1 = abs((z * (x / y_m)));
	tmp = 0.0;
	if (x <= -7e+127)
		tmp = t_0;
	elseif (x <= -1.85e-7)
		tmp = t_1;
	elseif (x <= 3.8e-112)
		tmp = abs((4.0 / y_m));
	elseif (x <= 5000000.0)
		tmp = abs(((x * z) / y_m));
	elseif (x <= 6.5e+166)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -7e+127], t$95$0, If[LessEqual[x, -1.85e-7], t$95$1, If[LessEqual[x, 3.8e-112], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5000000.0], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 6.5e+166], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.99999999999999956e127 or 5e6 < x < 6.5000000000000005e166

   1. Initial program 81.9%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in x around inf 98.4%

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

   6. Taylor expanded in z around 0 77.1%

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

      1. associate-*r/ 77.1%

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

      2. neg-mul-1 77.1%

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

   8. Simplified 77.1%

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

   if -6.99999999999999956e127 < x < -1.85000000000000002e-7 or 6.5000000000000005e166 < x

   1. Initial program 88.0%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in x around -inf 99.8%

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

   7. Simplified 87.0%

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

   8. Taylor expanded in z around inf 58.4%

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

      1. associate-*l/ 77.8%

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

   10. Applied egg-rr 77.8%

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

   if -1.85000000000000002e-7 < x < 3.79999999999999995e-112

   1. Initial program 96.8%

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

   3. Taylor expanded in x around 0 81.9%

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

   if 3.79999999999999995e-112 < x < 5e6

   1. Initial program 97.9%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in x around -inf 96.1%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around inf 70.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-7}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-112}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5000000:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+166}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-7} \lor \neg \left(x \leq 4.2 \cdot 10^{-111}\right):\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y_m} + \frac{-1}{y_m}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.75e-7) (not (<= x 4.2e-111)))
   (fabs (* x (+ (/ z y_m) (/ -1.0 y_m))))
   (fabs (/ (- x -4.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.75e-7) || !(x <= 4.2e-111)) {
		tmp = fabs((x * ((z / y_m) + (-1.0 / y_m))));
	} else {
		tmp = fabs(((x - -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 <= (-1.75d-7)) .or. (.not. (x <= 4.2d-111))) then
        tmp = abs((x * ((z / y_m) + ((-1.0d0) / y_m))))
    else
        tmp = abs(((x - (-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 <= -1.75e-7) || !(x <= 4.2e-111)) {
		tmp = Math.abs((x * ((z / y_m) + (-1.0 / y_m))));
	} else {
		tmp = Math.abs(((x - -4.0) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.75e-7) or not (x <= 4.2e-111):
		tmp = math.fabs((x * ((z / y_m) + (-1.0 / y_m))))
	else:
		tmp = math.fabs(((x - -4.0) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.75e-7) || !(x <= 4.2e-111))
		tmp = abs(Float64(x * Float64(Float64(z / y_m) + Float64(-1.0 / y_m))));
	else
		tmp = abs(Float64(Float64(x - -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 <= -1.75e-7) || ~((x <= 4.2e-111)))
		tmp = abs((x * ((z / y_m) + (-1.0 / y_m))));
	else
		tmp = abs(((x - -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, -1.75e-7], N[Not[LessEqual[x, 4.2e-111]], $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[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.74999999999999992e-7 or 4.1999999999999997e-111 < x

   1. Initial program 86.6%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in x around inf 94.1%

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

   if -1.74999999999999992e-7 < x < 4.1999999999999997e-111

   1. Initial program 96.8%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in z around 0 83.2%

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

      1. associate-*r/ 83.2%

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

      2. distribute-lft-in 83.2%

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

      3. metadata-eval 83.2%

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

      4. neg-mul-1 83.2%

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

      5. sub-neg 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-7} \lor \neg \left(x \leq 4.2 \cdot 10^{-111}\right):\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} + \frac{-1}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+40} \lor \neg \left(x \leq 75000000000000\right):\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y_m} + \frac{-1}{y_m}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1e+40) (not (<= x 75000000000000.0)))
   (fabs (* x (+ (/ z y_m) (/ -1.0 y_m))))
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1e+40) || !(x <= 75000000000000.0)) {
		tmp = fabs((x * ((z / y_m) + (-1.0 / y_m))));
	} else {
		tmp = fabs((((x + 4.0) - (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 ((x <= (-1d+40)) .or. (.not. (x <= 75000000000000.0d0))) then
        tmp = abs((x * ((z / y_m) + ((-1.0d0) / y_m))))
    else
        tmp = abs((((x + 4.0d0) - (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 ((x <= -1e+40) || !(x <= 75000000000000.0)) {
		tmp = Math.abs((x * ((z / y_m) + (-1.0 / y_m))));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1e+40) or not (x <= 75000000000000.0):
		tmp = math.fabs((x * ((z / y_m) + (-1.0 / y_m))))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1e+40) || !(x <= 75000000000000.0))
		tmp = abs(Float64(x * Float64(Float64(z / y_m) + Float64(-1.0 / y_m))));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1e+40) || ~((x <= 75000000000000.0)))
		tmp = abs((x * ((z / y_m) + (-1.0 / y_m))));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1e+40], N[Not[LessEqual[x, 75000000000000.0]], $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[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000003e40 or 7.5e13 < x

   1. Initial program 83.2%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in x around inf 99.9%

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

   if -1.00000000000000003e40 < x < 7.5e13

   1. Initial program 97.2%

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

   3. Taylor expanded in y around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+40} \lor \neg \left(x \leq 75000000000000\right):\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} + \frac{-1}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.14 \cdot 10^{-7} \lor \neg \left(x \leq 4.2 \cdot 10^{-111}\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y_m}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.14e-7) (not (<= x 4.2e-111)))
   (fabs (/ (+ z -1.0) (/ y_m x)))
   (fabs (/ (- x -4.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.14e-7) || !(x <= 4.2e-111)) {
		tmp = fabs(((z + -1.0) / (y_m / x)));
	} else {
		tmp = fabs(((x - -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 <= (-1.14d-7)) .or. (.not. (x <= 4.2d-111))) then
        tmp = abs(((z + (-1.0d0)) / (y_m / x)))
    else
        tmp = abs(((x - (-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 <= -1.14e-7) || !(x <= 4.2e-111)) {
		tmp = Math.abs(((z + -1.0) / (y_m / x)));
	} else {
		tmp = Math.abs(((x - -4.0) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.14e-7) or not (x <= 4.2e-111):
		tmp = math.fabs(((z + -1.0) / (y_m / x)))
	else:
		tmp = math.fabs(((x - -4.0) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.14e-7) || !(x <= 4.2e-111))
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y_m / x)));
	else
		tmp = abs(Float64(Float64(x - -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 <= -1.14e-7) || ~((x <= 4.2e-111)))
		tmp = abs(((z + -1.0) / (y_m / x)));
	else
		tmp = abs(((x - -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, -1.14e-7], N[Not[LessEqual[x, 4.2e-111]], $MachinePrecision]], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.14000000000000002e-7 or 4.1999999999999997e-111 < x

   1. Initial program 86.6%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in x around -inf 99.3%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in x around inf 85.6%

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

      1. sub-neg 85.6%

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

      2. metadata-eval 85.6%

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

      3. *-commutative 85.6%

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

      4. associate-/l* 93.9%

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

   10. Simplified 93.9%

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

   if -1.14000000000000002e-7 < x < 4.1999999999999997e-111

   1. Initial program 96.8%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in z around 0 83.2%

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

      1. associate-*r/ 83.2%

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

      2. distribute-lft-in 83.2%

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

      3. metadata-eval 83.2%

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

      4. neg-mul-1 83.2%

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

      5. sub-neg 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.14 \cdot 10^{-7} \lor \neg \left(x \leq 4.2 \cdot 10^{-111}\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -180000000000:\\ \;\;\;\;\left|x \cdot \frac{z}{y_m}\right|\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+30}:\\ \;\;\;\;\left|\frac{x - -4}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -180000000000.0)
   (fabs (* x (/ z y_m)))
   (if (<= z 1.8e+30) (fabs (/ (- x -4.0) y_m)) (fabs (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -180000000000.0) {
		tmp = fabs((x * (z / y_m)));
	} else if (z <= 1.8e+30) {
		tmp = fabs(((x - -4.0) / y_m));
	} else {
		tmp = fabs((z * (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 (z <= (-180000000000.0d0)) then
        tmp = abs((x * (z / y_m)))
    else if (z <= 1.8d+30) then
        tmp = abs(((x - (-4.0d0)) / y_m))
    else
        tmp = abs((z * (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 (z <= -180000000000.0) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (z <= 1.8e+30) {
		tmp = Math.abs(((x - -4.0) / y_m));
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -180000000000.0:
		tmp = math.fabs((x * (z / y_m)))
	elif z <= 1.8e+30:
		tmp = math.fabs(((x - -4.0) / y_m))
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -180000000000.0)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (z <= 1.8e+30)
		tmp = abs(Float64(Float64(x - -4.0) / y_m));
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -180000000000.0)
		tmp = abs((x * (z / y_m)));
	elseif (z <= 1.8e+30)
		tmp = abs(((x - -4.0) / y_m));
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -180000000000.0], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.8e+30], N[Abs[N[(N[(x - -4.0), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8e11

   1. Initial program 96.0%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in x around -inf 96.7%

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

   7. Simplified 88.7%

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

   8. Taylor expanded in z around inf 67.8%

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

      1. associate-*r/ 78.9%

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

   10. Simplified 78.9%

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

   if -1.8e11 < z < 1.8000000000000001e30

   1. Initial program 91.8%

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

   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 97.9%

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

      1. associate-*r/ 97.9%

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

      2. distribute-lft-in 97.9%

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

      3. metadata-eval 97.9%

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

      4. neg-mul-1 97.9%

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

      5. sub-neg 97.9%

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

   7. Simplified 97.9%

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

   if 1.8000000000000001e30 < z

   1. Initial program 83.3%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in x around -inf 86.5%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in z around inf 66.1%

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

      1. associate-*l/ 76.4%

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

   10. Applied egg-rr 76.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -180000000000:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+30}:\\ \;\;\;\;\left|\frac{x - -4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -10.2 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -10.2) (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 <= -10.2) || !(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 <= (-10.2d0)) .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 <= -10.2) || !(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 <= -10.2) 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 <= -10.2) || !(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 <= -10.2) || ~((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, -10.2], 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 < -10.199999999999999 or 4 < x

   1. Initial program 84.7%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in x around inf 98.8%

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

   6. Taylor expanded in z around 0 61.2%

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

      1. associate-*r/ 61.2%

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

      2. neg-mul-1 61.2%

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

   8. Simplified 61.2%

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

   if -10.199999999999999 < x < 4

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0 73.4%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.2 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.3% 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 90.7%

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

3. Taylor expanded in x around 0 38.2%

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

4. Final simplification 38.2%

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

Reproduce

herbie shell --seed 2024013 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))