fabs fraction 1

Percentage Accurate: 91.9% → 99.0%

Time: 5.9s

Alternatives: 8

Speedup: 1.0×

• 20.4% 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: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{-107}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 8.8e-107)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (+ (* x (/ z y)) (/ (- -4.0 x) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.8e-107) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(((x * (z / y)) + ((-4.0 - x) / y)));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 8.8d-107) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs(((x * (z / y)) + (((-4.0d0) - x) / y)))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.8e-107) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs(((x * (z / y)) + ((-4.0 - x) / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 8.8e-107:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs(((x * (z / y)) + ((-4.0 - x) / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 8.8e-107)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(x * Float64(z / y)) + Float64(Float64(-4.0 - x) / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.8e-107)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs(((x * (z / y)) + ((-4.0 - x) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 8.8e-107], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.8000000000000005e-107

   1. Initial program 87.4%

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

      1. associate-*l/ 88.6%

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

      2. sub-div 95.5%

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

   3. Applied egg-rr 95.5%

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

   if 8.8000000000000005e-107 < y

   1. Initial program 97.3%

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

   3. Simplified 94.2%

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

      1. fma-udef 94.2%

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

      2. distribute-rgt-in 94.2%

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

      3. associate-*r* 99.7%

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

      4. div-inv 99.8%

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

      5. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{-107}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 2: 68.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{+232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 300000000000 \lor \neg \left(x \leq 2.2 \cdot 10^{+106}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))) (t_1 (fabs (/ x y))))
   (if (<= x -6.8e+232)
     t_0
     (if (<= x -1.4e+132)
       t_1
       (if (<= x -2.05e-24)
         t_0
         (if (<= x 4e-65)
           (fabs (/ 4.0 y))
           (if (or (<= x 300000000000.0) (not (<= x 2.2e+106))) t_0 t_1)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -6.8e+232) {
		tmp = t_0;
	} else if (x <= -1.4e+132) {
		tmp = t_1;
	} else if (x <= -2.05e-24) {
		tmp = t_0;
	} else if (x <= 4e-65) {
		tmp = fabs((4.0 / y));
	} else if ((x <= 300000000000.0) || !(x <= 2.2e+106)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    t_1 = abs((x / y))
    if (x <= (-6.8d+232)) then
        tmp = t_0
    else if (x <= (-1.4d+132)) then
        tmp = t_1
    else if (x <= (-2.05d-24)) then
        tmp = t_0
    else if (x <= 4d-65) then
        tmp = abs((4.0d0 / y))
    else if ((x <= 300000000000.0d0) .or. (.not. (x <= 2.2d+106))) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z * (x / y)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -6.8e+232) {
		tmp = t_0;
	} else if (x <= -1.4e+132) {
		tmp = t_1;
	} else if (x <= -2.05e-24) {
		tmp = t_0;
	} else if (x <= 4e-65) {
		tmp = Math.abs((4.0 / y));
	} else if ((x <= 300000000000.0) || !(x <= 2.2e+106)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -6.8e+232:
		tmp = t_0
	elif x <= -1.4e+132:
		tmp = t_1
	elif x <= -2.05e-24:
		tmp = t_0
	elif x <= 4e-65:
		tmp = math.fabs((4.0 / y))
	elif (x <= 300000000000.0) or not (x <= 2.2e+106):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -6.8e+232)
		tmp = t_0;
	elseif (x <= -1.4e+132)
		tmp = t_1;
	elseif (x <= -2.05e-24)
		tmp = t_0;
	elseif (x <= 4e-65)
		tmp = abs(Float64(4.0 / y));
	elseif ((x <= 300000000000.0) || !(x <= 2.2e+106))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -6.8e+232)
		tmp = t_0;
	elseif (x <= -1.4e+132)
		tmp = t_1;
	elseif (x <= -2.05e-24)
		tmp = t_0;
	elseif (x <= 4e-65)
		tmp = abs((4.0 / y));
	elseif ((x <= 300000000000.0) || ~((x <= 2.2e+106)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.8e+232], t$95$0, If[LessEqual[x, -1.4e+132], t$95$1, If[LessEqual[x, -2.05e-24], t$95$0, If[LessEqual[x, 4e-65], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 300000000000.0], N[Not[LessEqual[x, 2.2e+106]], $MachinePrecision]], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.7999999999999996e232 or -1.4e132 < x < -2.05000000000000007e-24 or 3.99999999999999969e-65 < x < 3e11 or 2.19999999999999992e106 < x

   1. Initial program 88.8%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in z around inf 50.4%

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

      1. *-commutative 50.4%

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

      2. associate-*l/ 72.0%

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

      3. *-commutative 72.0%

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

   6. Simplified 72.0%

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

   if -6.7999999999999996e232 < x < -1.4e132 or 3e11 < x < 2.19999999999999992e106

   1. Initial program 83.7%

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

      1. associate-*l/ 84.1%

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

      2. sub-div 94.9%

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

   3. Applied egg-rr 94.9%

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

   4. Taylor expanded in x around inf 94.9%

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

      1. associate-/l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in z around 0 83.0%

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

   if -2.05000000000000007e-24 < x < 3.99999999999999969e-65

   1. Initial program 94.4%

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

   2. Taylor expanded in x around 0 79.6%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+232}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+132}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-24}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 300000000000 \lor \neg \left(x \leq 2.2 \cdot 10^{+106}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 68.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{z}{\frac{y}{x}}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -3.05 \cdot 10^{+230}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-63}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+15} \lor \neg \left(x \leq 6.8 \cdot 10^{+106}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ z (/ y x)))) (t_1 (fabs (/ x y))))
   (if (<= x -3.05e+230)
     (fabs (* z (/ x y)))
     (if (<= x -1.4e+132)
       t_1
       (if (<= x -3.2e-24)
         t_0
         (if (<= x 8.2e-63)
           (fabs (/ 4.0 y))
           (if (or (<= x 3.4e+15) (not (<= x 6.8e+106))) t_0 t_1)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z / (y / x)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -3.05e+230) {
		tmp = fabs((z * (x / y)));
	} else if (x <= -1.4e+132) {
		tmp = t_1;
	} else if (x <= -3.2e-24) {
		tmp = t_0;
	} else if (x <= 8.2e-63) {
		tmp = fabs((4.0 / y));
	} else if ((x <= 3.4e+15) || !(x <= 6.8e+106)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z / (y / x)))
    t_1 = abs((x / y))
    if (x <= (-3.05d+230)) then
        tmp = abs((z * (x / y)))
    else if (x <= (-1.4d+132)) then
        tmp = t_1
    else if (x <= (-3.2d-24)) then
        tmp = t_0
    else if (x <= 8.2d-63) then
        tmp = abs((4.0d0 / y))
    else if ((x <= 3.4d+15) .or. (.not. (x <= 6.8d+106))) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z / (y / x)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -3.05e+230) {
		tmp = Math.abs((z * (x / y)));
	} else if (x <= -1.4e+132) {
		tmp = t_1;
	} else if (x <= -3.2e-24) {
		tmp = t_0;
	} else if (x <= 8.2e-63) {
		tmp = Math.abs((4.0 / y));
	} else if ((x <= 3.4e+15) || !(x <= 6.8e+106)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z / (y / x)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -3.05e+230:
		tmp = math.fabs((z * (x / y)))
	elif x <= -1.4e+132:
		tmp = t_1
	elif x <= -3.2e-24:
		tmp = t_0
	elif x <= 8.2e-63:
		tmp = math.fabs((4.0 / y))
	elif (x <= 3.4e+15) or not (x <= 6.8e+106):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z / Float64(y / x)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -3.05e+230)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (x <= -1.4e+132)
		tmp = t_1;
	elseif (x <= -3.2e-24)
		tmp = t_0;
	elseif (x <= 8.2e-63)
		tmp = abs(Float64(4.0 / y));
	elseif ((x <= 3.4e+15) || !(x <= 6.8e+106))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z / (y / x)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -3.05e+230)
		tmp = abs((z * (x / y)));
	elseif (x <= -1.4e+132)
		tmp = t_1;
	elseif (x <= -3.2e-24)
		tmp = t_0;
	elseif (x <= 8.2e-63)
		tmp = abs((4.0 / y));
	elseif ((x <= 3.4e+15) || ~((x <= 6.8e+106)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.05e+230], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -1.4e+132], t$95$1, If[LessEqual[x, -3.2e-24], t$95$0, If[LessEqual[x, 8.2e-63], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 3.4e+15], N[Not[LessEqual[x, 6.8e+106]], $MachinePrecision]], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.04999999999999995e230

   1. Initial program 85.7%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in z around inf 46.6%

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

      1. *-commutative 46.6%

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

      2. associate-*l/ 86.3%

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

      3. *-commutative 86.3%

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

   6. Simplified 86.3%

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

   if -3.04999999999999995e230 < x < -1.4e132 or 3.4e15 < x < 6.79999999999999989e106

   1. Initial program 83.7%

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

      1. associate-*l/ 84.1%

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

      2. sub-div 94.9%

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

   3. Applied egg-rr 94.9%

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

   4. Taylor expanded in x around inf 94.9%

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

      1. associate-/l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in z around 0 83.0%

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

   if -1.4e132 < x < -3.20000000000000012e-24 or 8.1999999999999995e-63 < x < 3.4e15 or 6.79999999999999989e106 < x

   1. Initial program 89.6%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in z around inf 51.5%

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

      1. *-commutative 51.5%

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

      2. associate-*l/ 68.2%

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

      3. *-commutative 68.2%

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

   6. Simplified 68.2%

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

      1. clear-num 68.2%

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

      2. un-div-inv 68.3%

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

   8. Applied egg-rr 68.3%

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

   if -3.20000000000000012e-24 < x < 8.1999999999999995e-63

   1. Initial program 94.4%

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

   2. Taylor expanded in x around 0 79.6%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{+230}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+132}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-24}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-63}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+15} \lor \neg \left(x \leq 6.8 \cdot 10^{+106}\right):\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 69.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+265}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{+233}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{+131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-63}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 510000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))) (t_1 (fabs (* x (/ z y)))))
   (if (<= x -1.4e+265)
     t_0
     (if (<= x -2.1e+233)
       t_1
       (if (<= x -1.3e+131)
         t_0
         (if (<= x -7.6e-25)
           t_1
           (if (<= x 6e-63)
             (fabs (/ 4.0 y))
             (if (<= x 510000000000.0) t_1 t_0))))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double t_1 = fabs((x * (z / y)));
	double tmp;
	if (x <= -1.4e+265) {
		tmp = t_0;
	} else if (x <= -2.1e+233) {
		tmp = t_1;
	} else if (x <= -1.3e+131) {
		tmp = t_0;
	} else if (x <= -7.6e-25) {
		tmp = t_1;
	} else if (x <= 6e-63) {
		tmp = fabs((4.0 / y));
	} else if (x <= 510000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y))
    t_1 = abs((x * (z / y)))
    if (x <= (-1.4d+265)) then
        tmp = t_0
    else if (x <= (-2.1d+233)) then
        tmp = t_1
    else if (x <= (-1.3d+131)) then
        tmp = t_0
    else if (x <= (-7.6d-25)) then
        tmp = t_1
    else if (x <= 6d-63) then
        tmp = abs((4.0d0 / y))
    else if (x <= 510000000000.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double t_1 = Math.abs((x * (z / y)));
	double tmp;
	if (x <= -1.4e+265) {
		tmp = t_0;
	} else if (x <= -2.1e+233) {
		tmp = t_1;
	} else if (x <= -1.3e+131) {
		tmp = t_0;
	} else if (x <= -7.6e-25) {
		tmp = t_1;
	} else if (x <= 6e-63) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 510000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((x / y))
	t_1 = math.fabs((x * (z / y)))
	tmp = 0
	if x <= -1.4e+265:
		tmp = t_0
	elif x <= -2.1e+233:
		tmp = t_1
	elif x <= -1.3e+131:
		tmp = t_0
	elif x <= -7.6e-25:
		tmp = t_1
	elif x <= 6e-63:
		tmp = math.fabs((4.0 / y))
	elif x <= 510000000000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(x / y))
	t_1 = abs(Float64(x * Float64(z / y)))
	tmp = 0.0
	if (x <= -1.4e+265)
		tmp = t_0;
	elseif (x <= -2.1e+233)
		tmp = t_1;
	elseif (x <= -1.3e+131)
		tmp = t_0;
	elseif (x <= -7.6e-25)
		tmp = t_1;
	elseif (x <= 6e-63)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 510000000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	t_1 = abs((x * (z / y)));
	tmp = 0.0;
	if (x <= -1.4e+265)
		tmp = t_0;
	elseif (x <= -2.1e+233)
		tmp = t_1;
	elseif (x <= -1.3e+131)
		tmp = t_0;
	elseif (x <= -7.6e-25)
		tmp = t_1;
	elseif (x <= 6e-63)
		tmp = abs((4.0 / y));
	elseif (x <= 510000000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.4e+265], t$95$0, If[LessEqual[x, -2.1e+233], t$95$1, If[LessEqual[x, -1.3e+131], t$95$0, If[LessEqual[x, -7.6e-25], t$95$1, If[LessEqual[x, 6e-63], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 510000000000.0], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.39999999999999995e265 or -2.09999999999999997e233 < x < -1.3e131 or 5.1e11 < x

   1. Initial program 79.2%

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

      1. associate-*l/ 77.3%

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

      2. sub-div 91.9%

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

   3. Applied egg-rr 91.9%

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

   4. Taylor expanded in x around inf 91.9%

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

      1. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in z around 0 78.3%

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

   if -1.39999999999999995e265 < x < -2.09999999999999997e233 or -1.3e131 < x < -7.5999999999999996e-25 or 5.99999999999999959e-63 < x < 5.1e11

   1. Initial program 99.8%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in z around inf 56.2%

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

      1. associate-*l/ 64.7%

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

      2. *-commutative 64.7%

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

   6. Simplified 64.7%

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

   if -7.5999999999999996e-25 < x < 5.99999999999999959e-63

   1. Initial program 94.4%

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

   2. Taylor expanded in x around 0 79.6%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+265}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{+233}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{+131}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-25}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-63}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 510000000000:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 5: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+15} \lor \neg \left(x \leq 4.8 \cdot 10^{+66}\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.05e+15) (not (<= x 4.8e+66)))
   (fabs (/ (+ z -1.0) (/ y x)))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.05e+15) || !(x <= 4.8e+66)) {
		tmp = fabs(((z + -1.0) / (y / x)));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2.05d+15)) .or. (.not. (x <= 4.8d+66))) then
        tmp = abs(((z + (-1.0d0)) / (y / x)))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.05e+15) || !(x <= 4.8e+66)) {
		tmp = Math.abs(((z + -1.0) / (y / x)));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -2.05e+15) or not (x <= 4.8e+66):
		tmp = math.fabs(((z + -1.0) / (y / x)))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.05e+15) || !(x <= 4.8e+66))
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y / x)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.05e+15) || ~((x <= 4.8e+66)))
		tmp = abs(((z + -1.0) / (y / x)));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[x, -2.05e+15], N[Not[LessEqual[x, 4.8e+66]], $MachinePrecision]], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.05e15 or 4.8000000000000003e66 < x

   1. Initial program 84.6%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in x around inf 88.2%

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

      1. associate-/l* 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

   if -2.05e15 < x < 4.8000000000000003e66

   1. Initial program 94.9%

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

      1. associate-*l/ 99.2%

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

      2. sub-div 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+15} \lor \neg \left(x \leq 4.8 \cdot 10^{+66}\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 6: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+107} \lor \neg \left(z \leq 10^{+119}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6e+107) (not (<= z 1e+119)))
   (fabs (* x (/ z y)))
   (fabs (/ (- -4.0 x) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6e+107) || !(z <= 1e+119)) {
		tmp = fabs((x * (z / y)));
	} else {
		tmp = fabs(((-4.0 - x) / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-6d+107)) .or. (.not. (z <= 1d+119))) then
        tmp = abs((x * (z / y)))
    else
        tmp = abs((((-4.0d0) - x) / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6e+107) || !(z <= 1e+119)) {
		tmp = Math.abs((x * (z / y)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (z <= -6e+107) or not (z <= 1e+119):
		tmp = math.fabs((x * (z / y)))
	else:
		tmp = math.fabs(((-4.0 - x) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((z <= -6e+107) || !(z <= 1e+119))
		tmp = abs(Float64(x * Float64(z / y)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6e+107) || ~((z <= 1e+119)))
		tmp = abs((x * (z / y)));
	else
		tmp = abs(((-4.0 - x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[z, -6e+107], N[Not[LessEqual[z, 1e+119]], $MachinePrecision]], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.00000000000000046e107 or 9.99999999999999944e118 < z

   1. Initial program 83.1%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in z around inf 72.8%

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

      1. associate-*l/ 84.1%

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

      2. *-commutative 84.1%

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

   6. Simplified 84.1%

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

   if -6.00000000000000046e107 < z < 9.99999999999999944e118

   1. Initial program 93.9%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 90.9%

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

      1. associate-*r/ 90.9%

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

      2. distribute-lft-in 90.9%

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

      3. metadata-eval 90.9%

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

      4. neg-mul-1 90.9%

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

      5. sub-neg 90.9%

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

   6. Simplified 90.9%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+107} \lor \neg \left(z \leq 10^{+119}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 7: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = fabs((x / y));
	} else {
		tmp = fabs((4.0 / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = Math.abs((x / y));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -1.55) or not (x <= 4.0):
		tmp = math.fabs((x / y))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 4.0))
		tmp = abs(Float64(x / y));
	else
		tmp = abs(Float64(4.0 / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 4.0)))
		tmp = abs((x / y));
	else
		tmp = abs((4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 4 < x

   1. Initial program 85.4%

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

      1. associate-*l/ 79.3%

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

      2. sub-div 89.5%

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

   3. Applied egg-rr 89.5%

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

   4. Taylor expanded in x around inf 88.9%

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

      1. associate-/l* 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in z around 0 65.8%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 95.1%

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

   2. Taylor expanded in x around 0 73.8%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 8: 41.1% accurate, 1.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z) :precision binary64 (fabs (/ 4.0 y)))

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((4.0d0 / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 90.7%

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

2. Taylor expanded in x around 0 42.4%

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

3. Final simplification 42.4%

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

Reproduce

herbie shell --seed 2023202 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))