fabs fraction 1

Percentage Accurate: 92.2% → 99.8%

Time: 6.4s

Alternatives: 12

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+42}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}} - \frac{x + 4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 5e+42)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ x (/ y z)) (/ (+ x 4.0) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+42) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(((x / (y / z)) - ((x + 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 (y <= 5d+42) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs(((x / (y / z)) - ((x + 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 (y <= 5e+42) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs(((x / (y / z)) - ((x + 4.0) / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 5e+42:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs(((x / (y / z)) - ((x + 4.0) / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 5e+42)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(x / Float64(y / z)) - Float64(Float64(x + 4.0) / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e+42)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs(((x / (y / z)) - ((x + 4.0) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 5e+42], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.00000000000000007e42

   1. Initial program 90.0%

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

      1. associate-*l/ 88.7%

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

      2. sub-div 95.7%

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

   3. Applied egg-rr 95.7%

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

   if 5.00000000000000007e42 < y

   1. Initial program 99.9%

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

      1. associate-*l/ 93.4%

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

      2. associate-/l* 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+42}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}} - \frac{x + 4}{y}\right|\\ \end{array} \]

Alternative 2: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \frac{x + 4}{y} - z \cdot \frac{x}{y}\\ \mathbf{if}\;t_0 \leq 6 \cdot 10^{+305}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y) (* z (/ x y)))))
   (if (<= t_0 6e+305) (fabs t_0) (fabs (/ (- (+ x 4.0) (* x z)) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = ((x + 4.0) / y) - (z * (x / y));
	double tmp;
	if (t_0 <= 6e+305) {
		tmp = fabs(t_0);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((x + 4.0d0) / y) - (z * (x / y))
    if (t_0 <= 6d+305) then
        tmp = abs(t_0)
    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 t_0 = ((x + 4.0) / y) - (z * (x / y));
	double tmp;
	if (t_0 <= 6e+305) {
		tmp = Math.abs(t_0);
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = ((x + 4.0) / y) - (z * (x / y))
	tmp = 0
	if t_0 <= 6e+305:
		tmp = math.fabs(t_0)
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y) - Float64(z * Float64(x / y)))
	tmp = 0.0
	if (t_0 <= 6e+305)
		tmp = abs(t_0);
	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)
	t_0 = ((x + 4.0) / y) - (z * (x / y));
	tmp = 0.0;
	if (t_0 <= 6e+305)
		tmp = abs(t_0);
	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_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 6e+305], N[Abs[t$95$0], $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 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < 5.99999999999999964e305

   1. Initial program 99.1%

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

   if 5.99999999999999964e305 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))

   1. Initial program 51.4%

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

      1. associate-*l/ 62.2%

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

      2. sub-div 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq 6 \cdot 10^{+305}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 3: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+221}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-79}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+49} \lor \neg \left(x \leq 5.6 \cdot 10^{+137}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 -2.5e+221)
     t_0
     (if (<= x -6.4e-19)
       t_1
       (if (<= x 1.3e-79)
         (fabs (/ 4.0 y))
         (if (or (<= x 7e+49) (not (<= x 5.6e+137))) 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 <= -2.5e+221) {
		tmp = t_0;
	} else if (x <= -6.4e-19) {
		tmp = t_1;
	} else if (x <= 1.3e-79) {
		tmp = fabs((4.0 / y));
	} else if ((x <= 7e+49) || !(x <= 5.6e+137)) {
		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 <= (-2.5d+221)) then
        tmp = t_0
    else if (x <= (-6.4d-19)) then
        tmp = t_1
    else if (x <= 1.3d-79) then
        tmp = abs((4.0d0 / y))
    else if ((x <= 7d+49) .or. (.not. (x <= 5.6d+137))) 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 <= -2.5e+221) {
		tmp = t_0;
	} else if (x <= -6.4e-19) {
		tmp = t_1;
	} else if (x <= 1.3e-79) {
		tmp = Math.abs((4.0 / y));
	} else if ((x <= 7e+49) || !(x <= 5.6e+137)) {
		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 <= -2.5e+221:
		tmp = t_0
	elif x <= -6.4e-19:
		tmp = t_1
	elif x <= 1.3e-79:
		tmp = math.fabs((4.0 / y))
	elif (x <= 7e+49) or not (x <= 5.6e+137):
		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 <= -2.5e+221)
		tmp = t_0;
	elseif (x <= -6.4e-19)
		tmp = t_1;
	elseif (x <= 1.3e-79)
		tmp = abs(Float64(4.0 / y));
	elseif ((x <= 7e+49) || !(x <= 5.6e+137))
		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 <= -2.5e+221)
		tmp = t_0;
	elseif (x <= -6.4e-19)
		tmp = t_1;
	elseif (x <= 1.3e-79)
		tmp = abs((4.0 / y));
	elseif ((x <= 7e+49) || ~((x <= 5.6e+137)))
		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, -2.5e+221], t$95$0, If[LessEqual[x, -6.4e-19], t$95$1, If[LessEqual[x, 1.3e-79], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 7e+49], N[Not[LessEqual[x, 5.6e+137]], $MachinePrecision]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5000000000000001e221 or 6.9999999999999995e49 < x < 5.60000000000000002e137

   1. Initial program 87.1%

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

      1. fabs-neg 87.1%

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

      2. sub-neg 87.1%

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

      3. distribute-neg-in 87.1%

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

      4. sub-neg 87.1%

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

      5. distribute-neg-frac 87.1%

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

      6. associate-*l/ 80.2%

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

      7. distribute-neg-frac 80.2%

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

      8. neg-mul-1 80.2%

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

      9. associate-*l/ 80.1%

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

      10. neg-mul-1 80.1%

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

      11. associate-*l/ 80.1%

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

      12. distribute-lft-out-- 90.3%

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

      13. fabs-mul 90.3%

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

      14. fabs-sub 90.3%

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

      15. fabs-mul 90.3%

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

      16. associate-*l/ 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in x around inf 90.4%

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

      1. associate-/l* 99.6%

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

      2. associate-/r/ 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in z around 0 76.1%

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

   if -2.5000000000000001e221 < x < -6.39999999999999965e-19 or 1.29999999999999997e-79 < x < 6.9999999999999995e49 or 5.60000000000000002e137 < x

   1. Initial program 86.8%

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

   2. Taylor expanded in z around inf 62.0%

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

      1. associate-*r/ 62.0%

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

      2. mul-1-neg 62.0%

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

      3. *-commutative 62.0%

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

      4. distribute-lft-neg-in 62.0%

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

      5. associate-*l/ 76.2%

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

      6. *-commutative 76.2%

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

   4. Simplified 76.2%

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

      1. clear-num 76.1%

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

      2. un-div-inv 76.2%

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

      3. add-sqr-sqrt 33.5%

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

      4. sqrt-unprod 64.2%

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

      5. sqr-neg 64.2%

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

      6. sqrt-unprod 42.6%

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

      7. add-sqr-sqrt 76.2%

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

   6. Applied egg-rr 76.2%

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

      1. associate-/r/ 69.5%

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

   8. Applied egg-rr 69.5%

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

   if -6.39999999999999965e-19 < x < 1.29999999999999997e-79

   1. Initial program 98.4%

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

   2. Taylor expanded in x around 0 84.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+221}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-19}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-79}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+49} \lor \neg \left(x \leq 5.6 \cdot 10^{+137}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -3.25 \cdot 10^{+221}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-18}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-80}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+50} \lor \neg \left(x \leq 9.8 \cdot 10^{+137}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -3.25e+221)
     t_0
     (if (<= x -1.35e-18)
       (fabs (/ x (/ y z)))
       (if (<= x 9.6e-80)
         (fabs (/ 4.0 y))
         (if (or (<= x 1.2e+50) (not (<= x 9.8e+137)))
           (fabs (* x (/ z y)))
           t_0))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double tmp;
	if (x <= -3.25e+221) {
		tmp = t_0;
	} else if (x <= -1.35e-18) {
		tmp = fabs((x / (y / z)));
	} else if (x <= 9.6e-80) {
		tmp = fabs((4.0 / y));
	} else if ((x <= 1.2e+50) || !(x <= 9.8e+137)) {
		tmp = fabs((x * (z / y)));
	} 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) :: tmp
    t_0 = abs((x / y))
    if (x <= (-3.25d+221)) then
        tmp = t_0
    else if (x <= (-1.35d-18)) then
        tmp = abs((x / (y / z)))
    else if (x <= 9.6d-80) then
        tmp = abs((4.0d0 / y))
    else if ((x <= 1.2d+50) .or. (.not. (x <= 9.8d+137))) then
        tmp = abs((x * (z / y)))
    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 tmp;
	if (x <= -3.25e+221) {
		tmp = t_0;
	} else if (x <= -1.35e-18) {
		tmp = Math.abs((x / (y / z)));
	} else if (x <= 9.6e-80) {
		tmp = Math.abs((4.0 / y));
	} else if ((x <= 1.2e+50) || !(x <= 9.8e+137)) {
		tmp = Math.abs((x * (z / y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((x / y))
	tmp = 0
	if x <= -3.25e+221:
		tmp = t_0
	elif x <= -1.35e-18:
		tmp = math.fabs((x / (y / z)))
	elif x <= 9.6e-80:
		tmp = math.fabs((4.0 / y))
	elif (x <= 1.2e+50) or not (x <= 9.8e+137):
		tmp = math.fabs((x * (z / y)))
	else:
		tmp = t_0
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -3.25e+221)
		tmp = t_0;
	elseif (x <= -1.35e-18)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (x <= 9.6e-80)
		tmp = abs(Float64(4.0 / y));
	elseif ((x <= 1.2e+50) || !(x <= 9.8e+137))
		tmp = abs(Float64(x * Float64(z / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	tmp = 0.0;
	if (x <= -3.25e+221)
		tmp = t_0;
	elseif (x <= -1.35e-18)
		tmp = abs((x / (y / z)));
	elseif (x <= 9.6e-80)
		tmp = abs((4.0 / y));
	elseif ((x <= 1.2e+50) || ~((x <= 9.8e+137)))
		tmp = abs((x * (z / y)));
	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]}, If[LessEqual[x, -3.25e+221], t$95$0, If[LessEqual[x, -1.35e-18], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 9.6e-80], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 1.2e+50], N[Not[LessEqual[x, 9.8e+137]], $MachinePrecision]], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.2500000000000002e221 or 1.2000000000000001e50 < x < 9.80000000000000065e137

   1. Initial program 87.1%

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

      1. fabs-neg 87.1%

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

      2. sub-neg 87.1%

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

      3. distribute-neg-in 87.1%

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

      4. sub-neg 87.1%

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

      5. distribute-neg-frac 87.1%

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

      6. associate-*l/ 80.2%

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

      7. distribute-neg-frac 80.2%

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

      8. neg-mul-1 80.2%

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

      9. associate-*l/ 80.1%

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

      10. neg-mul-1 80.1%

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

      11. associate-*l/ 80.1%

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

      12. distribute-lft-out-- 90.3%

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

      13. fabs-mul 90.3%

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

      14. fabs-sub 90.3%

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

      15. fabs-mul 90.3%

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

      16. associate-*l/ 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in x around inf 90.4%

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

      1. associate-/l* 99.6%

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

      2. associate-/r/ 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in z around 0 76.1%

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

   if -3.2500000000000002e221 < x < -1.34999999999999994e-18

   1. Initial program 86.9%

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

   2. Taylor expanded in z around inf 62.0%

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

      1. associate-*r/ 62.0%

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

      2. mul-1-neg 62.0%

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

      3. *-commutative 62.0%

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

      4. distribute-lft-neg-in 62.0%

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

      5. associate-*l/ 72.2%

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

      6. *-commutative 72.2%

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

   4. Simplified 72.2%

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

      1. *-commutative 72.2%

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

      2. distribute-frac-neg 72.2%

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

      3. distribute-lft-neg-in 72.2%

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

      4. associate-/r/ 68.1%

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

      5. distribute-neg-frac 68.1%

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

      6. add-sqr-sqrt 68.0%

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

      7. sqrt-unprod 59.5%

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

      8. sqr-neg 59.5%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 68.1%

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

   6. Applied egg-rr 68.1%

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

   if -1.34999999999999994e-18 < x < 9.5999999999999996e-80

   1. Initial program 98.4%

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

   2. Taylor expanded in x around 0 84.2%

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

   if 9.5999999999999996e-80 < x < 1.2000000000000001e50 or 9.80000000000000065e137 < x

   1. Initial program 86.7%

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

   2. Taylor expanded in z around inf 61.9%

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

      1. associate-*r/ 61.9%

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

      2. mul-1-neg 61.9%

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

      3. *-commutative 61.9%

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

      4. distribute-lft-neg-in 61.9%

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

      5. associate-*l/ 79.7%

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

      6. *-commutative 79.7%

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

   4. Simplified 79.7%

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

      1. clear-num 79.6%

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

      2. un-div-inv 79.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 64.5%

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

      5. sqr-neg 64.5%

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

      6. sqrt-unprod 79.6%

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

      7. add-sqr-sqrt 79.8%

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

   6. Applied egg-rr 79.8%

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

      1. associate-/r/ 70.8%

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

   8. Applied egg-rr 70.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{+221}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-18}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-80}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+50} \lor \neg \left(x \leq 9.8 \cdot 10^{+137}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 5: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{-82}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+50} \lor \neg \left(x \leq 1.12 \cdot 10^{+138}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ z (/ y x)))))
   (if (<= x -2.15e-18)
     t_0
     (if (<= x 3.55e-82)
       (fabs (/ 4.0 y))
       (if (or (<= x 3.2e+50) (not (<= x 1.12e+138))) t_0 (fabs (/ x y)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z / (y / x)));
	double tmp;
	if (x <= -2.15e-18) {
		tmp = t_0;
	} else if (x <= 3.55e-82) {
		tmp = fabs((4.0 / y));
	} else if ((x <= 3.2e+50) || !(x <= 1.12e+138)) {
		tmp = t_0;
	} else {
		tmp = fabs((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) :: t_0
    real(8) :: tmp
    t_0 = abs((z / (y / x)))
    if (x <= (-2.15d-18)) then
        tmp = t_0
    else if (x <= 3.55d-82) then
        tmp = abs((4.0d0 / y))
    else if ((x <= 3.2d+50) .or. (.not. (x <= 1.12d+138))) then
        tmp = t_0
    else
        tmp = abs((x / y))
    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 tmp;
	if (x <= -2.15e-18) {
		tmp = t_0;
	} else if (x <= 3.55e-82) {
		tmp = Math.abs((4.0 / y));
	} else if ((x <= 3.2e+50) || !(x <= 1.12e+138)) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z / (y / x)))
	tmp = 0
	if x <= -2.15e-18:
		tmp = t_0
	elif x <= 3.55e-82:
		tmp = math.fabs((4.0 / y))
	elif (x <= 3.2e+50) or not (x <= 1.12e+138):
		tmp = t_0
	else:
		tmp = math.fabs((x / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z / Float64(y / x)))
	tmp = 0.0
	if (x <= -2.15e-18)
		tmp = t_0;
	elseif (x <= 3.55e-82)
		tmp = abs(Float64(4.0 / y));
	elseif ((x <= 3.2e+50) || !(x <= 1.12e+138))
		tmp = t_0;
	else
		tmp = abs(Float64(x / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z / (y / x)));
	tmp = 0.0;
	if (x <= -2.15e-18)
		tmp = t_0;
	elseif (x <= 3.55e-82)
		tmp = abs((4.0 / y));
	elseif ((x <= 3.2e+50) || ~((x <= 1.12e+138)))
		tmp = t_0;
	else
		tmp = abs((x / y));
	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]}, If[LessEqual[x, -2.15e-18], t$95$0, If[LessEqual[x, 3.55e-82], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 3.2e+50], N[Not[LessEqual[x, 1.12e+138]], $MachinePrecision]], t$95$0, N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1500000000000001e-18 or 3.55000000000000021e-82 < x < 3.19999999999999983e50 or 1.12e138 < x

   1. Initial program 84.5%

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

   2. Taylor expanded in z around inf 57.4%

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

      1. associate-*r/ 57.4%

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

      2. mul-1-neg 57.4%

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

      3. *-commutative 57.4%

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

      4. distribute-lft-neg-in 57.4%

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

      5. associate-*l/ 75.8%

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

      6. *-commutative 75.8%

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

   4. Simplified 75.8%

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

      1. clear-num 75.7%

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

      2. un-div-inv 75.8%

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

      3. add-sqr-sqrt 39.6%

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

      4. sqrt-unprod 62.5%

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

      5. sqr-neg 62.5%

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

      6. sqrt-unprod 36.0%

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

      7. add-sqr-sqrt 75.8%

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

   6. Applied egg-rr 75.8%

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

   if -2.1500000000000001e-18 < x < 3.55000000000000021e-82

   1. Initial program 98.4%

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

   2. Taylor expanded in x around 0 84.2%

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

   if 3.19999999999999983e50 < x < 1.12e138

   1. Initial program 99.9%

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

      1. fabs-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. associate-*l/ 99.9%

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

      7. distribute-neg-frac 99.9%

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

      8. neg-mul-1 99.9%

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

      9. associate-*l/ 99.7%

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

      10. neg-mul-1 99.7%

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

      11. associate-*l/ 99.7%

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

      12. distribute-lft-out-- 99.7%

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

      13. fabs-mul 99.7%

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

      14. fabs-sub 99.7%

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

      15. fabs-mul 99.7%

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

      16. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. associate-/l* 99.6%

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

      2. associate-/r/ 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in z around 0 73.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-18}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{-82}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+50} \lor \neg \left(x \leq 1.12 \cdot 10^{+138}\right):\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 6: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{-4 - x}{y}\right|\\ \mathbf{if}\;z \leq -8 \cdot 10^{+165}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq -6.9 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2300000000:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ (- -4.0 x) y))))
   (if (<= z -8e+165)
     (fabs (* x (/ z y)))
     (if (<= z -6.9e+90)
       t_0
       (if (<= z -2300000000.0)
         (fabs (/ x (/ y z)))
         (if (<= z 2.5e+38) t_0 (fabs (/ z (/ y x)))))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs(((-4.0 - x) / y));
	double tmp;
	if (z <= -8e+165) {
		tmp = fabs((x * (z / y)));
	} else if (z <= -6.9e+90) {
		tmp = t_0;
	} else if (z <= -2300000000.0) {
		tmp = fabs((x / (y / z)));
	} else if (z <= 2.5e+38) {
		tmp = t_0;
	} else {
		tmp = fabs((z / (y / x)));
	}
	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) :: tmp
    t_0 = abs((((-4.0d0) - x) / y))
    if (z <= (-8d+165)) then
        tmp = abs((x * (z / y)))
    else if (z <= (-6.9d+90)) then
        tmp = t_0
    else if (z <= (-2300000000.0d0)) then
        tmp = abs((x / (y / z)))
    else if (z <= 2.5d+38) then
        tmp = t_0
    else
        tmp = abs((z / (y / x)))
    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(((-4.0 - x) / y));
	double tmp;
	if (z <= -8e+165) {
		tmp = Math.abs((x * (z / y)));
	} else if (z <= -6.9e+90) {
		tmp = t_0;
	} else if (z <= -2300000000.0) {
		tmp = Math.abs((x / (y / z)));
	} else if (z <= 2.5e+38) {
		tmp = t_0;
	} else {
		tmp = Math.abs((z / (y / x)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs(((-4.0 - x) / y))
	tmp = 0
	if z <= -8e+165:
		tmp = math.fabs((x * (z / y)))
	elif z <= -6.9e+90:
		tmp = t_0
	elif z <= -2300000000.0:
		tmp = math.fabs((x / (y / z)))
	elif z <= 2.5e+38:
		tmp = t_0
	else:
		tmp = math.fabs((z / (y / x)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(Float64(-4.0 - x) / y))
	tmp = 0.0
	if (z <= -8e+165)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (z <= -6.9e+90)
		tmp = t_0;
	elseif (z <= -2300000000.0)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (z <= 2.5e+38)
		tmp = t_0;
	else
		tmp = abs(Float64(z / Float64(y / x)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs(((-4.0 - x) / y));
	tmp = 0.0;
	if (z <= -8e+165)
		tmp = abs((x * (z / y)));
	elseif (z <= -6.9e+90)
		tmp = t_0;
	elseif (z <= -2300000000.0)
		tmp = abs((x / (y / z)));
	elseif (z <= 2.5e+38)
		tmp = t_0;
	else
		tmp = abs((z / (y / x)));
	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[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -8e+165], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -6.9e+90], t$95$0, If[LessEqual[z, -2300000000.0], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.5e+38], t$95$0, N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.9999999999999992e165

   1. Initial program 92.0%

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

   2. Taylor expanded in z around inf 80.6%

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

      1. associate-*r/ 80.6%

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

      2. mul-1-neg 80.6%

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

      3. *-commutative 80.6%

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

      4. distribute-lft-neg-in 80.6%

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

      5. associate-*l/ 88.0%

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

      6. *-commutative 88.0%

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

   4. Simplified 88.0%

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

      1. clear-num 87.9%

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

      2. un-div-inv 88.1%

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

      3. add-sqr-sqrt 16.5%

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

      4. sqrt-unprod 72.5%

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

      5. sqr-neg 72.5%

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

      6. sqrt-unprod 71.2%

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

      7. add-sqr-sqrt 88.1%

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

   6. Applied egg-rr 88.1%

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

      1. associate-/r/ 92.1%

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

   8. Applied egg-rr 92.1%

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

   if -7.9999999999999992e165 < z < -6.89999999999999955e90 or -2.3e9 < z < 2.49999999999999985e38

   1. Initial program 95.9%

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

      1. fabs-sub 95.9%

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

      2. associate-*l/ 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-*l/ 98.6%

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

      5. *-commutative 98.6%

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

      6. fma-neg 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 95.6%

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

      1. associate-*r/ 95.6%

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

      2. distribute-lft-in 95.6%

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

      3. metadata-eval 95.6%

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

      4. neg-mul-1 95.6%

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

      5. sub-neg 95.6%

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

   6. Simplified 95.6%

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

   if -6.89999999999999955e90 < z < -2.3e9

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 77.9%

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

      1. associate-*r/ 77.9%

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

      2. mul-1-neg 77.9%

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

      3. *-commutative 77.9%

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

      4. distribute-lft-neg-in 77.9%

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

      5. associate-*l/ 77.9%

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

      6. *-commutative 77.9%

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

   4. Simplified 77.9%

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

      1. *-commutative 77.9%

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

      2. distribute-frac-neg 77.9%

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

      3. distribute-lft-neg-in 77.9%

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

      4. associate-/r/ 78.1%

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

      5. distribute-neg-frac 78.1%

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

      6. add-sqr-sqrt 35.5%

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

      7. sqrt-unprod 52.1%

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

      8. sqr-neg 52.1%

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

      9. sqrt-unprod 42.3%

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

      10. add-sqr-sqrt 78.1%

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

   6. Applied egg-rr 78.1%

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

   if 2.49999999999999985e38 < z

   1. Initial program 81.2%

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

   2. Taylor expanded in z around inf 70.4%

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

      1. associate-*r/ 70.4%

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

      2. mul-1-neg 70.4%

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

      3. *-commutative 70.4%

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

      4. distribute-lft-neg-in 70.4%

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

      5. associate-*l/ 83.6%

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

      6. *-commutative 83.6%

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

   4. Simplified 83.6%

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

      1. clear-num 83.5%

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

      2. un-div-inv 83.6%

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

      3. add-sqr-sqrt 44.7%

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

      4. sqrt-unprod 65.1%

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

      5. sqr-neg 65.1%

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

      6. sqrt-unprod 38.7%

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

      7. add-sqr-sqrt 83.6%

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

   6. Applied egg-rr 83.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+165}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq -6.9 \cdot 10^{+90}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{elif}\;z \leq -2300000000:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+38}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 7: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{-4 - x}{y}\right|\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+166}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq -2 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2300000000:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ (- -4.0 x) y))))
   (if (<= z -1.5e+166)
     (fabs (* x (/ z y)))
     (if (<= z -2e+90)
       t_0
       (if (<= z -2300000000.0)
         (fabs (/ x (/ y z)))
         (if (<= z 2.85e+38) t_0 (fabs (* z (/ x y)))))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs(((-4.0 - x) / y));
	double tmp;
	if (z <= -1.5e+166) {
		tmp = fabs((x * (z / y)));
	} else if (z <= -2e+90) {
		tmp = t_0;
	} else if (z <= -2300000000.0) {
		tmp = fabs((x / (y / z)));
	} else if (z <= 2.85e+38) {
		tmp = t_0;
	} else {
		tmp = fabs((z * (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) :: t_0
    real(8) :: tmp
    t_0 = abs((((-4.0d0) - x) / y))
    if (z <= (-1.5d+166)) then
        tmp = abs((x * (z / y)))
    else if (z <= (-2d+90)) then
        tmp = t_0
    else if (z <= (-2300000000.0d0)) then
        tmp = abs((x / (y / z)))
    else if (z <= 2.85d+38) then
        tmp = t_0
    else
        tmp = abs((z * (x / y)))
    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(((-4.0 - x) / y));
	double tmp;
	if (z <= -1.5e+166) {
		tmp = Math.abs((x * (z / y)));
	} else if (z <= -2e+90) {
		tmp = t_0;
	} else if (z <= -2300000000.0) {
		tmp = Math.abs((x / (y / z)));
	} else if (z <= 2.85e+38) {
		tmp = t_0;
	} else {
		tmp = Math.abs((z * (x / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs(((-4.0 - x) / y))
	tmp = 0
	if z <= -1.5e+166:
		tmp = math.fabs((x * (z / y)))
	elif z <= -2e+90:
		tmp = t_0
	elif z <= -2300000000.0:
		tmp = math.fabs((x / (y / z)))
	elif z <= 2.85e+38:
		tmp = t_0
	else:
		tmp = math.fabs((z * (x / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(Float64(-4.0 - x) / y))
	tmp = 0.0
	if (z <= -1.5e+166)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (z <= -2e+90)
		tmp = t_0;
	elseif (z <= -2300000000.0)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (z <= 2.85e+38)
		tmp = t_0;
	else
		tmp = abs(Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs(((-4.0 - x) / y));
	tmp = 0.0;
	if (z <= -1.5e+166)
		tmp = abs((x * (z / y)));
	elseif (z <= -2e+90)
		tmp = t_0;
	elseif (z <= -2300000000.0)
		tmp = abs((x / (y / z)));
	elseif (z <= 2.85e+38)
		tmp = t_0;
	else
		tmp = abs((z * (x / y)));
	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[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.5e+166], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -2e+90], t$95$0, If[LessEqual[z, -2300000000.0], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.85e+38], t$95$0, N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.49999999999999999e166

   1. Initial program 92.0%

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

   2. Taylor expanded in z around inf 80.6%

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

      1. associate-*r/ 80.6%

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

      2. mul-1-neg 80.6%

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

      3. *-commutative 80.6%

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

      4. distribute-lft-neg-in 80.6%

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

      5. associate-*l/ 88.0%

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

      6. *-commutative 88.0%

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

   4. Simplified 88.0%

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

      1. clear-num 87.9%

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

      2. un-div-inv 88.1%

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

      3. add-sqr-sqrt 16.5%

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

      4. sqrt-unprod 72.5%

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

      5. sqr-neg 72.5%

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

      6. sqrt-unprod 71.2%

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

      7. add-sqr-sqrt 88.1%

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

   6. Applied egg-rr 88.1%

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

      1. associate-/r/ 92.1%

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

   8. Applied egg-rr 92.1%

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

   if -1.49999999999999999e166 < z < -1.99999999999999993e90 or -2.3e9 < z < 2.8499999999999999e38

   1. Initial program 95.9%

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

      1. fabs-sub 95.9%

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

      2. associate-*l/ 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-*l/ 98.6%

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

      5. *-commutative 98.6%

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

      6. fma-neg 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 95.6%

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

      1. associate-*r/ 95.6%

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

      2. distribute-lft-in 95.6%

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

      3. metadata-eval 95.6%

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

      4. neg-mul-1 95.6%

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

      5. sub-neg 95.6%

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

   6. Simplified 95.6%

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

   if -1.99999999999999993e90 < z < -2.3e9

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 77.9%

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

      1. associate-*r/ 77.9%

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

      2. mul-1-neg 77.9%

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

      3. *-commutative 77.9%

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

      4. distribute-lft-neg-in 77.9%

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

      5. associate-*l/ 77.9%

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

      6. *-commutative 77.9%

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

   4. Simplified 77.9%

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

      1. *-commutative 77.9%

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

      2. distribute-frac-neg 77.9%

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

      3. distribute-lft-neg-in 77.9%

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

      4. associate-/r/ 78.1%

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

      5. distribute-neg-frac 78.1%

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

      6. add-sqr-sqrt 35.5%

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

      7. sqrt-unprod 52.1%

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

      8. sqr-neg 52.1%

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

      9. sqrt-unprod 42.3%

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

      10. add-sqr-sqrt 78.1%

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

   6. Applied egg-rr 78.1%

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

   if 2.8499999999999999e38 < z

   1. Initial program 81.2%

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

   2. Taylor expanded in z around inf 70.4%

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

      1. associate-*r/ 70.4%

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

      2. mul-1-neg 70.4%

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

      3. *-commutative 70.4%

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

      4. distribute-lft-neg-in 70.4%

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

      5. associate-*l/ 83.6%

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

      6. *-commutative 83.6%

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

   4. Simplified 83.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+166}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq -2 \cdot 10^{+90}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{elif}\;z \leq -2300000000:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{+38}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 8: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+64} \lor \neg \left(x \leq 10^{+125}\right):\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.5e+64) (not (<= x 1e+125)))
   (fabs (/ (- 1.0 z) (/ 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 <= -6.5e+64) || !(x <= 1e+125)) {
		tmp = fabs(((1.0 - z) / (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 <= (-6.5d+64)) .or. (.not. (x <= 1d+125))) then
        tmp = abs(((1.0d0 - z) / (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 <= -6.5e+64) || !(x <= 1e+125)) {
		tmp = Math.abs(((1.0 - z) / (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 <= -6.5e+64) or not (x <= 1e+125):
		tmp = math.fabs(((1.0 - z) / (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 <= -6.5e+64) || !(x <= 1e+125))
		tmp = abs(Float64(Float64(1.0 - z) / 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 <= -6.5e+64) || ~((x <= 1e+125)))
		tmp = abs(((1.0 - z) / (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, -6.5e+64], N[Not[LessEqual[x, 1e+125]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $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 < -6.50000000000000007e64 or 9.9999999999999992e124 < x

   1. Initial program 77.7%

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

      1. fabs-neg 77.7%

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

      2. sub-neg 77.7%

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

      3. distribute-neg-in 77.7%

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

      4. sub-neg 77.7%

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

      5. distribute-neg-frac 77.7%

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

      6. associate-*l/ 67.7%

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

      7. distribute-neg-frac 67.7%

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

      8. neg-mul-1 67.7%

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

      9. associate-*l/ 67.6%

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

      10. neg-mul-1 67.6%

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

      11. associate-*l/ 67.5%

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

      12. distribute-lft-out-- 84.8%

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

      13. fabs-mul 84.8%

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

      14. fabs-sub 84.8%

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

      15. fabs-mul 84.8%

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

      16. associate-*l/ 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in x around inf 85.0%

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

      1. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

   if -6.50000000000000007e64 < x < 9.9999999999999992e124

   1. Initial program 98.9%

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

      1. associate-*l/ 99.9%

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

      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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+64} \lor \neg \left(x \leq 10^{+125}\right):\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 9: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -76000 \lor \neg \left(x \leq 6.5 \cdot 10^{-80}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -76000.0) (not (<= x 6.5e-80)))
   (fabs (* x (/ (- 1.0 z) y)))
   (fabs (/ (- -4.0 x) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -76000.0) || !(x <= 6.5e-80)) {
		tmp = fabs((x * ((1.0 - 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 ((x <= (-76000.0d0)) .or. (.not. (x <= 6.5d-80))) then
        tmp = abs((x * ((1.0d0 - 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 ((x <= -76000.0) || !(x <= 6.5e-80)) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -76000.0) or not (x <= 6.5e-80):
		tmp = math.fabs((x * ((1.0 - 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 ((x <= -76000.0) || !(x <= 6.5e-80))
		tmp = abs(Float64(x * Float64(Float64(1.0 - 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 ((x <= -76000.0) || ~((x <= 6.5e-80)))
		tmp = abs((x * ((1.0 - 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[x, -76000.0], N[Not[LessEqual[x, 6.5e-80]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -76000 or 6.49999999999999984e-80 < x

   1. Initial program 86.4%

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

      1. fabs-neg 86.4%

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

      2. sub-neg 86.4%

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

      3. distribute-neg-in 86.4%

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

      4. sub-neg 86.4%

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

      5. distribute-neg-frac 86.4%

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

      6. associate-*l/ 80.3%

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

      7. distribute-neg-frac 80.3%

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

      8. neg-mul-1 80.3%

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

      9. associate-*l/ 80.2%

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

      10. neg-mul-1 80.2%

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

      11. associate-*l/ 80.1%

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

      12. distribute-lft-out-- 90.6%

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

      13. fabs-mul 90.6%

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

      14. fabs-sub 90.6%

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

      15. fabs-mul 90.6%

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

      16. associate-*l/ 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in x around inf 88.7%

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

      1. associate-/l* 97.8%

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

      2. associate-/r/ 97.0%

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

   6. Simplified 97.0%

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

   if -76000 < x < 6.49999999999999984e-80

   1. Initial program 98.4%

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

      1. fabs-sub 98.4%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l/ 93.8%

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

      5. *-commutative 93.8%

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

      6. fma-neg 93.8%

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

      7. distribute-neg-frac 93.8%

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

      8. +-commutative 93.8%

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

      9. distribute-neg-in 93.8%

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

      10. unsub-neg 93.8%

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

      11. metadata-eval 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in z around 0 83.3%

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

      1. associate-*r/ 83.3%

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

      2. distribute-lft-in 83.3%

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

      3. metadata-eval 83.3%

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

      4. neg-mul-1 83.3%

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

      5. sub-neg 83.3%

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

   6. Simplified 83.3%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -76000 \lor \neg \left(x \leq 6.5 \cdot 10^{-80}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 10: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -76000 \lor \neg \left(x \leq 5.9 \cdot 10^{-81}\right):\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -76000.0) (not (<= x 5.9e-81)))
   (fabs (/ (- 1.0 z) (/ y x)))
   (fabs (/ (- -4.0 x) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -76000.0) || !(x <= 5.9e-81)) {
		tmp = fabs(((1.0 - z) / (y / x)));
	} 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 ((x <= (-76000.0d0)) .or. (.not. (x <= 5.9d-81))) then
        tmp = abs(((1.0d0 - z) / (y / x)))
    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 ((x <= -76000.0) || !(x <= 5.9e-81)) {
		tmp = Math.abs(((1.0 - z) / (y / x)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -76000.0) or not (x <= 5.9e-81):
		tmp = math.fabs(((1.0 - z) / (y / x)))
	else:
		tmp = math.fabs(((-4.0 - x) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -76000.0) || !(x <= 5.9e-81))
		tmp = abs(Float64(Float64(1.0 - z) / Float64(y / x)));
	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 ((x <= -76000.0) || ~((x <= 5.9e-81)))
		tmp = abs(((1.0 - z) / (y / x)));
	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[x, -76000.0], N[Not[LessEqual[x, 5.9e-81]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -76000 or 5.90000000000000024e-81 < x

   1. Initial program 86.4%

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

      1. fabs-neg 86.4%

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

      2. sub-neg 86.4%

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

      3. distribute-neg-in 86.4%

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

      4. sub-neg 86.4%

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

      5. distribute-neg-frac 86.4%

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

      6. associate-*l/ 80.3%

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

      7. distribute-neg-frac 80.3%

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

      8. neg-mul-1 80.3%

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

      9. associate-*l/ 80.2%

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

      10. neg-mul-1 80.2%

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

      11. associate-*l/ 80.1%

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

      12. distribute-lft-out-- 90.6%

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

      13. fabs-mul 90.6%

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

      14. fabs-sub 90.6%

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

      15. fabs-mul 90.6%

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

      16. associate-*l/ 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in x around inf 88.7%

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

      1. associate-/l* 97.8%

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

   6. Simplified 97.8%

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

   if -76000 < x < 5.90000000000000024e-81

   1. Initial program 98.4%

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

      1. fabs-sub 98.4%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l/ 93.8%

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

      5. *-commutative 93.8%

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

      6. fma-neg 93.8%

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

      7. distribute-neg-frac 93.8%

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

      8. +-commutative 93.8%

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

      9. distribute-neg-in 93.8%

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

      10. unsub-neg 93.8%

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

      11. metadata-eval 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in z around 0 83.3%

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

      1. associate-*r/ 83.3%

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

      2. distribute-lft-in 83.3%

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

      3. metadata-eval 83.3%

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

      4. neg-mul-1 83.3%

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

      5. sub-neg 83.3%

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

   6. Simplified 83.3%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -76000 \lor \neg \left(x \leq 5.9 \cdot 10^{-81}\right):\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 11: 68.8% 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 84.8%

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

      1. fabs-neg 84.8%

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

      2. sub-neg 84.8%

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

      3. distribute-neg-in 84.8%

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

      4. sub-neg 84.8%

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

      5. distribute-neg-frac 84.8%

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

      6. associate-*l/ 78.0%

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

      7. distribute-neg-frac 78.0%

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

      8. neg-mul-1 78.0%

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

      9. associate-*l/ 77.8%

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

      10. neg-mul-1 77.8%

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

      11. associate-*l/ 77.8%

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

      12. distribute-lft-out-- 89.6%

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

      13. fabs-mul 89.6%

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

      14. fabs-sub 89.6%

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

      15. fabs-mul 89.6%

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

      16. associate-*l/ 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in x around inf 89.2%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in z around 0 60.4%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 98.6%

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

   2. Taylor expanded in x around 0 76.3%

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

4. Final simplification 68.9%

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

Alternative 12: 39.7% 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 92.2%

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

2. Taylor expanded in x around 0 43.3%

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

3. Final simplification 43.3%

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

Reproduce

herbie shell --seed 2023257 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))