fabs fraction 1

Percentage Accurate: 91.9% → 99.7%

Time: 6.0s

Alternatives: 8

Speedup: 1.0×

• 20.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-35}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5e-35)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (+ (* x (/ z y)) (/ (- -4.0 x) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e-35) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(((x * (z / y)) + ((-4.0 - x) / y)));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.5d-35) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs(((x * (z / y)) + (((-4.0d0) - x) / y)))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e-35) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs(((x * (z / y)) + ((-4.0 - x) / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 2.5e-35:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs(((x * (z / y)) + ((-4.0 - x) / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5e-35)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(x * Float64(z / y)) + Float64(Float64(-4.0 - x) / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.5e-35)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs(((x * (z / y)) + ((-4.0 - x) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 2.5e-35], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.49999999999999982e-35

   1. Initial program 92.2%

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

      1. associate-*l/ 90.8%

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

      2. sub-div 96.8%

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

   3. Applied egg-rr 96.8%

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

   if 2.49999999999999982e-35 < y

   1. Initial program 96.1%

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

   3. Simplified 92.1%

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

      1. fma-udef 92.1%

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

      2. distribute-rgt-in 92.1%

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

      3. associate-*r* 99.8%

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

      4. div-inv 99.8%

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

      5. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-35}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y} + \frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 2: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -5.7 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{-53}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 -5.7e+150)
     t_0
     (if (<= x -6.5e-47)
       t_1
       (if (<= x 6.1e-53) (fabs (/ 4.0 y)) (if (<= x 9.8e+51) 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 <= -5.7e+150) {
		tmp = t_0;
	} else if (x <= -6.5e-47) {
		tmp = t_1;
	} else if (x <= 6.1e-53) {
		tmp = fabs((4.0 / y));
	} else if (x <= 9.8e+51) {
		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 <= (-5.7d+150)) then
        tmp = t_0
    else if (x <= (-6.5d-47)) then
        tmp = t_1
    else if (x <= 6.1d-53) then
        tmp = abs((4.0d0 / y))
    else if (x <= 9.8d+51) 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 <= -5.7e+150) {
		tmp = t_0;
	} else if (x <= -6.5e-47) {
		tmp = t_1;
	} else if (x <= 6.1e-53) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 9.8e+51) {
		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 <= -5.7e+150:
		tmp = t_0
	elif x <= -6.5e-47:
		tmp = t_1
	elif x <= 6.1e-53:
		tmp = math.fabs((4.0 / y))
	elif x <= 9.8e+51:
		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 <= -5.7e+150)
		tmp = t_0;
	elseif (x <= -6.5e-47)
		tmp = t_1;
	elseif (x <= 6.1e-53)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 9.8e+51)
		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 <= -5.7e+150)
		tmp = t_0;
	elseif (x <= -6.5e-47)
		tmp = t_1;
	elseif (x <= 6.1e-53)
		tmp = abs((4.0 / y));
	elseif (x <= 9.8e+51)
		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, -5.7e+150], t$95$0, If[LessEqual[x, -6.5e-47], t$95$1, If[LessEqual[x, 6.1e-53], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 9.8e+51], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7000000000000002e150 or 9.79999999999999967e51 < x

   1. Initial program 87.3%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. mul-1-neg 99.7%

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

      4. distribute-lft-in 88.3%

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

      5. associate-*r/ 88.3%

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

      6. *-rgt-identity 88.3%

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

      7. mul-1-neg 88.3%

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

      8. distribute-rgt-neg-in 88.3%

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

      9. unsub-neg 88.3%

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

      10. *-lft-identity 88.3%

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

      11. associate-/l* 88.3%

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

      12. *-commutative 88.3%

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

      13. associate-/r/ 87.2%

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

      14. div-sub 99.8%

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

      15. associate-/r/ 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in z around 0 70.2%

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

   if -5.7000000000000002e150 < x < -6.5000000000000004e-47 or 6.0999999999999999e-53 < x < 9.79999999999999967e51

   1. Initial program 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in z around inf 62.8%

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

      1. associate-*l/ 66.3%

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

      2. *-commutative 66.3%

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

   6. Simplified 66.3%

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

   if -6.5000000000000004e-47 < x < 6.0999999999999999e-53

   1. Initial program 96.6%

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

   2. Taylor expanded in x around 0 83.6%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+150}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-47}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{-53}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+51}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+133} \lor \neg \left(x \leq 88000000\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.4e+133) (not (<= x 88000000.0)))
   (fabs (/ (+ z -1.0) (/ y x)))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.4e+133) || !(x <= 88000000.0)) {
		tmp = fabs(((z + -1.0) / (y / x)));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-4.4d+133)) .or. (.not. (x <= 88000000.0d0))) then
        tmp = abs(((z + (-1.0d0)) / (y / x)))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.4e+133) || !(x <= 88000000.0)) {
		tmp = Math.abs(((z + -1.0) / (y / x)));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -4.4e+133) or not (x <= 88000000.0):
		tmp = math.fabs(((z + -1.0) / (y / x)))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.4e+133) || !(x <= 88000000.0))
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y / x)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.4e+133) || ~((x <= 88000000.0)))
		tmp = abs(((z + -1.0) / (y / x)));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[x, -4.4e+133], N[Not[LessEqual[x, 88000000.0]], $MachinePrecision]], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4e133 or 8.8e7 < x

   1. Initial program 88.4%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in x around inf 88.2%

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

      1. associate-/l* 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

   if -4.4e133 < x < 8.8e7

   1. Initial program 96.2%

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

      1. associate-*l/ 99.2%

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

      2. sub-div 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+133} \lor \neg \left(x \leq 88000000\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 4: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-48} \lor \neg \left(x \leq 1.15 \cdot 10^{-51}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1e-48) (not (<= x 1.15e-51)))
   (fabs (* x (/ (- 1.0 z) y)))
   (fabs (/ 4.0 y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1e-48) || !(x <= 1.15e-51)) {
		tmp = fabs((x * ((1.0 - z) / 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 <= (-1d-48)) .or. (.not. (x <= 1.15d-51))) then
        tmp = abs((x * ((1.0d0 - z) / 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 <= -1e-48) || !(x <= 1.15e-51)) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -1e-48) or not (x <= 1.15e-51):
		tmp = math.fabs((x * ((1.0 - z) / y)))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -1e-48) || !(x <= 1.15e-51))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / 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 <= -1e-48) || ~((x <= 1.15e-51)))
		tmp = abs((x * ((1.0 - z) / 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, -1e-48], N[Not[LessEqual[x, 1.15e-51]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.9999999999999997e-49 or 1.15000000000000001e-51 < x

   1. Initial program 90.7%

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

   2. Taylor expanded in x around inf 94.5%

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

      1. *-commutative 94.5%

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

      2. sub-neg 94.5%

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

      3. mul-1-neg 94.5%

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

      4. distribute-lft-in 86.7%

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

      5. associate-*r/ 86.8%

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

      6. *-rgt-identity 86.8%

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

      7. mul-1-neg 86.8%

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

      8. distribute-rgt-neg-in 86.8%

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

      9. unsub-neg 86.8%

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

      10. *-lft-identity 86.8%

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

      11. associate-/l* 86.7%

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

      12. *-commutative 86.7%

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

      13. associate-/r/ 85.4%

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

      14. div-sub 93.8%

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

      15. associate-/r/ 94.6%

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

   4. Simplified 94.6%

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

   if -9.9999999999999997e-49 < x < 1.15000000000000001e-51

   1. Initial program 96.6%

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

   2. Taylor expanded in x around 0 83.6%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-48} \lor \neg \left(x \leq 1.15 \cdot 10^{-51}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 5: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-47}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x -3.9e-47)
   (fabs (* x (/ (- 1.0 z) y)))
   (if (<= x 2e-52) (fabs (/ 4.0 y)) (fabs (/ (+ z -1.0) (/ y x))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.9e-47) {
		tmp = fabs((x * ((1.0 - z) / y)));
	} else if (x <= 2e-52) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = fabs(((z + -1.0) / (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) :: tmp
    if (x <= (-3.9d-47)) then
        tmp = abs((x * ((1.0d0 - z) / y)))
    else if (x <= 2d-52) then
        tmp = abs((4.0d0 / y))
    else
        tmp = abs(((z + (-1.0d0)) / (y / x)))
    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 <= -3.9e-47) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} else if (x <= 2e-52) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = Math.abs(((z + -1.0) / (y / x)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if x <= -3.9e-47:
		tmp = math.fabs((x * ((1.0 - z) / y)))
	elif x <= 2e-52:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = math.fabs(((z + -1.0) / (y / x)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (x <= -3.9e-47)
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	elseif (x <= 2e-52)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y / x)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.9e-47)
		tmp = abs((x * ((1.0 - z) / y)));
	elseif (x <= 2e-52)
		tmp = abs((4.0 / y));
	else
		tmp = abs(((z + -1.0) / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, -3.9e-47], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2e-52], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.89999999999999978e-47

   1. Initial program 94.1%

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

   2. Taylor expanded in x around inf 94.0%

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

      1. *-commutative 94.0%

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

      2. sub-neg 94.0%

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

      3. mul-1-neg 94.0%

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

      4. distribute-lft-in 91.2%

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

      5. associate-*r/ 91.2%

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

      6. *-rgt-identity 91.2%

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

      7. mul-1-neg 91.2%

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

      8. distribute-rgt-neg-in 91.2%

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

      9. unsub-neg 91.2%

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

      10. *-lft-identity 91.2%

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

      11. associate-/l* 91.2%

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

      12. *-commutative 91.2%

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

      13. associate-/r/ 88.4%

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

      14. div-sub 92.6%

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

      15. associate-/r/ 94.0%

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

   4. Simplified 94.0%

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

   if -3.89999999999999978e-47 < x < 2e-52

   1. Initial program 96.6%

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

   2. Taylor expanded in x around 0 83.6%

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

   if 2e-52 < x

   1. Initial program 87.2%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in x around inf 87.3%

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

      1. associate-/l* 95.1%

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

      2. sub-neg 95.1%

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

      3. metadata-eval 95.1%

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

   6. Simplified 95.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-47}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 6: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+113}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+48}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -4.8e+113)
   (fabs (/ (* x z) y))
   (if (<= z 7.2e+48) (fabs (/ (- -4.0 x) y)) (fabs (* x (/ z y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e+113) {
		tmp = fabs(((x * z) / y));
	} else if (z <= 7.2e+48) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((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 (z <= (-4.8d+113)) then
        tmp = abs(((x * z) / y))
    else if (z <= 7.2d+48) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((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 (z <= -4.8e+113) {
		tmp = Math.abs(((x * z) / y));
	} else if (z <= 7.2e+48) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((x * (z / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -4.8e+113:
		tmp = math.fabs(((x * z) / y))
	elif z <= 7.2e+48:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((x * (z / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -4.8e+113)
		tmp = abs(Float64(Float64(x * z) / y));
	elseif (z <= 7.2e+48)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.8e+113)
		tmp = abs(((x * z) / y));
	elseif (z <= 7.2e+48)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((x * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -4.8e+113], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 7.2e+48], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.79999999999999966e113

   1. Initial program 90.8%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in z around inf 75.9%

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

   if -4.79999999999999966e113 < z < 7.19999999999999967e48

   1. Initial program 96.9%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in z around 0 92.8%

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

      1. associate-*r/ 92.8%

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

      2. distribute-lft-in 92.8%

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

      3. metadata-eval 92.8%

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

      4. neg-mul-1 92.8%

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

      5. sub-neg 92.8%

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

   6. Simplified 92.8%

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

   if 7.19999999999999967e48 < z

   1. Initial program 85.4%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in z around inf 71.0%

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

      1. associate-*l/ 80.3%

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

      2. *-commutative 80.3%

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

   6. Simplified 80.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+113}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+48}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 7: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.5) (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 <= -1.5) || !(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 <= (-1.5d0)) .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 <= -1.5) || !(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 <= -1.5) 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 <= -1.5) || !(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 <= -1.5) || ~((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, -1.5], 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 < -1.5 or 4 < x

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. *-commutative 98.9%

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

      2. sub-neg 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-lft-in 89.6%

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

      5. associate-*r/ 89.7%

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

      6. *-rgt-identity 89.7%

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

      7. mul-1-neg 89.7%

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

      8. distribute-rgt-neg-in 89.7%

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

      9. unsub-neg 89.7%

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

      10. *-lft-identity 89.7%

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

      11. associate-/l* 89.6%

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

      12. *-commutative 89.6%

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

      13. associate-/r/ 88.8%

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

      14. div-sub 98.9%

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

      15. associate-/r/ 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in z around 0 61.9%

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

   if -1.5 < x < 4

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0 75.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 8: 39.6% 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 93.3%

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

2. Taylor expanded in x around 0 42.5%

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

3. Final simplification 42.5%

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

Reproduce

herbie shell --seed 2023230 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))