fabs fraction 1

Percentage Accurate: 91.6% → 99.4%

Time: 6.6s

Alternatives: 9

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-85)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))))

C

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

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-85)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 5e-85], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000002e-85

   1. Initial program 90.5%

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

      1. associate-*l/ 90.6%

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

      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 5.0000000000000002e-85 < y

   1. Initial program 95.3%

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

   3. Simplified 99.9%

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

4. Final simplification 97.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+107} \lor \neg \left(x \leq 3.4 \cdot 10^{+66}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.3e+107) (not (<= x 3.4e+66)))
   (fabs (* (/ x y) (- 1.0 z)))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.3e+107) || !(x <= 3.4e+66)) {
		tmp = fabs(((x / y) * (1.0 - z)));
	} 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 <= (-1.3d+107)) .or. (.not. (x <= 3.4d+66))) then
        tmp = abs(((x / y) * (1.0d0 - z)))
    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 <= -1.3e+107) || !(x <= 3.4e+66)) {
		tmp = Math.abs(((x / y) * (1.0 - z)));
	} 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 <= -1.3e+107) or not (x <= 3.4e+66):
		tmp = math.fabs(((x / y) * (1.0 - z)))
	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 <= -1.3e+107) || !(x <= 3.4e+66))
		tmp = abs(Float64(Float64(x / y) * Float64(1.0 - z)));
	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 <= -1.3e+107) || ~((x <= 3.4e+66)))
		tmp = abs(((x / y) * (1.0 - z)));
	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, -1.3e+107], N[Not[LessEqual[x, 3.4e+66]], $MachinePrecision]], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $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 < -1.3000000000000001e107 or 3.4000000000000003e66 < x

   1. Initial program 86.3%

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

   2. Taylor expanded in x around inf 86.3%

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

      1. *-un-lft-identity 86.3%

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

      2. *-commutative 86.3%

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

      3. distribute-rgt-out-- 99.9%

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

   4. Applied egg-rr 99.9%

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

   if -1.3000000000000001e107 < x < 3.4000000000000003e66

   1. Initial program 95.0%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+107} \lor \neg \left(x \leq 3.4 \cdot 10^{+66}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.6e-49)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ (+ x 4.0) y) (/ x (/ y z))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.6e-49) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (x / (y / z))));
	}
	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 <= 1.6d-49) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((((x + 4.0d0) / y) - (x / (y / z))))
    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 <= 1.6e-49) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (x / (y / z))));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 1.6e-49], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.60000000000000001e-49

   1. Initial program 90.8%

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

      1. associate-*l/ 90.9%

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

      2. sub-div 96.9%

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

   3. Applied egg-rr 96.9%

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

   if 1.60000000000000001e-49 < y

   1. Initial program 95.0%

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

   3. Simplified 99.9%

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

4. Final simplification 97.8%

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

Alternative 4: 87.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -126000.0) (not (<= x 4.7e-16)))
   (fabs (* (/ x y) (- 1.0 z)))
   (fabs (/ (- -4.0 x) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -126000.0) || !(x <= 4.7e-16)) {
		tmp = fabs(((x / y) * (1.0 - z)));
	} 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 <= (-126000.0d0)) .or. (.not. (x <= 4.7d-16))) then
        tmp = abs(((x / y) * (1.0d0 - z)))
    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 <= -126000.0) || !(x <= 4.7e-16)) {
		tmp = Math.abs(((x / y) * (1.0 - z)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y));
	}
	return tmp;
}

Python

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

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -126000.0) || !(x <= 4.7e-16))
		tmp = abs(Float64(Float64(x / y) * Float64(1.0 - z)));
	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 <= -126000.0) || ~((x <= 4.7e-16)))
		tmp = abs(((x / y) * (1.0 - z)));
	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, -126000.0], N[Not[LessEqual[x, 4.7e-16]], $MachinePrecision]], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.1%

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

   2. Taylor expanded in x around inf 88.8%

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

      1. *-un-lft-identity 88.8%

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

      2. *-commutative 88.8%

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

      3. distribute-rgt-out-- 98.6%

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

   4. Applied egg-rr 98.6%

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

   if -126000 < x < 4.70000000000000044e-16

   1. Initial program 93.7%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in z around 0 76.2%

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

      1. associate-*r/ 76.2%

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

      2. distribute-lft-in 76.2%

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

      3. metadata-eval 76.2%

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

      4. neg-mul-1 76.2%

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

      5. sub-neg 76.2%

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

   6. Simplified 76.2%

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

4. Final simplification 86.9%

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

Alternative 5: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -4.65 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-23}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -4.65e+148)
     t_0
     (if (<= x -4e-23)
       (fabs (* x (/ z y)))
       (if (<= x 4.0) (fabs (/ 4.0 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 <= -4.65e+148) {
		tmp = t_0;
	} else if (x <= -4e-23) {
		tmp = fabs((x * (z / y)));
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / 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 <= (-4.65d+148)) then
        tmp = t_0
    else if (x <= (-4d-23)) then
        tmp = abs((x * (z / y)))
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / 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 <= -4.65e+148) {
		tmp = t_0;
	} else if (x <= -4e-23) {
		tmp = Math.abs((x * (z / y)));
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / 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 <= -4.65e+148:
		tmp = t_0
	elif x <= -4e-23:
		tmp = math.fabs((x * (z / y)))
	elif x <= 4.0:
		tmp = math.fabs((4.0 / 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 <= -4.65e+148)
		tmp = t_0;
	elseif (x <= -4e-23)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / 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 <= -4.65e+148)
		tmp = t_0;
	elseif (x <= -4e-23)
		tmp = abs((x * (z / y)));
	elseif (x <= 4.0)
		tmp = abs((4.0 / 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, -4.65e+148], t$95$0, If[LessEqual[x, -4e-23], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.64999999999999992e148 or 4 < x

   1. Initial program 87.8%

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

   2. Taylor expanded in x around inf 86.7%

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

   3. Taylor expanded in z around 0 75.2%

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

   if -4.64999999999999992e148 < x < -3.99999999999999984e-23

   1. Initial program 97.1%

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

   2. Taylor expanded in z around inf 52.5%

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

      1. mul-1-neg 52.5%

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

      2. associate-*r/ 62.7%

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

      3. distribute-rgt-neg-out 62.7%

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

      4. distribute-neg-frac 62.7%

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

   4. Simplified 62.7%

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

      1. clear-num 62.5%

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

      2. associate-/r/ 62.7%

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

      3. add-sqr-sqrt 29.9%

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

      4. sqrt-unprod 45.9%

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

      5. sqr-neg 45.9%

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

      6. sqrt-unprod 32.6%

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

      7. add-sqr-sqrt 62.7%

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

      8. associate-*l* 63.3%

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

      9. div-inv 63.4%

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

      10. expm1-log1p-u 35.7%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)\right)}\right| \]

      11. expm1-udef 34.9%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)} - 1}\right| \]

      12. associate-*l/ 28.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot z}{y}}\right)} - 1\right| \]

      13. div-inv 28.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot z\right) \cdot \frac{1}{y}}\right)} - 1\right| \]

      14. associate-*l* 31.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{x \cdot \left(z \cdot \frac{1}{y}\right)}\right)} - 1\right| \]

      15. div-inv 31.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{z}{y}}\right)} - 1\right| \]

   6. Applied egg-rr 31.5%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{z}{y}\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 35.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{z}{y}\right)\right)}\right| \]

      2. expm1-log1p 62.7%

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

   8. Simplified 62.7%

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

   if -3.99999999999999984e-23 < x < 4

   1. Initial program 94.1%

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

   2. Taylor expanded in x around 0 76.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.65 \cdot 10^{+148}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-23}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 6: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -4.9 \cdot 10^{+147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-9}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (<= x -4.9e+147)
     t_0
     (if (<= x -2.45e-9)
       (fabs (/ z (/ y x)))
       (if (<= x 4.0) (fabs (/ 4.0 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 <= -4.9e+147) {
		tmp = t_0;
	} else if (x <= -2.45e-9) {
		tmp = fabs((z / (y / x)));
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / 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 <= (-4.9d+147)) then
        tmp = t_0
    else if (x <= (-2.45d-9)) then
        tmp = abs((z / (y / x)))
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / 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 <= -4.9e+147) {
		tmp = t_0;
	} else if (x <= -2.45e-9) {
		tmp = Math.abs((z / (y / x)));
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / 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 <= -4.9e+147:
		tmp = t_0
	elif x <= -2.45e-9:
		tmp = math.fabs((z / (y / x)))
	elif x <= 4.0:
		tmp = math.fabs((4.0 / 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 <= -4.9e+147)
		tmp = t_0;
	elseif (x <= -2.45e-9)
		tmp = abs(Float64(z / Float64(y / x)));
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / 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 <= -4.9e+147)
		tmp = t_0;
	elseif (x <= -2.45e-9)
		tmp = abs((z / (y / x)));
	elseif (x <= 4.0)
		tmp = abs((4.0 / 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, -4.9e+147], t$95$0, If[LessEqual[x, -2.45e-9], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8999999999999998e147 or 4 < x

   1. Initial program 87.8%

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

   2. Taylor expanded in x around inf 86.7%

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

   3. Taylor expanded in z around 0 75.2%

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

   if -4.8999999999999998e147 < x < -2.45000000000000002e-9

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 52.4%

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

      1. mul-1-neg 52.4%

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

      2. associate-*r/ 63.3%

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

      3. distribute-rgt-neg-out 63.3%

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

      4. distribute-neg-frac 63.3%

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

   4. Simplified 63.3%

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

      1. *-commutative 63.3%

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

      2. div-inv 63.3%

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

      3. associate-*l* 66.9%

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

      4. add-sqr-sqrt 31.5%

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

      5. sqrt-unprod 48.6%

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

      6. sqr-neg 48.6%

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

      7. sqrt-unprod 35.1%

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

      8. add-sqr-sqrt 66.9%

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

      9. associate-/r/ 66.7%

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

      10. un-div-inv 66.9%

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

   6. Applied egg-rr 66.9%

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

   if -2.45000000000000002e-9 < x < 4

   1. Initial program 93.6%

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

   2. Taylor expanded in x around 0 75.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+147}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-9}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 7: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{+63}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -6.4e+15)
   (fabs (/ (* x z) y))
   (if (<= z 1.52e+63) (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 <= -6.4e+15) {
		tmp = fabs(((x * z) / y));
	} else if (z <= 1.52e+63) {
		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 <= (-6.4d+15)) then
        tmp = abs(((x * z) / y))
    else if (z <= 1.52d+63) 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 <= -6.4e+15) {
		tmp = Math.abs(((x * z) / y));
	} else if (z <= 1.52e+63) {
		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 <= -6.4e+15:
		tmp = math.fabs(((x * z) / y))
	elif z <= 1.52e+63:
		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 <= -6.4e+15)
		tmp = abs(Float64(Float64(x * z) / y));
	elseif (z <= 1.52e+63)
		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 <= -6.4e+15)
		tmp = abs(((x * z) / y));
	elseif (z <= 1.52e+63)
		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, -6.4e+15], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.52e+63], 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 < -6.4e15

   1. Initial program 91.3%

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

   2. Taylor expanded in z around inf 74.5%

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

      1. mul-1-neg 74.5%

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

      2. associate-*r/ 74.4%

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

      3. distribute-rgt-neg-out 74.4%

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

      4. distribute-neg-frac 74.4%

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

   4. Simplified 74.4%

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

      1. clear-num 74.3%

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

      2. associate-/r/ 74.3%

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

      3. add-sqr-sqrt 74.2%

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

      4. sqrt-unprod 59.7%

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

      5. sqr-neg 59.7%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 74.3%

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

      8. associate-*l* 71.9%

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

      9. div-inv 72.0%

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

      10. associate-*l/ 74.5%

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

   6. Applied egg-rr 74.5%

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

   if -6.4e15 < z < 1.51999999999999993e63

   1. Initial program 95.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 97.0%

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

      1. associate-*r/ 97.0%

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

      2. distribute-lft-in 97.0%

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

      3. metadata-eval 97.0%

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

      4. neg-mul-1 97.0%

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

      5. sub-neg 97.0%

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

   6. Simplified 97.0%

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

   if 1.51999999999999993e63 < z

   1. Initial program 80.3%

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

   2. Taylor expanded in z around inf 55.7%

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

      1. mul-1-neg 55.7%

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

      2. associate-*r/ 69.6%

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

      3. distribute-rgt-neg-out 69.6%

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

      4. distribute-neg-frac 69.6%

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

   4. Simplified 69.6%

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

      1. clear-num 69.5%

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

      2. associate-/r/ 69.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 38.6%

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

      5. sqr-neg 38.6%

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

      6. sqrt-unprod 69.2%

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

      7. add-sqr-sqrt 69.6%

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

      8. associate-*l* 66.8%

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

      9. div-inv 66.9%

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

      10. expm1-log1p-u 35.4%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)\right)}\right| \]

      11. expm1-udef 28.1%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)} - 1}\right| \]

      12. associate-*l/ 26.3%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot z}{y}}\right)} - 1\right| \]

      13. div-inv 26.3%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot z\right) \cdot \frac{1}{y}}\right)} - 1\right| \]

      14. associate-*l* 26.4%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{x \cdot \left(z \cdot \frac{1}{y}\right)}\right)} - 1\right| \]

      15. div-inv 26.4%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{z}{y}}\right)} - 1\right| \]

   6. Applied egg-rr 26.4%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{z}{y}\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 40.1%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{z}{y}\right)\right)}\right| \]

      2. expm1-log1p 69.6%

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

   8. Simplified 69.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{+63}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 8: 68.1% accurate, 1.0× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10.2) (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 <= -10.2) || !(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 <= (-10.2d0)) .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 <= -10.2) || !(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 <= -10.2) 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 <= -10.2) || !(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 <= -10.2) || ~((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, -10.2], 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 < -10.199999999999999 or 4 < x

   1. Initial program 90.1%

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

   2. Taylor expanded in x around inf 88.3%

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

   3. Taylor expanded in z around 0 67.7%

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

   if -10.199999999999999 < x < 4

   1. Initial program 93.7%

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

   2. Taylor expanded in x around 0 74.9%

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

4. Final simplification 71.4%

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

Alternative 9: 39.8% 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 92.0%

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

2. Taylor expanded in x around 0 41.4%

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

3. Final simplification 41.4%

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

Reproduce

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