fabs fraction 1

Percentage Accurate: 91.6% → 99.5%

Time: 7.8s

Alternatives: 13

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: 91.6% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+84}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+84)
   (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 <= 1e+84) {
		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 <= 1e+84)
		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, 1e+84], 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 < 1.00000000000000006e84

   1. Initial program 91.7%

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

      1. associate-*l/ 94.9%

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

      2. sub-div 98.9%

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

   3. Applied egg-rr 98.9%

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

   if 1.00000000000000006e84 < y

   1. Initial program 90.4%

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

      1. fabs-sub 90.4%

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

      2. associate-*l/ 88.8%

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

      3. *-commutative 88.8%

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

      4. associate-*l/ 99.8%

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

      5. *-commutative 99.8%

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

      6. fma-neg 99.8%

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

      7. distribute-neg-frac 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+84}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-40}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{\frac{\frac{y}{x}}{z + -1}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.2e-40)
   (fabs (/ (+ z -1.0) (/ y x)))
   (if (<= x 6.8e-13)
     (fabs (/ (- -4.0 x) y))
     (fabs (/ 1.0 (/ (/ y x) (+ z -1.0)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.2e-40) {
		tmp = fabs(((z + -1.0) / (y / x)));
	} else if (x <= 6.8e-13) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((1.0 / ((y / x) / (z + -1.0))));
	}
	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) :: tmp
    if (x <= (-1.2d-40)) then
        tmp = abs(((z + (-1.0d0)) / (y / x)))
    else if (x <= 6.8d-13) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((1.0d0 / ((y / x) / (z + (-1.0d0)))))
    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 <= -1.2e-40) {
		tmp = Math.abs(((z + -1.0) / (y / x)));
	} else if (x <= 6.8e-13) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((1.0 / ((y / x) / (z + -1.0))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if x <= -1.2e-40:
		tmp = math.fabs(((z + -1.0) / (y / x)))
	elif x <= 6.8e-13:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((1.0 / ((y / x) / (z + -1.0))))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (x <= -1.2e-40)
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y / x)));
	elseif (x <= 6.8e-13)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(1.0 / Float64(Float64(y / x) / Float64(z + -1.0))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.2e-40)
		tmp = abs(((z + -1.0) / (y / x)));
	elseif (x <= 6.8e-13)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((1.0 / ((y / x) / (z + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, -1.2e-40], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 6.8e-13], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(1.0 / N[(N[(y / x), $MachinePrecision] / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999996e-40

   1. Initial program 91.9%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in x around inf 85.5%

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

      1. associate-/l* 95.8%

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

      2. sub-neg 95.8%

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

      3. metadata-eval 95.8%

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

   6. Simplified 95.8%

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

   if -1.19999999999999996e-40 < x < 6.80000000000000031e-13

   1. Initial program 92.1%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. distribute-lft-in 78.0%

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

      3. metadata-eval 78.0%

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

      4. neg-mul-1 78.0%

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

      5. sub-neg 78.0%

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

   6. Simplified 78.0%

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

   if 6.80000000000000031e-13 < x

   1. Initial program 89.7%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in x around inf 95.0%

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

      1. associate-/l* 97.6%

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

      2. sub-neg 97.6%

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

      3. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

      1. clear-num 97.6%

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

      2. inv-pow 97.6%

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

   8. Applied egg-rr 97.6%

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

      1. unpow-1 97.6%

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

   10. Simplified 97.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-40}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{\frac{\frac{y}{x}}{z + -1}}\right|\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+108}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 5e+108)
   (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 <= 5e+108) {
		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 <= 5d+108) 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 <= 5e+108) {
		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 <= 5e+108:
		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 <= 5e+108)
		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 <= 5e+108)
		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, 5e+108], 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 < 4.99999999999999991e108

   1. Initial program 91.9%

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

      1. associate-*l/ 95.1%

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

      2. sub-div 99.0%

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

   3. Applied egg-rr 99.0%

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

   if 4.99999999999999991e108 < y

   1. Initial program 89.4%

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

      1. associate-*l/ 87.1%

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

      2. associate-/l* 99.8%

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

   3. Applied egg-rr 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+108}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 4: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -2 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* x (/ z y)))))
   (if (<= x -2e-41)
     t_0
     (if (<= x 8.5e-7)
       (fabs (/ 4.0 y))
       (if (<= x 1.3e+148) t_0 (fabs (/ x y)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((x * (z / y)));
	double tmp;
	if (x <= -2e-41) {
		tmp = t_0;
	} else if (x <= 8.5e-7) {
		tmp = fabs((4.0 / y));
	} else if (x <= 1.3e+148) {
		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((x * (z / y)))
    if (x <= (-2d-41)) then
        tmp = t_0
    else if (x <= 8.5d-7) then
        tmp = abs((4.0d0 / y))
    else if (x <= 1.3d+148) 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((x * (z / y)));
	double tmp;
	if (x <= -2e-41) {
		tmp = t_0;
	} else if (x <= 8.5e-7) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 1.3e+148) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((x * (z / y)))
	tmp = 0
	if x <= -2e-41:
		tmp = t_0
	elif x <= 8.5e-7:
		tmp = math.fabs((4.0 / y))
	elif x <= 1.3e+148:
		tmp = t_0
	else:
		tmp = math.fabs((x / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(x * Float64(z / y)))
	tmp = 0.0
	if (x <= -2e-41)
		tmp = t_0;
	elseif (x <= 8.5e-7)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 1.3e+148)
		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((x * (z / y)));
	tmp = 0.0;
	if (x <= -2e-41)
		tmp = t_0;
	elseif (x <= 8.5e-7)
		tmp = abs((4.0 / y));
	elseif (x <= 1.3e+148)
		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[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2e-41], t$95$0, If[LessEqual[x, 8.5e-7], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3e+148], t$95$0, N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000001e-41 or 8.50000000000000014e-7 < x < 1.3e148

   1. Initial program 93.6%

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

   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 z around inf 51.9%

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

      1. associate-*l/ 58.9%

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

      2. *-commutative 58.9%

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

   6. Simplified 58.9%

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

   if -2.00000000000000001e-41 < x < 8.50000000000000014e-7

   1. Initial program 92.2%

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

   2. Taylor expanded in x around 0 77.2%

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

   if 1.3e148 < x

   1. Initial program 81.8%

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

      1. clear-num 81.5%

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

      2. associate-*l/ 87.8%

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

      3. frac-sub 78.4%

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

      4. *-un-lft-identity 78.4%

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

   3. Applied egg-rr 78.4%

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

   4. Taylor expanded in x around inf 70.5%

      \[\leadsto \left|\color{blue}{\frac{\left(y - y \cdot z\right) \cdot x}{{y}^{2}}}\right| \]
   5. Step-by-step derivation

      1. associate-/l* 87.8%

         \[\leadsto \left|\color{blue}{\frac{y - y \cdot z}{\frac{{y}^{2}}{x}}}\right| \]

      2. unpow2 87.8%

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

   6. Simplified 87.8%

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

   7. Taylor expanded in z around 0 80.1%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-41}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+148}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 5: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))))
   (if (<= x -4.8e-42)
     t_0
     (if (<= x 6e-13)
       (fabs (/ 4.0 y))
       (if (<= x 1.95e+149) t_0 (fabs (/ x y)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double tmp;
	if (x <= -4.8e-42) {
		tmp = t_0;
	} else if (x <= 6e-13) {
		tmp = fabs((4.0 / y));
	} else if (x <= 1.95e+149) {
		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 * (x / y)))
    if (x <= (-4.8d-42)) then
        tmp = t_0
    else if (x <= 6d-13) then
        tmp = abs((4.0d0 / y))
    else if (x <= 1.95d+149) 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 * (x / y)));
	double tmp;
	if (x <= -4.8e-42) {
		tmp = t_0;
	} else if (x <= 6e-13) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 1.95e+149) {
		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 * (x / y)))
	tmp = 0
	if x <= -4.8e-42:
		tmp = t_0
	elif x <= 6e-13:
		tmp = math.fabs((4.0 / y))
	elif x <= 1.95e+149:
		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(x / y)))
	tmp = 0.0
	if (x <= -4.8e-42)
		tmp = t_0;
	elseif (x <= 6e-13)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 1.95e+149)
		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 * (x / y)));
	tmp = 0.0;
	if (x <= -4.8e-42)
		tmp = t_0;
	elseif (x <= 6e-13)
		tmp = abs((4.0 / y));
	elseif (x <= 1.95e+149)
		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[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4.8e-42], t$95$0, If[LessEqual[x, 6e-13], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.95e+149], t$95$0, N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.80000000000000005e-42 or 5.99999999999999968e-13 < x < 1.95e149

   1. Initial program 93.7%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in x around inf 88.3%

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

      1. associate-/l* 95.8%

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

      2. sub-neg 95.8%

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

      3. metadata-eval 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in z around inf 51.9%

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

      1. associate-*r/ 62.3%

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

   9. Simplified 62.3%

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

   if -4.80000000000000005e-42 < x < 5.99999999999999968e-13

   1. Initial program 92.1%

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

   2. Taylor expanded in x around 0 77.9%

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

   if 1.95e149 < x

   1. Initial program 81.8%

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

      1. clear-num 81.5%

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

      2. associate-*l/ 87.8%

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

      3. frac-sub 78.4%

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

      4. *-un-lft-identity 78.4%

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

   3. Applied egg-rr 78.4%

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

   4. Taylor expanded in x around inf 70.5%

      \[\leadsto \left|\color{blue}{\frac{\left(y - y \cdot z\right) \cdot x}{{y}^{2}}}\right| \]
   5. Step-by-step derivation

      1. associate-/l* 87.8%

         \[\leadsto \left|\color{blue}{\frac{y - y \cdot z}{\frac{{y}^{2}}{x}}}\right| \]

      2. unpow2 87.8%

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

   6. Simplified 87.8%

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

   7. Taylor expanded in z around 0 80.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-42}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+149}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 6: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-42}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+148}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x -3.1e-42)
   (fabs (* z (/ x y)))
   (if (<= x 6.5e-13)
     (fabs (/ 4.0 y))
     (if (<= x 6.5e+148) (fabs (/ z (/ y x))) (fabs (/ x y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1e-42) {
		tmp = fabs((z * (x / y)));
	} else if (x <= 6.5e-13) {
		tmp = fabs((4.0 / y));
	} else if (x <= 6.5e+148) {
		tmp = fabs((z / (y / x)));
	} 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) :: tmp
    if (x <= (-3.1d-42)) then
        tmp = abs((z * (x / y)))
    else if (x <= 6.5d-13) then
        tmp = abs((4.0d0 / y))
    else if (x <= 6.5d+148) then
        tmp = abs((z / (y / x)))
    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 tmp;
	if (x <= -3.1e-42) {
		tmp = Math.abs((z * (x / y)));
	} else if (x <= 6.5e-13) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 6.5e+148) {
		tmp = Math.abs((z / (y / x)));
	} else {
		tmp = Math.abs((x / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if x <= -3.1e-42:
		tmp = math.fabs((z * (x / y)))
	elif x <= 6.5e-13:
		tmp = math.fabs((4.0 / y))
	elif x <= 6.5e+148:
		tmp = math.fabs((z / (y / x)))
	else:
		tmp = math.fabs((x / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (x <= -3.1e-42)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (x <= 6.5e-13)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 6.5e+148)
		tmp = abs(Float64(z / Float64(y / x)));
	else
		tmp = abs(Float64(x / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.1e-42)
		tmp = abs((z * (x / y)));
	elseif (x <= 6.5e-13)
		tmp = abs((4.0 / y));
	elseif (x <= 6.5e+148)
		tmp = abs((z / (y / x)));
	else
		tmp = abs((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, -3.1e-42], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 6.5e-13], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 6.5e+148], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.1000000000000003e-42

   1. Initial program 91.9%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in x around inf 85.5%

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

      1. associate-/l* 95.8%

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

      2. sub-neg 95.8%

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

      3. metadata-eval 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in z around inf 46.7%

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

      1. associate-*r/ 61.6%

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

   9. Simplified 61.6%

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

   if -3.1000000000000003e-42 < x < 6.49999999999999957e-13

   1. Initial program 92.1%

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

   2. Taylor expanded in x around 0 77.9%

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

   if 6.49999999999999957e-13 < x < 6.49999999999999947e148

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf 61.0%

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

      1. associate-*r/ 61.0%

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

      2. mul-1-neg 61.0%

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

      3. distribute-rgt-neg-out 61.0%

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

      4. associate-*r/ 63.5%

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

      5. distribute-frac-neg 63.5%

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

      6. mul-1-neg 63.5%

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

      7. metadata-eval 63.5%

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

      8. times-frac 63.5%

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

      9. *-lft-identity 63.5%

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

      10. neg-mul-1 63.5%

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

   4. Simplified 63.5%

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

      1. clear-num 63.4%

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

      2. un-div-inv 63.7%

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

      3. add-sqr-sqrt 34.0%

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

      4. sqrt-unprod 47.2%

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

      5. sqr-neg 47.2%

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

      6. sqrt-unprod 29.5%

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

      7. add-sqr-sqrt 63.7%

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

   6. Applied egg-rr 63.7%

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

   if 6.49999999999999947e148 < x

   1. Initial program 81.8%

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

      1. clear-num 81.5%

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

      2. associate-*l/ 87.8%

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

      3. frac-sub 78.4%

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

      4. *-un-lft-identity 78.4%

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

   3. Applied egg-rr 78.4%

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

   4. Taylor expanded in x around inf 70.5%

      \[\leadsto \left|\color{blue}{\frac{\left(y - y \cdot z\right) \cdot x}{{y}^{2}}}\right| \]
   5. Step-by-step derivation

      1. associate-/l* 87.8%

         \[\leadsto \left|\color{blue}{\frac{y - y \cdot z}{\frac{{y}^{2}}{x}}}\right| \]

      2. unpow2 87.8%

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

   6. Simplified 87.8%

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

   7. Taylor expanded in z around 0 80.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-42}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+148}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 7: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-41} \lor \neg \left(x \leq 0.6\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.5e-41) (not (<= x 0.6)))
   (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 <= -6.5e-41) || !(x <= 0.6)) {
		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 <= (-6.5d-41)) .or. (.not. (x <= 0.6d0))) 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 <= -6.5e-41) || !(x <= 0.6)) {
		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 <= -6.5e-41) or not (x <= 0.6):
		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 <= -6.5e-41) || !(x <= 0.6))
		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 <= -6.5e-41) || ~((x <= 0.6)))
		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, -6.5e-41], N[Not[LessEqual[x, 0.6]], $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 < -6.5000000000000004e-41 or 0.599999999999999978 < x

   1. Initial program 90.6%

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

   2. Taylor expanded in x around inf 96.7%

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

      1. *-commutative 96.7%

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

      2. sub-neg 96.7%

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

      3. mul-1-neg 96.7%

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

      4. distribute-lft-in 90.6%

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

      5. associate-*r/ 90.8%

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

      6. *-rgt-identity 90.8%

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

      7. mul-1-neg 90.8%

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

      8. distribute-rgt-neg-in 90.8%

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

      9. unsub-neg 90.8%

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

      10. *-lft-identity 90.8%

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

      11. associate-/l* 90.7%

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

      12. *-commutative 90.7%

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

      13. associate-/r/ 88.9%

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

      14. div-sub 97.4%

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

      15. associate-/r/ 96.8%

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

   4. Simplified 96.8%

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

   if -6.5000000000000004e-41 < x < 0.599999999999999978

   1. Initial program 92.2%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 77.6%

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

      1. associate-*r/ 77.6%

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

      2. distribute-lft-in 77.6%

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

      3. metadata-eval 77.6%

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

      4. neg-mul-1 77.6%

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

      5. sub-neg 77.6%

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

   6. Simplified 77.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-41} \lor \neg \left(x \leq 0.6\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 8: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-44} \lor \neg \left(x \leq 1.05 \cdot 10^{-6}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.4e-44) (not (<= x 1.05e-6)))
   (fabs (* x (/ (- 1.0 z) y)))
   (fabs (+ (/ 4.0 y) (/ x y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.4e-44) || !(x <= 1.05e-6)) {
		tmp = fabs((x * ((1.0 - z) / y)));
	} else {
		tmp = fabs(((4.0 / y) + (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 <= (-3.4d-44)) .or. (.not. (x <= 1.05d-6))) then
        tmp = abs((x * ((1.0d0 - z) / y)))
    else
        tmp = abs(((4.0d0 / y) + (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 <= -3.4e-44) || !(x <= 1.05e-6)) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} else {
		tmp = Math.abs(((4.0 / y) + (x / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -3.4e-44) or not (x <= 1.05e-6):
		tmp = math.fabs((x * ((1.0 - z) / y)))
	else:
		tmp = math.fabs(((4.0 / y) + (x / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.4e-44) || !(x <= 1.05e-6))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	else
		tmp = abs(Float64(Float64(4.0 / y) + Float64(x / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.4e-44) || ~((x <= 1.05e-6)))
		tmp = abs((x * ((1.0 - z) / y)));
	else
		tmp = abs(((4.0 / y) + (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, -3.4e-44], N[Not[LessEqual[x, 1.05e-6]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 / y), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000016e-44 or 1.0499999999999999e-6 < x

   1. Initial program 90.6%

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

   2. Taylor expanded in x around inf 96.7%

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

      1. *-commutative 96.7%

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

      2. sub-neg 96.7%

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

      3. mul-1-neg 96.7%

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

      4. distribute-lft-in 90.6%

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

      5. associate-*r/ 90.8%

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

      6. *-rgt-identity 90.8%

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

      7. mul-1-neg 90.8%

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

      8. distribute-rgt-neg-in 90.8%

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

      9. unsub-neg 90.8%

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

      10. *-lft-identity 90.8%

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

      11. associate-/l* 90.7%

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

      12. *-commutative 90.7%

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

      13. associate-/r/ 88.9%

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

      14. div-sub 97.4%

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

      15. associate-/r/ 96.8%

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

   4. Simplified 96.8%

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

   if -3.40000000000000016e-44 < x < 1.0499999999999999e-6

   1. Initial program 92.2%

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

   2. Taylor expanded in z around 0 77.7%

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

      1. associate-*r/ 77.7%

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

      2. metadata-eval 77.7%

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

   4. Simplified 77.7%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-44} \lor \neg \left(x \leq 1.05 \cdot 10^{-6}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + \frac{x}{y}\right|\\ \end{array} \]

Alternative 9: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-41} \lor \neg \left(x \leq 1.8 \cdot 10^{-12}\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.8e-41) (not (<= x 1.8e-12)))
   (fabs (/ (+ z -1.0) (/ y x)))
   (fabs (/ (- -4.0 x) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.8e-41) || !(x <= 1.8e-12)) {
		tmp = fabs(((z + -1.0) / (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 <= (-1.8d-41)) .or. (.not. (x <= 1.8d-12))) then
        tmp = abs(((z + (-1.0d0)) / (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 <= -1.8e-41) || !(x <= 1.8e-12)) {
		tmp = Math.abs(((z + -1.0) / (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 <= -1.8e-41) or not (x <= 1.8e-12):
		tmp = math.fabs(((z + -1.0) / (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 <= -1.8e-41) || !(x <= 1.8e-12))
		tmp = abs(Float64(Float64(z + -1.0) / 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 <= -1.8e-41) || ~((x <= 1.8e-12)))
		tmp = abs(((z + -1.0) / (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, -1.8e-41], N[Not[LessEqual[x, 1.8e-12]], $MachinePrecision]], N[Abs[N[(N[(z + -1.0), $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 < -1.8e-41 or 1.8e-12 < x

   1. Initial program 90.7%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in x around inf 90.5%

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

      1. associate-/l* 96.7%

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

      2. sub-neg 96.7%

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

      3. metadata-eval 96.7%

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

   6. Simplified 96.7%

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

   if -1.8e-41 < x < 1.8e-12

   1. Initial program 92.1%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. distribute-lft-in 78.0%

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

      3. metadata-eval 78.0%

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

      4. neg-mul-1 78.0%

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

      5. sub-neg 78.0%

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

   6. Simplified 78.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-41} \lor \neg \left(x \leq 1.8 \cdot 10^{-12}\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 10: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+151}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x -1e+151)
   (fabs (* x (/ (- 1.0 z) y)))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e+151) {
		tmp = fabs((x * ((1.0 - z) / y)));
	} 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 <= (-1d+151)) then
        tmp = abs((x * ((1.0d0 - z) / y)))
    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 <= -1e+151) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} 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 <= -1e+151:
		tmp = math.fabs((x * ((1.0 - z) / y)))
	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 <= -1e+151)
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	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 <= -1e+151)
		tmp = abs((x * ((1.0 - z) / y)));
	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[LessEqual[x, -1e+151], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $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 < -1.00000000000000002e151

   1. Initial program 92.4%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. mul-1-neg 99.8%

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

      4. distribute-lft-in 92.4%

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

      5. associate-*r/ 92.5%

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

      6. *-rgt-identity 92.5%

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

      7. mul-1-neg 92.5%

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

      8. distribute-rgt-neg-in 92.5%

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

      9. unsub-neg 92.5%

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

      10. *-lft-identity 92.5%

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

      11. associate-/l* 92.5%

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

      12. *-commutative 92.5%

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

      13. associate-/r/ 92.4%

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

      14. div-sub 99.8%

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

      15. associate-/r/ 99.8%

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

   4. Simplified 99.8%

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

   if -1.00000000000000002e151 < x

   1. Initial program 91.2%

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

      1. associate-*l/ 96.4%

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

      2. sub-div 99.0%

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

   3. Applied egg-rr 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+151}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 11: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+109} \lor \neg \left(z \leq 5.4 \cdot 10^{+82}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.5e+109) (not (<= z 5.4e+82)))
   (fabs (* x (/ z y)))
   (fabs (/ (- -4.0 x) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.5e+109) || !(z <= 5.4e+82)) {
		tmp = fabs((x * (z / y)));
	} else {
		tmp = fabs(((-4.0 - x) / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-3.5d+109)) .or. (.not. (z <= 5.4d+82))) then
        tmp = abs((x * (z / y)))
    else
        tmp = abs((((-4.0d0) - x) / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.5e+109) || !(z <= 5.4e+82)) {
		tmp = Math.abs((x * (z / y)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (z <= -3.5e+109) or not (z <= 5.4e+82):
		tmp = math.fabs((x * (z / y)))
	else:
		tmp = math.fabs(((-4.0 - x) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.5e+109) || !(z <= 5.4e+82))
		tmp = abs(Float64(x * Float64(z / y)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.5e+109) || ~((z <= 5.4e+82)))
		tmp = abs((x * (z / y)));
	else
		tmp = abs(((-4.0 - x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[z, -3.5e+109], N[Not[LessEqual[z, 5.4e+82]], $MachinePrecision]], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.49999999999999983e109 or 5.3999999999999999e82 < z

   1. Initial program 83.9%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in z around inf 74.6%

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

      1. associate-*l/ 77.8%

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

      2. *-commutative 77.8%

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

   6. Simplified 77.8%

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

   if -3.49999999999999983e109 < z < 5.3999999999999999e82

   1. Initial program 95.6%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in z around 0 89.6%

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

      1. associate-*r/ 89.6%

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

      2. distribute-lft-in 89.6%

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

      3. metadata-eval 89.6%

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

      4. neg-mul-1 89.6%

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

      5. sub-neg 89.6%

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

   6. Simplified 89.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+109} \lor \neg \left(z \leq 5.4 \cdot 10^{+82}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 12: 69.5% 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 90.2%

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

      1. clear-num 90.0%

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

      2. associate-*l/ 86.4%

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

      3. frac-sub 62.4%

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

      4. *-un-lft-identity 62.4%

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

   3. Applied egg-rr 62.4%

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

   4. Taylor expanded in x around inf 55.3%

      \[\leadsto \left|\color{blue}{\frac{\left(y - y \cdot z\right) \cdot x}{{y}^{2}}}\right| \]
   5. Step-by-step derivation

      1. associate-/l* 67.3%

         \[\leadsto \left|\color{blue}{\frac{y - y \cdot z}{\frac{{y}^{2}}{x}}}\right| \]

      2. unpow2 67.3%

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

   6. Simplified 67.3%

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

   7. Taylor expanded in z around 0 59.2%

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

   if -10.199999999999999 < x < 4

   1. Initial program 92.5%

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

   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 67.3%

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

Alternative 13: 40.3% 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 91.4%

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

2. Taylor expanded in x around 0 41.1%

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

3. Final simplification 41.1%

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

Reproduce

herbie shell --seed 2023200 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))