fabs fraction 1

Percentage Accurate: 91.3% → 99.8%

Time: 7.1s

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: 91.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-29)
   (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 <= 2e-29) {
		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 <= 2e-29)
		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, 2e-29], 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 < 1.99999999999999989e-29

   1. Initial program 89.2%

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

      1. associate-*l/ 92.2%

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

      2. associate-*r/ 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in x around 0 90.5%

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

      1. sub-neg 90.5%

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

      2. +-commutative 90.5%

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

      3. distribute-lft-in 85.8%

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

      4. associate-+r+ 85.8%

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

      5. distribute-rgt-in 85.8%

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

      6. associate-*l/ 85.9%

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

      7. *-lft-identity 85.9%

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

      8. +-commutative 85.9%

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

      9. distribute-rgt-neg-out 85.9%

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

      10. sub-neg 85.9%

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

      11. associate-*r/ 92.2%

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

      12. div-sub 96.9%

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

   6. Simplified 96.9%

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

   if 1.99999999999999989e-29 < y

   1. Initial program 98.2%

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

   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 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{+159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.24 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))) (t_1 (fabs (/ x y))))
   (if (<= x -5.2e+159)
     t_0
     (if (<= x -3.2e+34)
       t_1
       (if (<= x -1.24e-76)
         t_0
         (if (<= x 3.05e-13) (fabs (/ 4.0 y)) (if (<= x 5.1e+85) t_0 t_1)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -5.2e+159) {
		tmp = t_0;
	} else if (x <= -3.2e+34) {
		tmp = t_1;
	} else if (x <= -1.24e-76) {
		tmp = t_0;
	} else if (x <= 3.05e-13) {
		tmp = fabs((4.0 / y));
	} else if (x <= 5.1e+85) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    t_1 = abs((x / y))
    if (x <= (-5.2d+159)) then
        tmp = t_0
    else if (x <= (-3.2d+34)) then
        tmp = t_1
    else if (x <= (-1.24d-76)) then
        tmp = t_0
    else if (x <= 3.05d-13) then
        tmp = abs((4.0d0 / y))
    else if (x <= 5.1d+85) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z * (x / y)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -5.2e+159) {
		tmp = t_0;
	} else if (x <= -3.2e+34) {
		tmp = t_1;
	} else if (x <= -1.24e-76) {
		tmp = t_0;
	} else if (x <= 3.05e-13) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 5.1e+85) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -5.2e+159:
		tmp = t_0
	elif x <= -3.2e+34:
		tmp = t_1
	elif x <= -1.24e-76:
		tmp = t_0
	elif x <= 3.05e-13:
		tmp = math.fabs((4.0 / y))
	elif x <= 5.1e+85:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -5.2e+159)
		tmp = t_0;
	elseif (x <= -3.2e+34)
		tmp = t_1;
	elseif (x <= -1.24e-76)
		tmp = t_0;
	elseif (x <= 3.05e-13)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 5.1e+85)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -5.2e+159)
		tmp = t_0;
	elseif (x <= -3.2e+34)
		tmp = t_1;
	elseif (x <= -1.24e-76)
		tmp = t_0;
	elseif (x <= 3.05e-13)
		tmp = abs((4.0 / y));
	elseif (x <= 5.1e+85)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -5.2e+159], t$95$0, If[LessEqual[x, -3.2e+34], t$95$1, If[LessEqual[x, -1.24e-76], t$95$0, If[LessEqual[x, 3.05e-13], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5.1e+85], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.2000000000000001e159 or -3.1999999999999998e34 < x < -1.24000000000000003e-76 or 3.0500000000000001e-13 < x < 5.0999999999999998e85

   1. Initial program 87.3%

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

      1. associate-*l/ 87.9%

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

      2. associate-*r/ 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in z around inf 64.3%

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

      1. associate-*r/ 64.3%

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

      2. neg-mul-1 64.3%

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

      3. distribute-lft-neg-in 64.3%

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

      4. associate-*l/ 75.8%

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

      5. *-commutative 75.8%

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

   6. Simplified 75.8%

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

      1. *-commutative 75.8%

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

      2. add-sqr-sqrt 28.1%

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

      3. sqrt-unprod 70.5%

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

      4. distribute-frac-neg 70.5%

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

      5. distribute-frac-neg 70.5%

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

      6. sqr-neg 70.5%

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

      7. sqrt-unprod 47.4%

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

      8. add-sqr-sqrt 75.8%

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

      9. clear-num 75.1%

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

      10. associate-*l/ 73.9%

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

      11. *-un-lft-identity 73.9%

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

   8. Applied egg-rr 73.9%

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

      1. clear-num 73.9%

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

      2. associate-/r/ 75.1%

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

      3. clear-num 75.8%

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

   10. Applied egg-rr 75.8%

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

   if -5.2000000000000001e159 < x < -3.1999999999999998e34 or 5.0999999999999998e85 < x

   1. Initial program 87.1%

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

      1. associate-*l/ 83.6%

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

      2. associate-*r/ 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in x around inf 94.2%

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

   5. Taylor expanded in z around 0 72.7%

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

   if -1.24000000000000003e-76 < x < 3.0500000000000001e-13

   1. Initial program 97.0%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in x around 0 79.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+159}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.24 \cdot 10^{-76}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+85}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+86}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))) (t_1 (fabs (/ x y))))
   (if (<= x -5.6e+159)
     t_0
     (if (<= x -2.6e+34)
       t_1
       (if (<= x -2.6e-64)
         t_0
         (if (<= x 2.9e-13)
           (fabs (/ 4.0 y))
           (if (<= x 1.35e+86) (fabs (/ x (/ y z))) t_1)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -5.6e+159) {
		tmp = t_0;
	} else if (x <= -2.6e+34) {
		tmp = t_1;
	} else if (x <= -2.6e-64) {
		tmp = t_0;
	} else if (x <= 2.9e-13) {
		tmp = fabs((4.0 / y));
	} else if (x <= 1.35e+86) {
		tmp = fabs((x / (y / z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    t_1 = abs((x / y))
    if (x <= (-5.6d+159)) then
        tmp = t_0
    else if (x <= (-2.6d+34)) then
        tmp = t_1
    else if (x <= (-2.6d-64)) then
        tmp = t_0
    else if (x <= 2.9d-13) then
        tmp = abs((4.0d0 / y))
    else if (x <= 1.35d+86) then
        tmp = abs((x / (y / z)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z * (x / y)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -5.6e+159) {
		tmp = t_0;
	} else if (x <= -2.6e+34) {
		tmp = t_1;
	} else if (x <= -2.6e-64) {
		tmp = t_0;
	} else if (x <= 2.9e-13) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 1.35e+86) {
		tmp = Math.abs((x / (y / z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -5.6e+159:
		tmp = t_0
	elif x <= -2.6e+34:
		tmp = t_1
	elif x <= -2.6e-64:
		tmp = t_0
	elif x <= 2.9e-13:
		tmp = math.fabs((4.0 / y))
	elif x <= 1.35e+86:
		tmp = math.fabs((x / (y / z)))
	else:
		tmp = t_1
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -5.6e+159)
		tmp = t_0;
	elseif (x <= -2.6e+34)
		tmp = t_1;
	elseif (x <= -2.6e-64)
		tmp = t_0;
	elseif (x <= 2.9e-13)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 1.35e+86)
		tmp = abs(Float64(x / Float64(y / z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -5.6e+159)
		tmp = t_0;
	elseif (x <= -2.6e+34)
		tmp = t_1;
	elseif (x <= -2.6e-64)
		tmp = t_0;
	elseif (x <= 2.9e-13)
		tmp = abs((4.0 / y));
	elseif (x <= 1.35e+86)
		tmp = abs((x / (y / z)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -5.6e+159], t$95$0, If[LessEqual[x, -2.6e+34], t$95$1, If[LessEqual[x, -2.6e-64], t$95$0, If[LessEqual[x, 2.9e-13], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.35e+86], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.6000000000000002e159 or -2.59999999999999997e34 < x < -2.6e-64

   1. Initial program 83.8%

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

      1. associate-*l/ 84.6%

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

      2. associate-*r/ 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in z around inf 59.3%

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

      1. associate-*r/ 59.3%

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

      2. neg-mul-1 59.3%

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

      3. distribute-lft-neg-in 59.3%

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

      4. associate-*l/ 74.0%

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

      5. *-commutative 74.0%

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

   6. Simplified 74.0%

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

      1. *-commutative 74.0%

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

      2. add-sqr-sqrt 28.7%

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

      3. sqrt-unprod 72.2%

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

      4. distribute-frac-neg 72.2%

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

      5. distribute-frac-neg 72.2%

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

      6. sqr-neg 72.2%

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

      7. sqrt-unprod 45.1%

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

      8. add-sqr-sqrt 74.0%

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

      9. clear-num 73.3%

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

      10. associate-*l/ 71.7%

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

      11. *-un-lft-identity 71.7%

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

   8. Applied egg-rr 71.7%

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

      1. clear-num 71.6%

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

      2. associate-/r/ 73.3%

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

      3. clear-num 74.0%

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

   10. Applied egg-rr 74.0%

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

   if -5.6000000000000002e159 < x < -2.59999999999999997e34 or 1.35000000000000009e86 < x

   1. Initial program 87.1%

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

      1. associate-*l/ 83.6%

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

      2. associate-*r/ 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in x around inf 94.2%

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

   5. Taylor expanded in z around 0 72.7%

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

   if -2.6e-64 < x < 2.8999999999999998e-13

   1. Initial program 97.0%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in x around 0 79.2%

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

   if 2.8999999999999998e-13 < x < 1.35000000000000009e86

   1. Initial program 99.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 82.2%

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

      1. associate-*r/ 82.2%

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

      2. neg-mul-1 82.2%

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

      3. distribute-lft-neg-in 82.2%

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

      4. associate-*l/ 82.1%

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

      5. *-commutative 82.1%

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

   6. Simplified 82.1%

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

      1. *-commutative 82.1%

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

      2. add-sqr-sqrt 25.9%

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

      3. sqrt-unprod 64.7%

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

      4. distribute-frac-neg 64.7%

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

      5. distribute-frac-neg 64.7%

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

      6. sqr-neg 64.7%

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

      7. sqrt-unprod 55.9%

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

      8. add-sqr-sqrt 82.1%

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

      9. associate-/r/ 82.2%

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

   8. Applied egg-rr 82.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+159}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-64}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+86}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -84000000000000 \lor \neg \left(x \leq 2.2 \cdot 10^{+127}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -84000000000000.0) (not (<= x 2.2e+127)))
   (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 <= -84000000000000.0) || !(x <= 2.2e+127)) {
		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 <= (-84000000000000.0d0)) .or. (.not. (x <= 2.2d+127))) 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 <= -84000000000000.0) || !(x <= 2.2e+127)) {
		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 <= -84000000000000.0) or not (x <= 2.2e+127):
		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 <= -84000000000000.0) || !(x <= 2.2e+127))
		tmp = abs(Float64(x / Float64(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 <= -84000000000000.0) || ~((x <= 2.2e+127)))
		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, -84000000000000.0], N[Not[LessEqual[x, 2.2e+127]], $MachinePrecision]], N[Abs[N[(x / N[(y / N[(1.0 - z), $MachinePrecision]), $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 < -8.4e13 or 2.2000000000000002e127 < x

   1. Initial program 82.2%

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

      1. associate-*l/ 80.0%

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

      2. associate-*r/ 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-lft-in 90.9%

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

      4. associate-+r+ 90.9%

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

      5. distribute-rgt-in 90.9%

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

      6. associate-*l/ 91.1%

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

      7. *-lft-identity 91.1%

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

      8. +-commutative 91.1%

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

      9. distribute-rgt-neg-out 91.1%

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

      10. sub-neg 91.1%

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

      11. associate-*r/ 80.0%

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

      12. div-sub 88.9%

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

   6. Simplified 88.9%

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

   7. Taylor expanded in x around inf 88.8%

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

      1. associate-/l* 99.9%

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

   9. Simplified 99.9%

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

   if -8.4e13 < x < 2.2000000000000002e127

   1. Initial program 97.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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. associate-*r/ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 88.4%

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

      1. sub-neg 88.4%

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

      2. +-commutative 88.4%

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

      3. distribute-lft-in 88.4%

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

      4. associate-+r+ 88.4%

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

      5. distribute-rgt-in 88.4%

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

      6. associate-*l/ 88.4%

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

      7. *-lft-identity 88.4%

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

      8. +-commutative 88.4%

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

      9. distribute-rgt-neg-out 88.4%

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

      10. sub-neg 88.4%

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

      11. associate-*r/ 99.9%

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

      12. div-sub 99.9%

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

   6. Simplified 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -84000000000000 \lor \neg \left(x \leq 2.2 \cdot 10^{+127}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 5: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.45 \cdot 10^{-18}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 3.45e-18)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ (+ x 4.0) y) (* x (/ z y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.45e-18) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (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 (y <= 3.45d-18) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((((x + 4.0d0) / y) - (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 (y <= 3.45e-18) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (x * (z / y))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 3.45e-18:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((((x + 4.0) / y) - (x * (z / y))))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 3.45e-18)
		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(z / y))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.45e-18)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((((x + 4.0) / y) - (x * (z / y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 3.45e-18], 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[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4500000000000001e-18

   1. Initial program 89.3%

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

      1. associate-*l/ 92.3%

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

      2. associate-*r/ 86.0%

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

   3. Simplified 86.0%

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

   4. Taylor expanded in x around 0 90.6%

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

      1. sub-neg 90.6%

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

      2. +-commutative 90.6%

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

      3. distribute-lft-in 86.0%

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

      4. associate-+r+ 86.0%

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

      5. distribute-rgt-in 86.0%

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

      6. associate-*l/ 86.0%

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

      7. *-lft-identity 86.0%

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

      8. +-commutative 86.0%

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

      9. distribute-rgt-neg-out 86.0%

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

      10. sub-neg 86.0%

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

      11. associate-*r/ 92.3%

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

      12. div-sub 97.0%

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

   6. Simplified 97.0%

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

   if 3.4500000000000001e-18 < y

   1. Initial program 98.2%

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

      1. associate-*l/ 91.0%

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

      2. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.45 \cdot 10^{-18}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 6: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-38} \lor \neg \left(x \leq 3.5 \cdot 10^{-13}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.8e-38) (not (<= x 3.5e-13)))
   (fabs (/ x (/ y (- 1.0 z))))
   (fabs (/ (+ x 4.0) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.8e-38) || !(x <= 3.5e-13)) {
		tmp = fabs((x / (y / (1.0 - z))));
	} else {
		tmp = fabs(((x + 4.0) / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-4.8d-38)) .or. (.not. (x <= 3.5d-13))) then
        tmp = abs((x / (y / (1.0d0 - z))))
    else
        tmp = abs(((x + 4.0d0) / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.8e-38) || !(x <= 3.5e-13)) {
		tmp = Math.abs((x / (y / (1.0 - z))));
	} else {
		tmp = Math.abs(((x + 4.0) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -4.8e-38) or not (x <= 3.5e-13):
		tmp = math.fabs((x / (y / (1.0 - z))))
	else:
		tmp = math.fabs(((x + 4.0) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.8e-38) || !(x <= 3.5e-13))
		tmp = abs(Float64(x / Float64(y / Float64(1.0 - z))));
	else
		tmp = abs(Float64(Float64(x + 4.0) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.8e-38) || ~((x <= 3.5e-13)))
		tmp = abs((x / (y / (1.0 - z))));
	else
		tmp = abs(((x + 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, -4.8e-38], N[Not[LessEqual[x, 3.5e-13]], $MachinePrecision]], N[Abs[N[(x / N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.80000000000000044e-38 or 3.5000000000000002e-13 < x

   1. Initial program 86.9%

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

      1. associate-*l/ 85.4%

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

      2. associate-*r/ 92.2%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in x around 0 98.5%

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

      1. sub-neg 98.5%

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

      2. +-commutative 98.5%

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

      3. distribute-lft-in 92.0%

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

      4. associate-+r+ 92.0%

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

      5. distribute-rgt-in 92.1%

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

      6. associate-*l/ 92.2%

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

      7. *-lft-identity 92.2%

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

      8. +-commutative 92.2%

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

      9. distribute-rgt-neg-out 92.2%

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

      10. sub-neg 92.2%

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

      11. associate-*r/ 85.4%

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

      12. div-sub 91.9%

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

   6. Simplified 91.9%

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

   7. Taylor expanded in x around inf 88.6%

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

      1. associate-/l* 95.9%

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

   9. Simplified 95.9%

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

   if -4.80000000000000044e-38 < x < 3.5000000000000002e-13

   1. Initial program 97.1%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in z around 0 78.4%

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

      1. +-commutative 78.4%

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

      2. *-rgt-identity 78.4%

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

      3. associate-*r/ 78.4%

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

      4. distribute-rgt-in 78.3%

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

      5. associate-*l/ 78.4%

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

      6. *-lft-identity 78.4%

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

   6. Simplified 78.4%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-38} \lor \neg \left(x \leq 3.5 \cdot 10^{-13}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \]

Alternative 7: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+81}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+41}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -7e+81)
   (fabs (/ z (/ y x)))
   (if (<= z 6.8e+41) (fabs (/ (+ x 4.0) y)) (fabs (* z (/ x y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -7e+81) {
		tmp = fabs((z / (y / x)));
	} else if (z <= 6.8e+41) {
		tmp = fabs(((x + 4.0) / y));
	} else {
		tmp = fabs((z * (x / y)));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-7d+81)) then
        tmp = abs((z / (y / x)))
    else if (z <= 6.8d+41) then
        tmp = abs(((x + 4.0d0) / y))
    else
        tmp = abs((z * (x / y)))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7e+81) {
		tmp = Math.abs((z / (y / x)));
	} else if (z <= 6.8e+41) {
		tmp = Math.abs(((x + 4.0) / y));
	} else {
		tmp = Math.abs((z * (x / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -7e+81:
		tmp = math.fabs((z / (y / x)))
	elif z <= 6.8e+41:
		tmp = math.fabs(((x + 4.0) / y))
	else:
		tmp = math.fabs((z * (x / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -7e+81)
		tmp = abs(Float64(z / Float64(y / x)));
	elseif (z <= 6.8e+41)
		tmp = abs(Float64(Float64(x + 4.0) / y));
	else
		tmp = abs(Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7e+81)
		tmp = abs((z / (y / x)));
	elseif (z <= 6.8e+41)
		tmp = abs(((x + 4.0) / y));
	else
		tmp = abs((z * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -7e+81], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 6.8e+41], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.0000000000000001e81

   1. Initial program 96.1%

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

      1. associate-*l/ 90.5%

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

      2. associate-*r/ 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in z around inf 77.7%

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

      1. associate-*r/ 77.7%

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

      2. neg-mul-1 77.7%

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

      3. distribute-lft-neg-in 77.7%

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

      4. associate-*l/ 83.2%

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

      5. *-commutative 83.2%

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

   6. Simplified 83.2%

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

      1. *-commutative 83.2%

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

      2. add-sqr-sqrt 22.7%

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

      3. sqrt-unprod 74.8%

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

      4. distribute-frac-neg 74.8%

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

      5. distribute-frac-neg 74.8%

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

      6. sqr-neg 74.8%

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

      7. sqrt-unprod 60.4%

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

      8. add-sqr-sqrt 83.2%

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

      9. clear-num 83.1%

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

      10. associate-*l/ 83.2%

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

      11. *-un-lft-identity 83.2%

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

   8. Applied egg-rr 83.2%

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

   if -7.0000000000000001e81 < z < 6.79999999999999996e41

   1. Initial program 91.4%

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

      1. associate-*l/ 97.8%

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

      2. associate-*r/ 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in z around 0 93.6%

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

      1. +-commutative 93.6%

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

      2. *-rgt-identity 93.6%

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

      3. associate-*r/ 93.5%

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

      4. distribute-rgt-in 93.5%

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

      5. associate-*l/ 93.6%

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

      6. *-lft-identity 93.6%

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

   6. Simplified 93.6%

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

   if 6.79999999999999996e41 < z

   1. Initial program 88.0%

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

      1. associate-*l/ 79.7%

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

      2. associate-*r/ 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in z around inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. neg-mul-1 65.1%

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

      3. distribute-lft-neg-in 65.1%

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

      4. associate-*l/ 73.3%

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

      5. *-commutative 73.3%

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

   6. Simplified 73.3%

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

      1. *-commutative 73.3%

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

      2. add-sqr-sqrt 29.5%

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

      3. sqrt-unprod 61.7%

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

      4. distribute-frac-neg 61.7%

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

      5. distribute-frac-neg 61.7%

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

      6. sqr-neg 61.7%

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

      7. sqrt-unprod 43.9%

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

      8. add-sqr-sqrt 73.3%

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

      9. clear-num 72.4%

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

      10. associate-*l/ 72.5%

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

      11. *-un-lft-identity 72.5%

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

   8. Applied egg-rr 72.5%

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

      1. clear-num 72.5%

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

      2. associate-/r/ 72.4%

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

      3. clear-num 73.3%

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

   10. Applied egg-rr 73.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+81}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+41}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 8: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = fabs((x / y));
	} else {
		tmp = fabs((4.0 / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-10.5d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = Math.abs((x / y));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -10.5) or not (x <= 4.0):
		tmp = math.fabs((x / y))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -10.5) || !(x <= 4.0))
		tmp = abs(Float64(x / y));
	else
		tmp = abs(Float64(4.0 / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -10.5) || ~((x <= 4.0)))
		tmp = abs((x / y));
	else
		tmp = abs((4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[x, -10.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 85.5%

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

      1. associate-*l/ 83.7%

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

      2. associate-*r/ 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in x around inf 92.3%

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

   5. Taylor expanded in z around 0 61.8%

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

   if -10.5 < x < 4

   1. Initial program 97.3%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in x around 0 71.6%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 9: 39.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. associate-*l/ 92.0%

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

   2. associate-*r/ 89.5%

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

3. Simplified 89.5%

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

4. Taylor expanded in x around 0 39.1%

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

5. Final simplification 39.1%

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

Reproduce

herbie shell --seed 2023268 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))