fabs fraction 1

Percentage Accurate: 91.7% → 99.5%

Time: 9.1s

Alternatives: 11

Speedup: 1.0×

• 21.0% 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: 91.7% 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

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{-87}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1e-87)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1e-87) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 1e-87)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(Float64(-4.0 - x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 1e-87], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000002e-87

   1. Initial program 90.7%

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

      1. associate-*l/ 92.7%

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

      2. sub-div 97.3%

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

   4. Applied egg-rr 97.3%

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

   if 1.00000000000000002e-87 < y

   1. Initial program 93.1%

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

      1. fabs-sub 93.1%

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

      2. associate-*l/ 94.2%

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

      3. associate-*r/ 99.8%

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

      4. fma-neg 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-neg-in 99.8%

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

      8. unsub-neg 99.8%

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

      9. 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%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-87}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ t_1 := \left|\frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{if}\;x \leq -2.95 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+74} \lor \neg \left(x \leq 2.9 \cdot 10^{+113}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (/ z (/ y_m x)))))
   (if (<= x -2.95e+60)
     t_0
     (if (<= x -2.2e+38)
       t_1
       (if (<= x -1.55)
         t_0
         (if (<= x 4.0)
           (fabs (/ 4.0 y_m))
           (if (or (<= x 7.8e+74) (not (<= x 2.9e+113))) t_0 t_1)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((z / (y_m / x)));
	double tmp;
	if (x <= -2.95e+60) {
		tmp = t_0;
	} else if (x <= -2.2e+38) {
		tmp = t_1;
	} else if (x <= -1.55) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 7.8e+74) || !(x <= 2.9e+113)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y_m))
    t_1 = abs((z / (y_m / x)))
    if (x <= (-2.95d+60)) then
        tmp = t_0
    else if (x <= (-2.2d+38)) then
        tmp = t_1
    else if (x <= (-1.55d0)) then
        tmp = t_0
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 7.8d+74) .or. (.not. (x <= 2.9d+113))) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double t_1 = Math.abs((z / (y_m / x)));
	double tmp;
	if (x <= -2.95e+60) {
		tmp = t_0;
	} else if (x <= -2.2e+38) {
		tmp = t_1;
	} else if (x <= -1.55) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 7.8e+74) || !(x <= 2.9e+113)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	t_1 = math.fabs((z / (y_m / x)))
	tmp = 0
	if x <= -2.95e+60:
		tmp = t_0
	elif x <= -2.2e+38:
		tmp = t_1
	elif x <= -1.55:
		tmp = t_0
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 7.8e+74) or not (x <= 2.9e+113):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(z / Float64(y_m / x)))
	tmp = 0.0
	if (x <= -2.95e+60)
		tmp = t_0;
	elseif (x <= -2.2e+38)
		tmp = t_1;
	elseif (x <= -1.55)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 7.8e+74) || !(x <= 2.9e+113))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	t_1 = abs((z / (y_m / x)));
	tmp = 0.0;
	if (x <= -2.95e+60)
		tmp = t_0;
	elseif (x <= -2.2e+38)
		tmp = t_1;
	elseif (x <= -1.55)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 7.8e+74) || ~((x <= 2.9e+113)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.95e+60], t$95$0, If[LessEqual[x, -2.2e+38], t$95$1, If[LessEqual[x, -1.55], t$95$0, If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 7.8e+74], N[Not[LessEqual[x, 2.9e+113]], $MachinePrecision]], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9500000000000001e60 or -2.20000000000000006e38 < x < -1.55000000000000004 or 4 < x < 7.80000000000000015e74 or 2.89999999999999984e113 < x

   1. Initial program 86.3%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in z around 0 74.9%

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

      1. distribute-lft-in 74.9%

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

      2. metadata-eval 74.9%

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

      3. neg-mul-1 74.9%

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

      4. sub-neg 74.9%

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

   7. Simplified 74.9%

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

   8. Taylor expanded in x around inf 74.3%

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

   if -2.9500000000000001e60 < x < -2.20000000000000006e38 or 7.80000000000000015e74 < x < 2.89999999999999984e113

   1. Initial program 99.8%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in z around inf 60.7%

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

      1. associate-*r/ 60.7%

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

      2. neg-mul-1 60.7%

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

      3. distribute-rgt-neg-in 60.7%

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

   7. Simplified 60.7%

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

      1. distribute-rgt-neg-out 60.7%

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

      2. *-commutative 60.7%

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

      3. distribute-rgt-neg-in 60.7%

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

      4. associate-*l/ 86.4%

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

      5. add-sqr-sqrt 42.9%

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

      6. sqrt-unprod 86.4%

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

      7. sqr-neg 86.4%

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

      8. sqrt-unprod 43.4%

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

      9. add-sqr-sqrt 86.4%

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

   9. Applied egg-rr 86.4%

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

   10. Taylor expanded in z around 0 60.7%

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

       1. associate-*r/ 86.4%

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

       2. *-commutative 86.4%

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

       3. associate-/r/ 93.0%

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

   12. Simplified 93.0%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 94.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 74.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{+60}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{+38}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq -1.55:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+74} \lor \neg \left(x \leq 2.9 \cdot 10^{+113}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+70}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;z \leq -1.42 \cdot 10^{+60}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;z \leq -30500000000:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+135}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -2.75e+70)
   (fabs (* x (/ z y_m)))
   (if (<= z -1.42e+60)
     (fabs (/ 4.0 y_m))
     (if (<= z -30500000000.0)
       (fabs (/ z (/ y_m x)))
       (if (<= z 4.1e+135)
         (fabs (/ (- -4.0 x) y_m))
         (fabs (/ x (/ y_m z))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -2.75e+70) {
		tmp = fabs((x * (z / y_m)));
	} else if (z <= -1.42e+60) {
		tmp = fabs((4.0 / y_m));
	} else if (z <= -30500000000.0) {
		tmp = fabs((z / (y_m / x)));
	} else if (z <= 4.1e+135) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((x / (y_m / z)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.75d+70)) then
        tmp = abs((x * (z / y_m)))
    else if (z <= (-1.42d+60)) then
        tmp = abs((4.0d0 / y_m))
    else if (z <= (-30500000000.0d0)) then
        tmp = abs((z / (y_m / x)))
    else if (z <= 4.1d+135) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((x / (y_m / z)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -2.75e+70) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (z <= -1.42e+60) {
		tmp = Math.abs((4.0 / y_m));
	} else if (z <= -30500000000.0) {
		tmp = Math.abs((z / (y_m / x)));
	} else if (z <= 4.1e+135) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x / (y_m / z)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -2.75e+70:
		tmp = math.fabs((x * (z / y_m)))
	elif z <= -1.42e+60:
		tmp = math.fabs((4.0 / y_m))
	elif z <= -30500000000.0:
		tmp = math.fabs((z / (y_m / x)))
	elif z <= 4.1e+135:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x / (y_m / z)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -2.75e+70)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (z <= -1.42e+60)
		tmp = abs(Float64(4.0 / y_m));
	elseif (z <= -30500000000.0)
		tmp = abs(Float64(z / Float64(y_m / x)));
	elseif (z <= 4.1e+135)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x / Float64(y_m / z)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -2.75e+70)
		tmp = abs((x * (z / y_m)));
	elseif (z <= -1.42e+60)
		tmp = abs((4.0 / y_m));
	elseif (z <= -30500000000.0)
		tmp = abs((z / (y_m / x)));
	elseif (z <= 4.1e+135)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x / (y_m / z)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -2.75e+70], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -1.42e+60], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -30500000000.0], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4.1e+135], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.74999999999999993e70

   1. Initial program 94.0%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in z around inf 76.8%

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

      1. associate-*r/ 76.8%

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

      2. neg-mul-1 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

   7. Simplified 76.8%

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

      1. distribute-rgt-neg-out 76.8%

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

      2. *-commutative 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

      4. associate-*l/ 80.6%

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

      5. add-sqr-sqrt 52.5%

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

      6. sqrt-unprod 75.3%

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

      7. sqr-neg 75.3%

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

      8. sqrt-unprod 27.9%

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

      9. add-sqr-sqrt 80.6%

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

   9. Applied egg-rr 80.6%

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

   if -2.74999999999999993e70 < z < -1.42000000000000001e60

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 89.0%

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

   if -1.42000000000000001e60 < z < -3.05e10

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in z around inf 78.2%

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

      1. associate-*r/ 78.2%

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

      2. neg-mul-1 78.2%

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

      3. distribute-rgt-neg-in 78.2%

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

   7. Simplified 78.2%

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

      1. distribute-rgt-neg-out 78.2%

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

      2. *-commutative 78.2%

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

      3. distribute-rgt-neg-in 78.2%

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

      4. associate-*l/ 78.1%

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

      5. add-sqr-sqrt 30.2%

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

      6. sqrt-unprod 61.6%

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

      7. sqr-neg 61.6%

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

      8. sqrt-unprod 47.7%

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

      9. add-sqr-sqrt 78.1%

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

   9. Applied egg-rr 78.1%

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

   10. Taylor expanded in z around 0 78.2%

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

       1. associate-*r/ 78.1%

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

       2. *-commutative 78.1%

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

       3. associate-/r/ 78.2%

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

   12. Simplified 78.2%

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

   if -3.05e10 < z < 4.1e135

   1. Initial program 91.4%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 94.7%

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

      1. +-commutative 94.7%

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

      2. rem-square-sqrt 40.1%

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

      3. fabs-sqr 40.1%

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

      4. rem-square-sqrt 94.7%

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

      5. fabs-neg 94.7%

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

      6. distribute-neg-frac 94.7%

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

      7. distribute-neg-in 94.7%

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

      8. metadata-eval 94.7%

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

      9. +-commutative 94.7%

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

      10. sub-neg 94.7%

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

      11. rem-square-sqrt 54.0%

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

      12. fabs-sqr 54.0%

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

      13. rem-square-sqrt 94.7%

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

   7. Simplified 94.7%

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

   if 4.1e135 < z

   1. Initial program 83.6%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in z around inf 72.1%

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

      1. mul-1-neg 72.1%

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

      2. associate-*r/ 84.3%

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

      3. *-commutative 84.3%

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

      4. distribute-rgt-neg-in 84.3%

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

   7. Simplified 84.3%

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

      1. *-commutative 84.3%

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

      2. clear-num 84.3%

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

      3. un-div-inv 84.9%

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

      4. add-sqr-sqrt 48.0%

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

      5. sqrt-unprod 71.8%

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

      6. sqr-neg 71.8%

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

      7. sqrt-unprod 36.6%

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

      8. add-sqr-sqrt 84.9%

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

   9. Applied egg-rr 84.9%

      \[\leadsto \left|\color{blue}{\frac{x}{\frac{y}{z}}}\right| \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+70}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq -1.42 \cdot 10^{+60}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq -30500000000:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+135}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+70}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;z \leq -27:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+135}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -8.5e+70)
   (fabs (* x (/ z y_m)))
   (if (<= z -3.2e+49)
     (fabs (/ 4.0 y_m))
     (if (<= z -27.0)
       (fabs (* (- 1.0 z) (/ x y_m)))
       (if (<= z 4.1e+135)
         (fabs (/ (- -4.0 x) y_m))
         (fabs (/ x (/ y_m z))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -8.5e+70) {
		tmp = fabs((x * (z / y_m)));
	} else if (z <= -3.2e+49) {
		tmp = fabs((4.0 / y_m));
	} else if (z <= -27.0) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else if (z <= 4.1e+135) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((x / (y_m / z)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-8.5d+70)) then
        tmp = abs((x * (z / y_m)))
    else if (z <= (-3.2d+49)) then
        tmp = abs((4.0d0 / y_m))
    else if (z <= (-27.0d0)) then
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    else if (z <= 4.1d+135) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((x / (y_m / z)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -8.5e+70) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (z <= -3.2e+49) {
		tmp = Math.abs((4.0 / y_m));
	} else if (z <= -27.0) {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	} else if (z <= 4.1e+135) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x / (y_m / z)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -8.5e+70:
		tmp = math.fabs((x * (z / y_m)))
	elif z <= -3.2e+49:
		tmp = math.fabs((4.0 / y_m))
	elif z <= -27.0:
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	elif z <= 4.1e+135:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x / (y_m / z)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -8.5e+70)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (z <= -3.2e+49)
		tmp = abs(Float64(4.0 / y_m));
	elseif (z <= -27.0)
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	elseif (z <= 4.1e+135)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x / Float64(y_m / z)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -8.5e+70)
		tmp = abs((x * (z / y_m)));
	elseif (z <= -3.2e+49)
		tmp = abs((4.0 / y_m));
	elseif (z <= -27.0)
		tmp = abs(((1.0 - z) * (x / y_m)));
	elseif (z <= 4.1e+135)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x / (y_m / z)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -8.5e+70], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -3.2e+49], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -27.0], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4.1e+135], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -8.4999999999999996e70

   1. Initial program 94.0%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in z around inf 76.8%

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

      1. associate-*r/ 76.8%

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

      2. neg-mul-1 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

   7. Simplified 76.8%

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

      1. distribute-rgt-neg-out 76.8%

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

      2. *-commutative 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

      4. associate-*l/ 80.6%

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

      5. add-sqr-sqrt 52.5%

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

      6. sqrt-unprod 75.3%

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

      7. sqr-neg 75.3%

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

      8. sqrt-unprod 27.9%

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

      9. add-sqr-sqrt 80.6%

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

   9. Applied egg-rr 80.6%

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

   if -8.4999999999999996e70 < z < -3.20000000000000014e49

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 89.0%

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

   if -3.20000000000000014e49 < z < -27

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 79.6%

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

      1. associate-*r/ 79.6%

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

      2. *-commutative 79.6%

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

      3. associate-*r* 79.6%

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

      4. sub-neg 79.6%

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

      5. metadata-eval 79.6%

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

      6. distribute-lft-in 79.6%

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

      7. neg-mul-1 79.6%

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

      8. metadata-eval 79.6%

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

      9. +-commutative 79.6%

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

      10. neg-mul-1 79.6%

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

      11. associate-/l* 79.4%

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

      12. neg-mul-1 79.4%

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

      13. unsub-neg 79.4%

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

   7. Simplified 79.4%

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

   if -27 < z < 4.1e135

   1. Initial program 91.2%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 96.2%

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

      1. +-commutative 96.2%

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

      2. rem-square-sqrt 40.4%

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

      3. fabs-sqr 40.4%

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

      4. rem-square-sqrt 96.2%

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

      5. fabs-neg 96.2%

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

      6. distribute-neg-frac 96.2%

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

      7. distribute-neg-in 96.2%

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

      8. metadata-eval 96.2%

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

      9. +-commutative 96.2%

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

      10. sub-neg 96.2%

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

      11. rem-square-sqrt 55.2%

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

      12. fabs-sqr 55.2%

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

      13. rem-square-sqrt 96.2%

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

   7. Simplified 96.2%

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

   if 4.1e135 < z

   1. Initial program 83.6%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in z around inf 72.1%

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

      1. mul-1-neg 72.1%

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

      2. associate-*r/ 84.3%

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

      3. *-commutative 84.3%

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

      4. distribute-rgt-neg-in 84.3%

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

   7. Simplified 84.3%

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

      1. *-commutative 84.3%

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

      2. clear-num 84.3%

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

      3. un-div-inv 84.9%

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

      4. add-sqr-sqrt 48.0%

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

      5. sqrt-unprod 71.8%

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

      6. sqr-neg 71.8%

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

      7. sqrt-unprod 36.6%

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

      8. add-sqr-sqrt 84.9%

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

   9. Applied egg-rr 84.9%

      \[\leadsto \left|\color{blue}{\frac{x}{\frac{y}{z}}}\right| \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+70}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq -27:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+135}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+72}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;z \leq -39:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y\_m}{x}}\right|\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+135}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -1.05e+72)
   (fabs (* x (/ z y_m)))
   (if (<= z -4e+60)
     (fabs (/ 4.0 y_m))
     (if (<= z -39.0)
       (fabs (/ (- 1.0 z) (/ y_m x)))
       (if (<= z 4.1e+135)
         (fabs (/ (- -4.0 x) y_m))
         (fabs (/ x (/ y_m z))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.05e+72) {
		tmp = fabs((x * (z / y_m)));
	} else if (z <= -4e+60) {
		tmp = fabs((4.0 / y_m));
	} else if (z <= -39.0) {
		tmp = fabs(((1.0 - z) / (y_m / x)));
	} else if (z <= 4.1e+135) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((x / (y_m / z)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.05d+72)) then
        tmp = abs((x * (z / y_m)))
    else if (z <= (-4d+60)) then
        tmp = abs((4.0d0 / y_m))
    else if (z <= (-39.0d0)) then
        tmp = abs(((1.0d0 - z) / (y_m / x)))
    else if (z <= 4.1d+135) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((x / (y_m / z)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.05e+72) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (z <= -4e+60) {
		tmp = Math.abs((4.0 / y_m));
	} else if (z <= -39.0) {
		tmp = Math.abs(((1.0 - z) / (y_m / x)));
	} else if (z <= 4.1e+135) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x / (y_m / z)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -1.05e+72:
		tmp = math.fabs((x * (z / y_m)))
	elif z <= -4e+60:
		tmp = math.fabs((4.0 / y_m))
	elif z <= -39.0:
		tmp = math.fabs(((1.0 - z) / (y_m / x)))
	elif z <= 4.1e+135:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x / (y_m / z)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -1.05e+72)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (z <= -4e+60)
		tmp = abs(Float64(4.0 / y_m));
	elseif (z <= -39.0)
		tmp = abs(Float64(Float64(1.0 - z) / Float64(y_m / x)));
	elseif (z <= 4.1e+135)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x / Float64(y_m / z)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -1.05e+72)
		tmp = abs((x * (z / y_m)));
	elseif (z <= -4e+60)
		tmp = abs((4.0 / y_m));
	elseif (z <= -39.0)
		tmp = abs(((1.0 - z) / (y_m / x)));
	elseif (z <= 4.1e+135)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x / (y_m / z)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -1.05e+72], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -4e+60], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -39.0], N[Abs[N[(N[(1.0 - z), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4.1e+135], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.0500000000000001e72

   1. Initial program 94.0%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in z around inf 76.8%

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

      1. associate-*r/ 76.8%

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

      2. neg-mul-1 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

   7. Simplified 76.8%

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

      1. distribute-rgt-neg-out 76.8%

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

      2. *-commutative 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

      4. associate-*l/ 80.6%

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

      5. add-sqr-sqrt 52.5%

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

      6. sqrt-unprod 75.3%

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

      7. sqr-neg 75.3%

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

      8. sqrt-unprod 27.9%

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

      9. add-sqr-sqrt 80.6%

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

   9. Applied egg-rr 80.6%

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

   if -1.0500000000000001e72 < z < -3.9999999999999998e60

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 89.0%

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

   if -3.9999999999999998e60 < z < -39

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 79.6%

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

      1. associate-*r/ 79.6%

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

      2. *-commutative 79.6%

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

      3. associate-*r* 79.6%

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

      4. sub-neg 79.6%

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

      5. metadata-eval 79.6%

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

      6. distribute-lft-in 79.6%

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

      7. neg-mul-1 79.6%

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

      8. metadata-eval 79.6%

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

      9. +-commutative 79.6%

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

      10. neg-mul-1 79.6%

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

      11. associate-/l* 79.4%

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

      12. neg-mul-1 79.4%

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

      13. unsub-neg 79.4%

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

   7. Simplified 79.4%

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

      1. clear-num 79.2%

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

      2. un-div-inv 79.7%

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

   9. Applied egg-rr 79.7%

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

   if -39 < z < 4.1e135

   1. Initial program 91.2%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 96.2%

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

      1. +-commutative 96.2%

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

      2. rem-square-sqrt 40.4%

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

      3. fabs-sqr 40.4%

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

      4. rem-square-sqrt 96.2%

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

      5. fabs-neg 96.2%

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

      6. distribute-neg-frac 96.2%

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

      7. distribute-neg-in 96.2%

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

      8. metadata-eval 96.2%

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

      9. +-commutative 96.2%

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

      10. sub-neg 96.2%

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

      11. rem-square-sqrt 55.2%

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

      12. fabs-sqr 55.2%

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

      13. rem-square-sqrt 96.2%

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

   7. Simplified 96.2%

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

   if 4.1e135 < z

   1. Initial program 83.6%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in z around inf 72.1%

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

      1. mul-1-neg 72.1%

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

      2. associate-*r/ 84.3%

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

      3. *-commutative 84.3%

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

      4. distribute-rgt-neg-in 84.3%

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

   7. Simplified 84.3%

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

      1. *-commutative 84.3%

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

      2. clear-num 84.3%

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

      3. un-div-inv 84.9%

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

      4. add-sqr-sqrt 48.0%

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

      5. sqrt-unprod 71.8%

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

      6. sqr-neg 71.8%

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

      7. sqrt-unprod 36.6%

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

      8. add-sqr-sqrt 84.9%

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

   9. Applied egg-rr 84.9%

      \[\leadsto \left|\color{blue}{\frac{x}{\frac{y}{z}}}\right| \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+72}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq -39:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+135}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{+36}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -1.5e+58)
     t_0
     (if (<= x -1.1e+36)
       (fabs (* x (/ z y_m)))
       (if (or (<= x -1.55) (not (<= x 4.0))) t_0 (fabs (/ 4.0 y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double tmp;
	if (x <= -1.5e+58) {
		tmp = t_0;
	} else if (x <= -1.1e+36) {
		tmp = fabs((x * (z / y_m)));
	} else if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = t_0;
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y_m))
    if (x <= (-1.5d+58)) then
        tmp = t_0
    else if (x <= (-1.1d+36)) then
        tmp = abs((x * (z / y_m)))
    else if ((x <= (-1.55d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = t_0
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double tmp;
	if (x <= -1.5e+58) {
		tmp = t_0;
	} else if (x <= -1.1e+36) {
		tmp = Math.abs((x * (z / y_m)));
	} else if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = t_0;
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	tmp = 0
	if x <= -1.5e+58:
		tmp = t_0
	elif x <= -1.1e+36:
		tmp = math.fabs((x * (z / y_m)))
	elif (x <= -1.55) or not (x <= 4.0):
		tmp = t_0
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -1.5e+58)
		tmp = t_0;
	elseif (x <= -1.1e+36)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif ((x <= -1.55) || !(x <= 4.0))
		tmp = t_0;
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -1.5e+58)
		tmp = t_0;
	elseif (x <= -1.1e+36)
		tmp = abs((x * (z / y_m)));
	elseif ((x <= -1.55) || ~((x <= 4.0)))
		tmp = t_0;
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.5e+58], t$95$0, If[LessEqual[x, -1.1e+36], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], t$95$0, N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5000000000000001e58 or -1.1e36 < x < -1.55000000000000004 or 4 < x

   1. Initial program 87.2%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in z around 0 71.8%

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

      1. distribute-lft-in 71.8%

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

      2. metadata-eval 71.8%

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

      3. neg-mul-1 71.8%

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

      4. sub-neg 71.8%

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

   7. Simplified 71.8%

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

   8. Taylor expanded in x around inf 71.2%

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

   if -1.5000000000000001e58 < x < -1.1e36

   1. Initial program 99.7%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in z around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. neg-mul-1 85.0%

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

      3. distribute-rgt-neg-in 85.0%

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

   7. Simplified 85.0%

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

      1. distribute-rgt-neg-out 85.0%

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

      2. *-commutative 85.0%

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

      3. distribute-rgt-neg-in 85.0%

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

      4. associate-*l/ 99.7%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 99.7%

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

      7. sqr-neg 99.7%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 99.7%

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

   9. Applied egg-rr 99.7%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 94.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 74.7%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+58}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{+36}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{+28}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -6.5e+56)
     t_0
     (if (<= x -3.4e+28)
       (fabs (/ x (/ y_m z)))
       (if (or (<= x -1.55) (not (<= x 4.0))) t_0 (fabs (/ 4.0 y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double tmp;
	if (x <= -6.5e+56) {
		tmp = t_0;
	} else if (x <= -3.4e+28) {
		tmp = fabs((x / (y_m / z)));
	} else if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = t_0;
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y_m))
    if (x <= (-6.5d+56)) then
        tmp = t_0
    else if (x <= (-3.4d+28)) then
        tmp = abs((x / (y_m / z)))
    else if ((x <= (-1.55d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = t_0
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double tmp;
	if (x <= -6.5e+56) {
		tmp = t_0;
	} else if (x <= -3.4e+28) {
		tmp = Math.abs((x / (y_m / z)));
	} else if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = t_0;
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	tmp = 0
	if x <= -6.5e+56:
		tmp = t_0
	elif x <= -3.4e+28:
		tmp = math.fabs((x / (y_m / z)))
	elif (x <= -1.55) or not (x <= 4.0):
		tmp = t_0
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -6.5e+56)
		tmp = t_0;
	elseif (x <= -3.4e+28)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif ((x <= -1.55) || !(x <= 4.0))
		tmp = t_0;
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -6.5e+56)
		tmp = t_0;
	elseif (x <= -3.4e+28)
		tmp = abs((x / (y_m / z)));
	elseif ((x <= -1.55) || ~((x <= 4.0)))
		tmp = t_0;
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.5e+56], t$95$0, If[LessEqual[x, -3.4e+28], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], t$95$0, N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5000000000000001e56 or -3.4e28 < x < -1.55000000000000004 or 4 < x

   1. Initial program 87.0%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in z around 0 72.1%

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

      1. distribute-lft-in 72.1%

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

      2. metadata-eval 72.1%

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

      3. neg-mul-1 72.1%

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

      4. sub-neg 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in x around inf 71.6%

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

   if -6.5000000000000001e56 < x < -3.4e28

   1. Initial program 99.8%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in z around inf 76.8%

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

      1. mul-1-neg 76.8%

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

      2. associate-*r/ 87.8%

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

      3. *-commutative 87.8%

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

      4. distribute-rgt-neg-in 87.8%

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

   7. Simplified 87.8%

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

      1. *-commutative 87.8%

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

      2. clear-num 87.8%

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

      3. un-div-inv 88.0%

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

      4. add-sqr-sqrt 88.0%

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

      5. sqrt-unprod 88.0%

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

      6. sqr-neg 88.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 88.0%

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

   9. Applied egg-rr 88.0%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 94.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 74.7%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+56}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{+28}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 3.9\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot \left(\frac{4}{z} - x\right)}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 3.9)))
   (fabs (* (- 1.0 z) (/ x y_m)))
   (fabs (/ (* z (- (/ 4.0 z) x)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.5) || !(x <= 3.9)) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = fabs(((z * ((4.0 / z) - x)) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.5d0)) .or. (.not. (x <= 3.9d0))) then
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    else
        tmp = abs(((z * ((4.0d0 / z) - x)) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.5) || !(x <= 3.9)) {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = Math.abs(((z * ((4.0 / z) - x)) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.5) or not (x <= 3.9):
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	else:
		tmp = math.fabs(((z * ((4.0 / z) - x)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.5) || !(x <= 3.9))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	else
		tmp = abs(Float64(Float64(z * Float64(Float64(4.0 / z) - x)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1.5) || ~((x <= 3.9)))
		tmp = abs(((1.0 - z) * (x / y_m)));
	else
		tmp = abs(((z * ((4.0 / z) - x)) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1.5], N[Not[LessEqual[x, 3.9]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * N[(N[(4.0 / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 3.89999999999999991 < x

   1. Initial program 87.8%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in x around inf 91.8%

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

      1. associate-*r/ 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*r* 91.8%

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

      4. sub-neg 91.8%

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

      5. metadata-eval 91.8%

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

      6. distribute-lft-in 91.8%

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

      7. neg-mul-1 91.8%

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

      8. metadata-eval 91.8%

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

      9. +-commutative 91.8%

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

      10. neg-mul-1 91.8%

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

      11. associate-/l* 99.2%

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

      12. neg-mul-1 99.2%

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

      13. unsub-neg 99.2%

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

   7. Simplified 99.2%

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

   if -1.5 < x < 3.89999999999999991

   1. Initial program 94.9%

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

      1. associate-*l/ 99.9%

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

      2. sub-div 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-neg-in 99.8%

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

      3. distribute-lft-out-- 99.8%

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

      4. mul-1-neg 99.8%

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

      5. remove-double-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 3.9\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot \left(\frac{4}{z} - x\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 1.15e+14)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (* (- 1.0 z) (/ x y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 1.15e+14) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.15d+14) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 1.15e+14) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 1.15e+14:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 1.15e+14)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 1.15e+14)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs(((1.0 - z) * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 1.15e+14], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e14

   1. Initial program 93.2%

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

      1. associate-*l/ 96.6%

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

      2. sub-div 98.0%

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

   4. Applied egg-rr 98.0%

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

   if 1.15e14 < x

   1. Initial program 85.1%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in x around inf 89.7%

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

      1. associate-*r/ 89.7%

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

      2. *-commutative 89.7%

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

      3. associate-*r* 89.7%

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

      4. sub-neg 89.7%

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

      5. metadata-eval 89.7%

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

      6. distribute-lft-in 89.7%

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

      7. neg-mul-1 89.7%

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

      8. metadata-eval 89.7%

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

      9. +-commutative 89.7%

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

      10. neg-mul-1 89.7%

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

      11. associate-/l* 99.9%

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

      12. neg-mul-1 99.9%

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

      13. unsub-neg 99.9%

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

   7. Simplified 99.9%

      \[\leadsto \left|\color{blue}{\left(1 - z\right) \cdot \frac{x}{y}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.1% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 4.0))) (fabs (/ x y_m)) (fabs (/ 4.0 y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = fabs((x / y_m));
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y_m))
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = Math.abs((x / y_m));
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.55) or not (x <= 4.0):
		tmp = math.fabs((x / y_m))
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 4.0))
		tmp = abs(Float64(x / y_m));
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 4.0)))
		tmp = abs((x / y_m));
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 4 < x

   1. Initial program 87.8%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in z around 0 68.5%

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

      1. distribute-lft-in 68.5%

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

      2. metadata-eval 68.5%

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

      3. neg-mul-1 68.5%

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

      4. sub-neg 68.5%

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

   7. Simplified 68.5%

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

   8. Taylor expanded in x around inf 68.0%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 94.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 74.7%

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

4. Final simplification 71.5%

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

Alternative 11: 40.5% accurate, 1.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (fabs (/ 4.0 y_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.5%

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

3. Simplified 96.2%

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

5. Taylor expanded in x around 0 41.1%

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

6. Final simplification 41.1%

   \[\leadsto \left|\frac{4}{y}\right| \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))