fabs fraction 1

Percentage Accurate: 92.0% → 99.9%

Time: 8.8s

Alternatives: 11

Speedup: 1.0×

• 20.6% 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: 92.0% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 400000:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 400000.0)
   (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 <= 400000.0) {
		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 <= 400000.0)
		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, 400000.0], 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 < 4e5

   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/ 90.4%

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

      2. sub-div 96.9%

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

   4. Applied egg-rr 96.9%

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

   if 4e5 < y

   1. Initial program 98.6%

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

      1. fabs-sub 98.6%

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

      2. associate-*l/ 90.7%

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

      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 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 400000:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ t_1 := \left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-40}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* z (/ x y_m)))))
   (if (<= x -1.65e+78)
     t_0
     (if (<= x -1.5e-64)
       t_1
       (if (<= x 2.7e-40) (fabs (/ 4.0 y_m)) (if (<= x 7.2e+30) t_1 t_0))))))

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 * (x / y_m)));
	double tmp;
	if (x <= -1.65e+78) {
		tmp = t_0;
	} else if (x <= -1.5e-64) {
		tmp = t_1;
	} else if (x <= 2.7e-40) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 7.2e+30) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 * (x / y_m)))
    if (x <= (-1.65d+78)) then
        tmp = t_0
    else if (x <= (-1.5d-64)) then
        tmp = t_1
    else if (x <= 2.7d-40) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 7.2d+30) then
        tmp = t_1
    else
        tmp = t_0
    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 * (x / y_m)));
	double tmp;
	if (x <= -1.65e+78) {
		tmp = t_0;
	} else if (x <= -1.5e-64) {
		tmp = t_1;
	} else if (x <= 2.7e-40) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 7.2e+30) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 * (x / y_m)))
	tmp = 0
	if x <= -1.65e+78:
		tmp = t_0
	elif x <= -1.5e-64:
		tmp = t_1
	elif x <= 2.7e-40:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 7.2e+30:
		tmp = t_1
	else:
		tmp = t_0
	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(x / y_m)))
	tmp = 0.0
	if (x <= -1.65e+78)
		tmp = t_0;
	elseif (x <= -1.5e-64)
		tmp = t_1;
	elseif (x <= 2.7e-40)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 7.2e+30)
		tmp = t_1;
	else
		tmp = t_0;
	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 * (x / y_m)));
	tmp = 0.0;
	if (x <= -1.65e+78)
		tmp = t_0;
	elseif (x <= -1.5e-64)
		tmp = t_1;
	elseif (x <= 2.7e-40)
		tmp = abs((4.0 / y_m));
	elseif (x <= 7.2e+30)
		tmp = t_1;
	else
		tmp = t_0;
	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[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.65e+78], t$95$0, If[LessEqual[x, -1.5e-64], t$95$1, If[LessEqual[x, 2.7e-40], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 7.2e+30], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65e78 or 7.2000000000000004e30 < x

   1. Initial program 86.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in x around inf 90.4%

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

      1. associate-/l* 99.8%

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

      2. div-sub 99.7%

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

      3. associate-*r* 99.7%

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

      4. neg-mul-1 99.7%

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

      5. *-commutative 99.7%

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

      6. div-sub 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

      9. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around 0 67.6%

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

   if -1.65e78 < x < -1.5e-64 or 2.7e-40 < x < 7.2000000000000004e30

   1. Initial program 97.6%

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

      1. associate-*l/ 97.8%

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

      2. sub-div 97.8%

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in z around inf 51.8%

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

      1. mul-1-neg 51.8%

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

      2. distribute-rgt-neg-in 51.8%

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

   7. Simplified 51.8%

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

      1. distribute-rgt-neg-out 51.8%

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

      2. distribute-lft-neg-in 51.8%

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

      3. associate-*l/ 57.8%

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

      4. add-sqr-sqrt 40.0%

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

      5. sqrt-unprod 57.8%

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

      6. sqr-neg 57.8%

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

      7. sqrt-unprod 17.6%

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

      8. add-sqr-sqrt 57.8%

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

   9. Applied egg-rr 57.8%

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

   if -1.5e-64 < x < 2.7e-40

   1. Initial program 98.9%

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

   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 85.8%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+78}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-64}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-40}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+30}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-64}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-40}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+30}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -1.65e+78)
     t_0
     (if (<= x -1.8e-64)
       (fabs (* z (/ x y_m)))
       (if (<= x 3.05e-40)
         (fabs (/ 4.0 y_m))
         (if (<= x 2.45e+30) (fabs (/ z (/ y_m x))) t_0))))))

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 <= -1.65e+78) {
		tmp = t_0;
	} else if (x <= -1.8e-64) {
		tmp = fabs((z * (x / y_m)));
	} else if (x <= 3.05e-40) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 2.45e+30) {
		tmp = fabs((z / (y_m / x)));
	} else {
		tmp = t_0;
	}
	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 <= (-1.65d+78)) then
        tmp = t_0
    else if (x <= (-1.8d-64)) then
        tmp = abs((z * (x / y_m)))
    else if (x <= 3.05d-40) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 2.45d+30) then
        tmp = abs((z / (y_m / x)))
    else
        tmp = t_0
    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 <= -1.65e+78) {
		tmp = t_0;
	} else if (x <= -1.8e-64) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (x <= 3.05e-40) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 2.45e+30) {
		tmp = Math.abs((z / (y_m / x)));
	} else {
		tmp = t_0;
	}
	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 <= -1.65e+78:
		tmp = t_0
	elif x <= -1.8e-64:
		tmp = math.fabs((z * (x / y_m)))
	elif x <= 3.05e-40:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 2.45e+30:
		tmp = math.fabs((z / (y_m / x)))
	else:
		tmp = t_0
	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 <= -1.65e+78)
		tmp = t_0;
	elseif (x <= -1.8e-64)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (x <= 3.05e-40)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 2.45e+30)
		tmp = abs(Float64(z / Float64(y_m / x)));
	else
		tmp = t_0;
	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 <= -1.65e+78)
		tmp = t_0;
	elseif (x <= -1.8e-64)
		tmp = abs((z * (x / y_m)));
	elseif (x <= 3.05e-40)
		tmp = abs((4.0 / y_m));
	elseif (x <= 2.45e+30)
		tmp = abs((z / (y_m / x)));
	else
		tmp = t_0;
	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, -1.65e+78], t$95$0, If[LessEqual[x, -1.8e-64], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.05e-40], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.45e+30], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.65e78 or 2.44999999999999992e30 < x

   1. Initial program 86.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in x around inf 90.4%

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

      1. associate-/l* 99.8%

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

      2. div-sub 99.7%

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

      3. associate-*r* 99.7%

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

      4. neg-mul-1 99.7%

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

      5. *-commutative 99.7%

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

      6. div-sub 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

      9. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around 0 67.6%

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

   if -1.65e78 < x < -1.7999999999999999e-64

   1. Initial program 96.9%

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

      1. associate-*l/ 97.1%

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

      2. sub-div 97.1%

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

   4. Applied egg-rr 97.1%

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

   5. Taylor expanded in z around inf 46.1%

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

      1. mul-1-neg 46.1%

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

      2. distribute-rgt-neg-in 46.1%

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

   7. Simplified 46.1%

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

      1. distribute-rgt-neg-out 46.1%

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

      2. distribute-lft-neg-in 46.1%

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

      3. associate-*l/ 54.2%

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

      4. add-sqr-sqrt 54.1%

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

      5. sqrt-unprod 54.2%

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

      6. sqr-neg 54.2%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 54.2%

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

   9. Applied egg-rr 54.2%

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

   if -1.7999999999999999e-64 < x < 3.0500000000000002e-40

   1. Initial program 98.9%

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

   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 85.8%

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

   if 3.0500000000000002e-40 < x < 2.44999999999999992e30

   1. Initial program 99.7%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in z around inf 68.0%

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

      1. associate-*r/ 68.0%

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

      2. neg-mul-1 68.0%

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

      3. distribute-lft-neg-in 68.0%

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

      4. *-commutative 68.0%

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

      5. associate-*r/ 67.9%

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

   7. Simplified 67.9%

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

      1. clear-num 67.9%

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

      2. un-div-inv 68.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 68.0%

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

      5. sqr-neg 68.0%

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

      6. sqrt-unprod 67.4%

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

      7. add-sqr-sqrt 68.0%

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

   9. Applied egg-rr 68.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+78}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-64}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-40}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+30}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-64}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-48}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+30}:\\ \;\;\;\;\left|\frac{x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -2.4e+78)
     t_0
     (if (<= x -1.8e-64)
       (fabs (* z (/ x y_m)))
       (if (<= x 9.2e-48)
         (fabs (/ 4.0 y_m))
         (if (<= x 2.7e+30) (fabs (/ (* x z) y_m)) t_0))))))

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 <= -2.4e+78) {
		tmp = t_0;
	} else if (x <= -1.8e-64) {
		tmp = fabs((z * (x / y_m)));
	} else if (x <= 9.2e-48) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 2.7e+30) {
		tmp = fabs(((x * z) / y_m));
	} else {
		tmp = t_0;
	}
	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 <= (-2.4d+78)) then
        tmp = t_0
    else if (x <= (-1.8d-64)) then
        tmp = abs((z * (x / y_m)))
    else if (x <= 9.2d-48) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 2.7d+30) then
        tmp = abs(((x * z) / y_m))
    else
        tmp = t_0
    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 <= -2.4e+78) {
		tmp = t_0;
	} else if (x <= -1.8e-64) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (x <= 9.2e-48) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 2.7e+30) {
		tmp = Math.abs(((x * z) / y_m));
	} else {
		tmp = t_0;
	}
	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 <= -2.4e+78:
		tmp = t_0
	elif x <= -1.8e-64:
		tmp = math.fabs((z * (x / y_m)))
	elif x <= 9.2e-48:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 2.7e+30:
		tmp = math.fabs(((x * z) / y_m))
	else:
		tmp = t_0
	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 <= -2.4e+78)
		tmp = t_0;
	elseif (x <= -1.8e-64)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (x <= 9.2e-48)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 2.7e+30)
		tmp = abs(Float64(Float64(x * z) / y_m));
	else
		tmp = t_0;
	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 <= -2.4e+78)
		tmp = t_0;
	elseif (x <= -1.8e-64)
		tmp = abs((z * (x / y_m)));
	elseif (x <= 9.2e-48)
		tmp = abs((4.0 / y_m));
	elseif (x <= 2.7e+30)
		tmp = abs(((x * z) / y_m));
	else
		tmp = t_0;
	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, -2.4e+78], t$95$0, If[LessEqual[x, -1.8e-64], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 9.2e-48], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.7e+30], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.3999999999999999e78 or 2.6999999999999999e30 < x

   1. Initial program 86.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in x around inf 90.4%

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

      1. associate-/l* 99.8%

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

      2. div-sub 99.7%

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

      3. associate-*r* 99.7%

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

      4. neg-mul-1 99.7%

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

      5. *-commutative 99.7%

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

      6. div-sub 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

      9. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around 0 67.6%

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

   if -2.3999999999999999e78 < x < -1.7999999999999999e-64

   1. Initial program 96.9%

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

      1. associate-*l/ 97.1%

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

      2. sub-div 97.1%

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

   4. Applied egg-rr 97.1%

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

   5. Taylor expanded in z around inf 46.1%

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

      1. mul-1-neg 46.1%

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

      2. distribute-rgt-neg-in 46.1%

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

   7. Simplified 46.1%

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

      1. distribute-rgt-neg-out 46.1%

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

      2. distribute-lft-neg-in 46.1%

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

      3. associate-*l/ 54.2%

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

      4. add-sqr-sqrt 54.1%

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

      5. sqrt-unprod 54.2%

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

      6. sqr-neg 54.2%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 54.2%

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

   9. Applied egg-rr 54.2%

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

   if -1.7999999999999999e-64 < x < 9.2000000000000003e-48

   1. Initial program 98.9%

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

   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 85.8%

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

   if 9.2000000000000003e-48 < x < 2.6999999999999999e30

   1. Initial program 99.7%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in z around inf 68.0%

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

      1. associate-*r/ 68.0%

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

      2. neg-mul-1 68.0%

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

      3. distribute-lft-neg-in 68.0%

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

      4. *-commutative 68.0%

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

      5. associate-*r/ 67.9%

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

   7. Simplified 67.9%

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

      1. associate-*r/ 68.0%

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

      2. distribute-rgt-neg-out 68.0%

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

      3. distribute-lft-neg-out 68.0%

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

      4. add-sqr-sqrt 17.2%

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

      5. sqrt-unprod 60.1%

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

      6. sqr-neg 60.1%

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

      7. sqrt-unprod 50.0%

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

      8. add-sqr-sqrt 68.0%

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

   9. Applied egg-rr 68.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+78}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-64}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-48}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+30}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.5 \cdot 10^{-58}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1.5e-58)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (* z (/ x y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.5e-58) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (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 (y_m <= 1.5d-58) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (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 (y_m <= 1.5e-58) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (z * (x / y_m))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 1.5e-58:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (z * (x / y_m))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.5e-58)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(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 (y_m <= 1.5e-58)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((((x + 4.0) / y_m) - (z * (x / y_m))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 1.5e-58], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000004e-58

   1. Initial program 90.0%

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

      1. associate-*l/ 89.7%

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

      2. sub-div 96.7%

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

   4. Applied egg-rr 96.7%

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

   if 1.50000000000000004e-58 < y

   1. Initial program 98.8%

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

4. Final simplification 97.4%

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

Alternative 6: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - \frac{z}{\frac{y\_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 5e-142)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (/ z (/ y_m x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5e-142) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (z / (y_m / x))));
	}
	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 (y_m <= 5d-142) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (z / (y_m / x))))
    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 (y_m <= 5e-142) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (z / (y_m / x))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 5e-142:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (z / (y_m / x))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 5e-142)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(z / Float64(y_m / x))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 5e-142)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((((x + 4.0) / y_m) - (z / (y_m / x))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 5e-142], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000002e-142

   1. Initial program 88.3%

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

      1. associate-*l/ 88.0%

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

      2. sub-div 96.2%

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

   4. Applied egg-rr 96.2%

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

   if 5.0000000000000002e-142 < y

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. clear-num 99.0%

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

      3. un-div-inv 99.1%

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

   4. Applied egg-rr 99.1%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -20000000:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+23}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -20000000.0)
   (fabs (* z (/ x y_m)))
   (if (<= z 1.1e+23) (fabs (/ (- -4.0 x) y_m)) (fabs (/ z (/ y_m x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -20000000.0) {
		tmp = fabs((z * (x / y_m)));
	} else if (z <= 1.1e+23) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((z / (y_m / x)));
	}
	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 <= (-20000000.0d0)) then
        tmp = abs((z * (x / y_m)))
    else if (z <= 1.1d+23) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((z / (y_m / x)))
    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 <= -20000000.0) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (z <= 1.1e+23) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((z / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -20000000.0:
		tmp = math.fabs((z * (x / y_m)))
	elif z <= 1.1e+23:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((z / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -20000000.0)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (z <= 1.1e+23)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(z / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -20000000.0)
		tmp = abs((z * (x / y_m)));
	elseif (z <= 1.1e+23)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((z / (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -20000000.0], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.1e+23], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2e7

   1. Initial program 98.2%

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

      1. associate-*l/ 85.9%

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

      2. sub-div 86.0%

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

   4. Applied egg-rr 86.0%

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

   5. Taylor expanded in z around inf 60.8%

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

      1. mul-1-neg 60.8%

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

      2. distribute-rgt-neg-in 60.8%

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

   7. Simplified 60.8%

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

      1. distribute-rgt-neg-out 60.8%

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

      2. distribute-lft-neg-in 60.8%

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

      3. associate-*l/ 72.9%

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

      4. add-sqr-sqrt 48.1%

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

      5. sqrt-unprod 53.0%

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

      6. sqr-neg 53.0%

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

      7. sqrt-unprod 24.7%

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

      8. add-sqr-sqrt 72.9%

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

   9. Applied egg-rr 72.9%

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

   if -2e7 < z < 1.10000000000000004e23

   1. Initial program 93.7%

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

      1. fabs-sub 93.7%

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

      2. associate-*l/ 97.2%

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

      3. associate-*r/ 97.2%

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

      4. fma-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 98.0%

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

      1. associate-*r/ 98.0%

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

      2. distribute-lft-in 98.0%

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

      3. metadata-eval 98.0%

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

      4. neg-mul-1 98.0%

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

      5. sub-neg 98.0%

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

   7. Simplified 98.0%

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

   if 1.10000000000000004e23 < z

   1. Initial program 86.2%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in z around inf 70.1%

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

      1. associate-*r/ 70.1%

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

      2. neg-mul-1 70.1%

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

      3. distribute-lft-neg-in 70.1%

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

      4. *-commutative 70.1%

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

      5. associate-*r/ 77.8%

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

   7. Simplified 77.8%

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

      1. clear-num 77.7%

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

      2. un-div-inv 77.8%

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

      3. add-sqr-sqrt 43.4%

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

      4. sqrt-unprod 66.1%

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

      5. sqr-neg 66.1%

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

      6. sqrt-unprod 34.2%

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

      7. add-sqr-sqrt 77.8%

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

   9. Applied egg-rr 77.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -20000000:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+23}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0045:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+23}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -0.0045)
   (fabs (* (/ x y_m) (- 1.0 z)))
   (if (<= z 1.15e+23) (fabs (/ (- -4.0 x) y_m)) (fabs (/ z (/ y_m x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -0.0045) {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	} else if (z <= 1.15e+23) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((z / (y_m / x)));
	}
	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 <= (-0.0045d0)) then
        tmp = abs(((x / y_m) * (1.0d0 - z)))
    else if (z <= 1.15d+23) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((z / (y_m / x)))
    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 <= -0.0045) {
		tmp = Math.abs(((x / y_m) * (1.0 - z)));
	} else if (z <= 1.15e+23) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((z / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -0.0045:
		tmp = math.fabs(((x / y_m) * (1.0 - z)))
	elif z <= 1.15e+23:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((z / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -0.0045)
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	elseif (z <= 1.15e+23)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(z / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -0.0045)
		tmp = abs(((x / y_m) * (1.0 - z)));
	elseif (z <= 1.15e+23)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((z / (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -0.0045], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.15e+23], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.00449999999999999966

   1. Initial program 98.2%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in x around -inf 63.8%

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

      1. *-commutative 63.8%

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

      2. associate-/l* 75.4%

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

      3. neg-mul-1 75.4%

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

      4. unsub-neg 75.4%

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

   7. Simplified 75.4%

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

   if -0.00449999999999999966 < z < 1.15e23

   1. Initial program 93.6%

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

      1. fabs-sub 93.6%

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

      2. associate-*l/ 97.1%

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

      3. associate-*r/ 97.1%

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

      4. fma-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 99.0%

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

      1. associate-*r/ 99.0%

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

      2. distribute-lft-in 99.0%

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

      3. metadata-eval 99.0%

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

      4. neg-mul-1 99.0%

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

      5. sub-neg 99.0%

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

   7. Simplified 99.0%

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

   if 1.15e23 < z

   1. Initial program 86.2%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in z around inf 70.1%

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

      1. associate-*r/ 70.1%

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

      2. neg-mul-1 70.1%

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

      3. distribute-lft-neg-in 70.1%

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

      4. *-commutative 70.1%

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

      5. associate-*r/ 77.8%

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

   7. Simplified 77.8%

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

      1. clear-num 77.7%

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

      2. un-div-inv 77.8%

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

      3. add-sqr-sqrt 43.4%

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

      4. sqrt-unprod 66.1%

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

      5. sqr-neg 66.1%

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

      6. sqrt-unprod 34.2%

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

      7. add-sqr-sqrt 77.8%

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

   9. Applied egg-rr 77.8%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0045:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+23}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+45}:\\ \;\;\;\;\left|\frac{x}{y\_m} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4e+45)
   (fabs (* (/ x y_m) (- 1.0 z)))
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4e+45) {
		tmp = fabs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / 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 <= (-4d+45)) then
        tmp = abs(((x / y_m) * (1.0d0 - z)))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / 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 <= -4e+45) {
		tmp = Math.abs(((x / y_m) * (1.0 - z)));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -4e+45:
		tmp = math.fabs(((x / y_m) * (1.0 - z)))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4e+45)
		tmp = abs(Float64(Float64(x / y_m) * Float64(1.0 - z)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -4e+45)
		tmp = abs(((x / y_m) * (1.0 - z)));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -4e+45], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999999999997e45

   1. Initial program 83.2%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in x around -inf 85.8%

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

      1. *-commutative 85.8%

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

      2. associate-/l* 99.9%

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

      3. neg-mul-1 99.9%

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

      4. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -3.9999999999999997e45 < x

   1. Initial program 96.3%

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

      1. associate-*l/ 96.3%

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

      2. sub-div 98.4%

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

   4. Applied egg-rr 98.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+45}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.52) (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 <= -1.52) || !(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 <= (-1.52d0)) .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 <= -1.52) || !(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 <= -1.52) 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 <= -1.52) || !(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 <= -1.52) || ~((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, -1.52], 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 < -1.52 or 4 < x

   1. Initial program 88.0%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in x around inf 89.0%

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

      1. associate-/l* 97.4%

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

      2. div-sub 97.4%

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

      3. associate-*r* 97.4%

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

      4. neg-mul-1 97.4%

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

      5. *-commutative 97.4%

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

      6. div-sub 97.4%

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

      7. sub-neg 97.4%

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

      8. metadata-eval 97.4%

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

      9. +-commutative 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in z around 0 63.3%

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

   if -1.52 < x < 4

   1. Initial program 99.1%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 76.0%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.1% 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 92.9%

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

3. Simplified 95.0%

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

5. Taylor expanded in x around 0 36.8%

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

6. Final simplification 36.8%

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

Reproduce

herbie shell --seed 2024041 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))