fabs fraction 1

Percentage Accurate: 91.5% → 99.9%

Time: 8.6s

Alternatives: 8

Speedup: 1.0×

• 21.2% 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: 91.5% 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, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 5e-10)
   (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
   (fabs (- (/ (+ 4.0 x) y_m) (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5e-10) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs((((4.0 + x) / y_m) - (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 (y_m <= 5d-10) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else
        tmp = abs((((4.0d0 + x) / y_m) - (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 (y_m <= 5e-10) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((4.0 + x) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 5e-10:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((4.0 + x) / y_m) - (x / (y_m / z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 5e-10)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) / y_m) - 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 (y_m <= 5e-10)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	else
		tmp = abs((((4.0 + x) / y_m) - (x / (y_m / z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 5e-10], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.00000000000000031e-10

   1. Initial program 89.2%

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

   3. Taylor expanded in y around 0 96.7%

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

   if 5.00000000000000031e-10 < y

   1. Initial program 98.1%

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

      1. associate-/r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 97.4%

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

Alternative 2: 62.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{\frac{y_m}{z}}\right|\\ t_1 := \left|\frac{4}{y_m}\right|\\ \mathbf{if}\;z \leq -980000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-255}:\\ \;\;\;\;\left|\frac{x}{y_m}\right|\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-203}:\\ \;\;\;\;\left|x \cdot \frac{1}{y_m}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x (/ y_m z)))) (t_1 (fabs (/ 4.0 y_m))))
   (if (<= z -980000000.0)
     t_0
     (if (<= z -1.25e-206)
       t_1
       (if (<= z -3.7e-255)
         (fabs (/ x y_m))
         (if (<= z -2.85e-299)
           t_1
           (if (<= z 5.2e-203)
             (fabs (* x (/ 1.0 y_m)))
             (if (<= z 2.5e-9) 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 / z)));
	double t_1 = fabs((4.0 / y_m));
	double tmp;
	if (z <= -980000000.0) {
		tmp = t_0;
	} else if (z <= -1.25e-206) {
		tmp = t_1;
	} else if (z <= -3.7e-255) {
		tmp = fabs((x / y_m));
	} else if (z <= -2.85e-299) {
		tmp = t_1;
	} else if (z <= 5.2e-203) {
		tmp = fabs((x * (1.0 / y_m)));
	} else if (z <= 2.5e-9) {
		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 / z)))
    t_1 = abs((4.0d0 / y_m))
    if (z <= (-980000000.0d0)) then
        tmp = t_0
    else if (z <= (-1.25d-206)) then
        tmp = t_1
    else if (z <= (-3.7d-255)) then
        tmp = abs((x / y_m))
    else if (z <= (-2.85d-299)) then
        tmp = t_1
    else if (z <= 5.2d-203) then
        tmp = abs((x * (1.0d0 / y_m)))
    else if (z <= 2.5d-9) 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 / z)));
	double t_1 = Math.abs((4.0 / y_m));
	double tmp;
	if (z <= -980000000.0) {
		tmp = t_0;
	} else if (z <= -1.25e-206) {
		tmp = t_1;
	} else if (z <= -3.7e-255) {
		tmp = Math.abs((x / y_m));
	} else if (z <= -2.85e-299) {
		tmp = t_1;
	} else if (z <= 5.2e-203) {
		tmp = Math.abs((x * (1.0 / y_m)));
	} else if (z <= 2.5e-9) {
		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 / z)))
	t_1 = math.fabs((4.0 / y_m))
	tmp = 0
	if z <= -980000000.0:
		tmp = t_0
	elif z <= -1.25e-206:
		tmp = t_1
	elif z <= -3.7e-255:
		tmp = math.fabs((x / y_m))
	elif z <= -2.85e-299:
		tmp = t_1
	elif z <= 5.2e-203:
		tmp = math.fabs((x * (1.0 / y_m)))
	elif z <= 2.5e-9:
		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 / Float64(y_m / z)))
	t_1 = abs(Float64(4.0 / y_m))
	tmp = 0.0
	if (z <= -980000000.0)
		tmp = t_0;
	elseif (z <= -1.25e-206)
		tmp = t_1;
	elseif (z <= -3.7e-255)
		tmp = abs(Float64(x / y_m));
	elseif (z <= -2.85e-299)
		tmp = t_1;
	elseif (z <= 5.2e-203)
		tmp = abs(Float64(x * Float64(1.0 / y_m)));
	elseif (z <= 2.5e-9)
		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 / z)));
	t_1 = abs((4.0 / y_m));
	tmp = 0.0;
	if (z <= -980000000.0)
		tmp = t_0;
	elseif (z <= -1.25e-206)
		tmp = t_1;
	elseif (z <= -3.7e-255)
		tmp = abs((x / y_m));
	elseif (z <= -2.85e-299)
		tmp = t_1;
	elseif (z <= 5.2e-203)
		tmp = abs((x * (1.0 / y_m)));
	elseif (z <= 2.5e-9)
		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 / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -980000000.0], t$95$0, If[LessEqual[z, -1.25e-206], t$95$1, If[LessEqual[z, -3.7e-255], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -2.85e-299], t$95$1, If[LessEqual[z, 5.2e-203], N[Abs[N[(x * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.5e-9], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9.8e8 or 2.5000000000000001e-9 < z

   1. Initial program 87.6%

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

   3. Taylor expanded in z around inf 72.2%

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

      1. mul-1-neg 72.2%

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

      2. associate-*l/ 77.8%

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

      3. distribute-rgt-neg-out 77.8%

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

   5. Simplified 77.8%

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

      1. add-sqr-sqrt 40.5%

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

      2. sqrt-unprod 55.0%

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

      3. sqr-neg 55.0%

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

      4. sqrt-unprod 37.2%

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

      5. add-sqr-sqrt 77.8%

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

      6. associate-/r/ 77.3%

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

   7. Applied egg-rr 77.3%

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

   if -9.8e8 < z < -1.25e-206 or -3.7000000000000002e-255 < z < -2.84999999999999993e-299 or 5.19999999999999951e-203 < z < 2.5000000000000001e-9

   1. Initial program 97.7%

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

   3. Taylor expanded in x around 0 71.7%

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

   if -1.25e-206 < z < -3.7000000000000002e-255

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sub-div 100.0%

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

      3. clear-num 99.5%

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

      4. associate--l+ 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in z around 0 99.5%

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

   6. Taylor expanded in x around inf 90.7%

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

   if -2.84999999999999993e-299 < z < 5.19999999999999951e-203

   1. Initial program 82.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 67.7%

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

   6. Taylor expanded in z around 0 67.7%

      \[\leadsto \left|x \cdot \color{blue}{\frac{-1}{y}}\right| \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -980000000:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-206}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-255}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{-299}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-203}:\\ \;\;\;\;\left|x \cdot \frac{1}{y}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{4}{y_m}\right|\\ \mathbf{if}\;z \leq -120000000:\\ \;\;\;\;\left|\frac{z}{\frac{y_m}{x}}\right|\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-208}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-254}:\\ \;\;\;\;\left|\frac{x}{y_m}\right|\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-299}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-205}:\\ \;\;\;\;\left|x \cdot \frac{1}{y_m}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ 4.0 y_m))))
   (if (<= z -120000000.0)
     (fabs (/ z (/ y_m x)))
     (if (<= z -7.6e-208)
       t_0
       (if (<= z -1.7e-254)
         (fabs (/ x y_m))
         (if (<= z -2.6e-299)
           t_0
           (if (<= z 2.8e-205)
             (fabs (* x (/ 1.0 y_m)))
             (if (<= z 2.5e-9) t_0 (fabs (/ x (/ y_m z)))))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((4.0 / y_m));
	double tmp;
	if (z <= -120000000.0) {
		tmp = fabs((z / (y_m / x)));
	} else if (z <= -7.6e-208) {
		tmp = t_0;
	} else if (z <= -1.7e-254) {
		tmp = fabs((x / y_m));
	} else if (z <= -2.6e-299) {
		tmp = t_0;
	} else if (z <= 2.8e-205) {
		tmp = fabs((x * (1.0 / y_m)));
	} else if (z <= 2.5e-9) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = abs((4.0d0 / y_m))
    if (z <= (-120000000.0d0)) then
        tmp = abs((z / (y_m / x)))
    else if (z <= (-7.6d-208)) then
        tmp = t_0
    else if (z <= (-1.7d-254)) then
        tmp = abs((x / y_m))
    else if (z <= (-2.6d-299)) then
        tmp = t_0
    else if (z <= 2.8d-205) then
        tmp = abs((x * (1.0d0 / y_m)))
    else if (z <= 2.5d-9) then
        tmp = t_0
    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 t_0 = Math.abs((4.0 / y_m));
	double tmp;
	if (z <= -120000000.0) {
		tmp = Math.abs((z / (y_m / x)));
	} else if (z <= -7.6e-208) {
		tmp = t_0;
	} else if (z <= -1.7e-254) {
		tmp = Math.abs((x / y_m));
	} else if (z <= -2.6e-299) {
		tmp = t_0;
	} else if (z <= 2.8e-205) {
		tmp = Math.abs((x * (1.0 / y_m)));
	} else if (z <= 2.5e-9) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x / (y_m / z)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((4.0 / y_m))
	tmp = 0
	if z <= -120000000.0:
		tmp = math.fabs((z / (y_m / x)))
	elif z <= -7.6e-208:
		tmp = t_0
	elif z <= -1.7e-254:
		tmp = math.fabs((x / y_m))
	elif z <= -2.6e-299:
		tmp = t_0
	elif z <= 2.8e-205:
		tmp = math.fabs((x * (1.0 / y_m)))
	elif z <= 2.5e-9:
		tmp = t_0
	else:
		tmp = math.fabs((x / (y_m / z)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(4.0 / y_m))
	tmp = 0.0
	if (z <= -120000000.0)
		tmp = abs(Float64(z / Float64(y_m / x)));
	elseif (z <= -7.6e-208)
		tmp = t_0;
	elseif (z <= -1.7e-254)
		tmp = abs(Float64(x / y_m));
	elseif (z <= -2.6e-299)
		tmp = t_0;
	elseif (z <= 2.8e-205)
		tmp = abs(Float64(x * Float64(1.0 / y_m)));
	elseif (z <= 2.5e-9)
		tmp = t_0;
	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)
	t_0 = abs((4.0 / y_m));
	tmp = 0.0;
	if (z <= -120000000.0)
		tmp = abs((z / (y_m / x)));
	elseif (z <= -7.6e-208)
		tmp = t_0;
	elseif (z <= -1.7e-254)
		tmp = abs((x / y_m));
	elseif (z <= -2.6e-299)
		tmp = t_0;
	elseif (z <= 2.8e-205)
		tmp = abs((x * (1.0 / y_m)));
	elseif (z <= 2.5e-9)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -120000000.0], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -7.6e-208], t$95$0, If[LessEqual[z, -1.7e-254], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, -2.6e-299], t$95$0, If[LessEqual[z, 2.8e-205], N[Abs[N[(x * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.5e-9], t$95$0, N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.2e8

   1. Initial program 96.8%

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

   3. Taylor expanded in z around inf 75.8%

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

      1. mul-1-neg 75.8%

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

      2. associate-*l/ 83.0%

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

      3. distribute-rgt-neg-out 83.0%

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

   5. Simplified 83.0%

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

      1. clear-num 83.0%

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

      2. associate-*l/ 83.1%

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

      3. *-un-lft-identity 83.1%

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

      4. add-sqr-sqrt 82.9%

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

      5. sqrt-unprod 63.6%

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

      6. sqr-neg 63.6%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 83.1%

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

   7. Applied egg-rr 83.1%

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

   if -1.2e8 < z < -7.60000000000000023e-208 or -1.69999999999999996e-254 < z < -2.5999999999999999e-299 or 2.79999999999999991e-205 < z < 2.5000000000000001e-9

   1. Initial program 97.7%

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

   3. Taylor expanded in x around 0 71.7%

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

   if -7.60000000000000023e-208 < z < -1.69999999999999996e-254

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sub-div 100.0%

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

      3. clear-num 99.5%

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

      4. associate--l+ 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in z around 0 99.5%

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

   6. Taylor expanded in x around inf 90.7%

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

   if -2.5999999999999999e-299 < z < 2.79999999999999991e-205

   1. Initial program 82.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 67.7%

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

   6. Taylor expanded in z around 0 67.7%

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

   if 2.5000000000000001e-9 < z

   1. Initial program 78.8%

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

   3. Taylor expanded in z around inf 68.7%

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

      1. mul-1-neg 68.7%

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

      2. associate-*l/ 72.8%

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

      3. distribute-rgt-neg-out 72.8%

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

   5. Simplified 72.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 46.9%

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

      3. sqr-neg 46.9%

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

      4. sqrt-unprod 72.6%

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

      5. add-sqr-sqrt 72.8%

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

      6. associate-/r/ 74.7%

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

   7. Applied egg-rr 74.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -120000000:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-208}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-254}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-299}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-205}:\\ \;\;\;\;\left|x \cdot \frac{1}{y}\right|\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+29}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y_m}{x}}\right|\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+32}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(\frac{1}{y_m} - \frac{z}{y_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -4e+29)
   (fabs (/ (+ z -1.0) (/ y_m x)))
   (if (<= x 5e+32)
     (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
     (fabs (* x (- (/ 1.0 y_m) (/ z y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -4e+29) {
		tmp = fabs(((z + -1.0) / (y_m / x)));
	} else if (x <= 5e+32) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs((x * ((1.0 / y_m) - (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+29)) then
        tmp = abs(((z + (-1.0d0)) / (y_m / x)))
    else if (x <= 5d+32) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else
        tmp = abs((x * ((1.0d0 / y_m) - (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+29) {
		tmp = Math.abs(((z + -1.0) / (y_m / x)));
	} else if (x <= 5e+32) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((x * ((1.0 / y_m) - (z / y_m))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -4e+29:
		tmp = math.fabs(((z + -1.0) / (y_m / x)))
	elif x <= 5e+32:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	else:
		tmp = math.fabs((x * ((1.0 / y_m) - (z / y_m))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -4e+29)
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y_m / x)));
	elseif (x <= 5e+32)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(x * Float64(Float64(1.0 / y_m) - Float64(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+29)
		tmp = abs(((z + -1.0) / (y_m / x)));
	elseif (x <= 5e+32)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	else
		tmp = abs((x * ((1.0 / y_m) - (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+29], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5e+32], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(1.0 / y$95$m), $MachinePrecision] - N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999966e29

   1. Initial program 83.2%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in x around inf 99.6%

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

   6. Taylor expanded in y around 0 88.5%

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

      1. sub-neg 88.5%

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

      2. metadata-eval 88.5%

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

      3. *-commutative 88.5%

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

      4. associate-/l* 99.8%

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

   8. Simplified 99.8%

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

   if -3.99999999999999966e29 < x < 4.9999999999999997e32

   1. Initial program 96.5%

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

   3. Taylor expanded in y around 0 99.9%

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

   if 4.9999999999999997e32 < x

   1. Initial program 85.8%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in x around inf 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+29}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+32}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -18000000000:\\ \;\;\;\;\left|\frac{z}{\frac{y_m}{x}}\right|\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+117}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -18000000000.0)
   (fabs (/ z (/ y_m x)))
   (if (<= z 4.8e+117) (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 <= -18000000000.0) {
		tmp = fabs((z / (y_m / x)));
	} else if (z <= 4.8e+117) {
		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 <= (-18000000000.0d0)) then
        tmp = abs((z / (y_m / x)))
    else if (z <= 4.8d+117) 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 <= -18000000000.0) {
		tmp = Math.abs((z / (y_m / x)));
	} else if (z <= 4.8e+117) {
		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 <= -18000000000.0:
		tmp = math.fabs((z / (y_m / x)))
	elif z <= 4.8e+117:
		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 <= -18000000000.0)
		tmp = abs(Float64(z / Float64(y_m / x)));
	elseif (z <= 4.8e+117)
		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 <= -18000000000.0)
		tmp = abs((z / (y_m / x)));
	elseif (z <= 4.8e+117)
		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, -18000000000.0], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4.8e+117], 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 3 regimes

2. if z < -1.8e10

   1. Initial program 96.8%

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

   3. Taylor expanded in z around inf 75.8%

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

      1. mul-1-neg 75.8%

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

      2. associate-*l/ 83.0%

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

      3. distribute-rgt-neg-out 83.0%

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

   5. Simplified 83.0%

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

      1. clear-num 83.0%

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

      2. associate-*l/ 83.1%

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

      3. *-un-lft-identity 83.1%

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

      4. add-sqr-sqrt 82.9%

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

      5. sqrt-unprod 63.6%

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

      6. sqr-neg 63.6%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 83.1%

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

   7. Applied egg-rr 83.1%

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

   if -1.8e10 < z < 4.7999999999999998e117

   1. Initial program 90.7%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in z around 0 94.8%

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

      1. associate-*r/ 94.8%

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

      2. distribute-lft-in 94.8%

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

      3. metadata-eval 94.8%

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

      4. neg-mul-1 94.8%

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

      5. sub-neg 94.8%

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

   7. Simplified 94.8%

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

   if 4.7999999999999998e117 < z

   1. Initial program 83.2%

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

   3. Taylor expanded in z around inf 76.9%

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

      1. mul-1-neg 76.9%

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

      2. associate-*l/ 83.5%

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

      3. distribute-rgt-neg-out 83.5%

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

   5. Simplified 83.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 41.8%

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

      3. sqr-neg 41.8%

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

      4. sqrt-unprod 83.3%

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

      5. add-sqr-sqrt 83.5%

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

      6. associate-/r/ 86.4%

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

   7. Applied egg-rr 86.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -18000000000:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+117}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -24000000000:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y_m}{x}}\right|\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+119}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -24000000000.0)
   (fabs (/ (+ z -1.0) (/ y_m x)))
   (if (<= z 4.2e+119) (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 <= -24000000000.0) {
		tmp = fabs(((z + -1.0) / (y_m / x)));
	} else if (z <= 4.2e+119) {
		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 <= (-24000000000.0d0)) then
        tmp = abs(((z + (-1.0d0)) / (y_m / x)))
    else if (z <= 4.2d+119) 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 <= -24000000000.0) {
		tmp = Math.abs(((z + -1.0) / (y_m / x)));
	} else if (z <= 4.2e+119) {
		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 <= -24000000000.0:
		tmp = math.fabs(((z + -1.0) / (y_m / x)))
	elif z <= 4.2e+119:
		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 <= -24000000000.0)
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y_m / x)));
	elseif (z <= 4.2e+119)
		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 <= -24000000000.0)
		tmp = abs(((z + -1.0) / (y_m / x)));
	elseif (z <= 4.2e+119)
		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, -24000000000.0], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4.2e+119], 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 3 regimes

2. if z < -2.4e10

   1. Initial program 96.8%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in x around inf 80.9%

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

   6. Taylor expanded in y around 0 76.6%

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

      1. sub-neg 76.6%

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

      2. metadata-eval 76.6%

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

      3. *-commutative 76.6%

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

      4. associate-/l* 83.9%

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

   8. Simplified 83.9%

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

   if -2.4e10 < z < 4.19999999999999966e119

   1. Initial program 90.7%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in z around 0 94.8%

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

      1. associate-*r/ 94.8%

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

      2. distribute-lft-in 94.8%

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

      3. metadata-eval 94.8%

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

      4. neg-mul-1 94.8%

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

      5. sub-neg 94.8%

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

   7. Simplified 94.8%

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

   if 4.19999999999999966e119 < z

   1. Initial program 83.2%

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

   3. Taylor expanded in z around inf 76.9%

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

      1. mul-1-neg 76.9%

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

      2. associate-*l/ 83.5%

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

      3. distribute-rgt-neg-out 83.5%

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

   5. Simplified 83.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 41.8%

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

      3. sqr-neg 41.8%

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

      4. sqrt-unprod 83.3%

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

      5. add-sqr-sqrt 83.5%

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

      6. associate-/r/ 86.4%

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

   7. Applied egg-rr 86.4%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -24000000000:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+119}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.4% 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 85.3%

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

      1. associate-*l/ 80.4%

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

      2. sub-div 90.1%

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

      3. clear-num 89.9%

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

      4. associate--l+ 89.9%

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

   4. Applied egg-rr 89.9%

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

   5. Taylor expanded in z around 0 60.1%

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

   6. Taylor expanded in x around inf 59.6%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 0 72.3%

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

4. Final simplification 66.2%

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

Alternative 8: 40.9% 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 91.0%

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

3. Taylor expanded in x around 0 39.7%

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

4. Final simplification 39.7%

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

Reproduce

herbie shell --seed 2024019 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))