fabs fraction 1

Percentage Accurate: 92.2% → 99.3%

Time: 7.0s

Alternatives: 10

Speedup: 1.0×

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: 92.2% 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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 4e-84)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ (+ x 4.0) 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 <= 4e-84) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / 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 <= 4d-84) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / 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 <= 4e-84) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / 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 <= 4e-84:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / 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 <= 4e-84)
		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(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 <= 4e-84)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((((x + 4.0) / 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, 4e-84], 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[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.0000000000000001e-84

   1. Initial program 92.5%

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

      1. associate-*l/ 92.8%

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

      2. sub-div 96.5%

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

   4. Applied egg-rr 96.5%

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

   if 4.0000000000000001e-84 < y

   1. Initial program 94.5%

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

      1. associate-*l/ 95.9%

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

      2. associate-*r/ 99.9%

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

      3. clear-num 99.8%

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

      4. un-div-inv 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. Add Preprocessing

Alternative 2: 68.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{+260}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;x \leq -1.52:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+172} \lor \neg \left(x \leq 1.6 \cdot 10^{+217}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -7.6e+260)
     (fabs (/ x (/ y_m z)))
     (if (<= x -1.52)
       t_0
       (if (<= x 4.0)
         (fabs (/ 4.0 y_m))
         (if (or (<= x 2.25e+172) (not (<= x 1.6e+217)))
           t_0
           (fabs (* z (/ x y_m)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double tmp;
	if (x <= -7.6e+260) {
		tmp = fabs((x / (y_m / z)));
	} else if (x <= -1.52) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 2.25e+172) || !(x <= 1.6e+217)) {
		tmp = t_0;
	} else {
		tmp = fabs((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) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y_m))
    if (x <= (-7.6d+260)) then
        tmp = abs((x / (y_m / z)))
    else if (x <= (-1.52d0)) then
        tmp = t_0
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 2.25d+172) .or. (.not. (x <= 1.6d+217))) then
        tmp = t_0
    else
        tmp = abs((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 t_0 = Math.abs((x / y_m));
	double tmp;
	if (x <= -7.6e+260) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (x <= -1.52) {
		tmp = t_0;
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 2.25e+172) || !(x <= 1.6e+217)) {
		tmp = t_0;
	} else {
		tmp = Math.abs((z * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	tmp = 0
	if x <= -7.6e+260:
		tmp = math.fabs((x / (y_m / z)))
	elif x <= -1.52:
		tmp = t_0
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 2.25e+172) or not (x <= 1.6e+217):
		tmp = t_0
	else:
		tmp = math.fabs((z * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -7.6e+260)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (x <= -1.52)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 2.25e+172) || !(x <= 1.6e+217))
		tmp = t_0;
	else
		tmp = abs(Float64(z * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -7.6e+260)
		tmp = abs((x / (y_m / z)));
	elseif (x <= -1.52)
		tmp = t_0;
	elseif (x <= 4.0)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 2.25e+172) || ~((x <= 1.6e+217)))
		tmp = t_0;
	else
		tmp = abs((z * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -7.6e+260], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -1.52], t$95$0, If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 2.25e+172], N[Not[LessEqual[x, 1.6e+217]], $MachinePrecision]], t$95$0, N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.5999999999999995e260

   1. Initial program 99.8%

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

   3. Simplified 60.2%

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

   5. Taylor expanded in z around inf 46.7%

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

      1. mul-1-neg 46.7%

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

      2. distribute-frac-neg2 46.7%

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

      3. associate-/l* 86.3%

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

   7. Simplified 86.3%

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

      1. clear-num 86.3%

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

      2. un-div-inv 86.5%

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

      3. add-sqr-sqrt 29.1%

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

      4. sqrt-unprod 58.5%

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

      5. sqr-neg 58.5%

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

      6. sqrt-unprod 56.7%

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

      7. add-sqr-sqrt 86.5%

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

   9. Applied egg-rr 86.5%

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

   if -7.5999999999999995e260 < x < -1.52 or 4 < x < 2.2500000000000001e172 or 1.6e217 < x

   1. Initial program 93.1%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in x around inf 94.0%

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

      1. mul-1-neg 94.0%

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

      2. *-commutative 94.0%

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

      3. associate-/l* 99.4%

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

      4. distribute-lft-neg-in 99.4%

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

      5. neg-sub0 99.4%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 99.4%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 99.4%

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

      8. +-commutative 99.4%

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

      9. unsub-neg 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in z around 0 73.4%

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

   if -1.52 < x < 4

   1. Initial program 93.2%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 75.5%

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

   if 2.2500000000000001e172 < x < 1.6e217

   1. Initial program 88.7%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in z around inf 78.7%

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

      1. mul-1-neg 78.7%

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

      2. distribute-frac-neg2 78.7%

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

      3. associate-/l* 89.4%

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

   7. Simplified 89.4%

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

      1. clear-num 89.2%

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

      2. un-div-inv 89.2%

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

      3. add-sqr-sqrt 55.8%

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

      4. sqrt-unprod 89.1%

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

      5. sqr-neg 89.1%

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

      6. sqrt-unprod 33.3%

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

      7. add-sqr-sqrt 89.2%

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

   9. Applied egg-rr 89.2%

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

       1. associate-/r/ 99.8%

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

       2. *-commutative 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+260}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq -1.52:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+172} \lor \neg \left(x \leq 1.6 \cdot 10^{+217}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y\_m}\right|\\ t_1 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+259}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.52:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+172} \lor \neg \left(x \leq 8.2 \cdot 10^{+216}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -4.8e+259)
     t_0
     (if (<= x -1.52)
       t_1
       (if (<= x 4.0)
         (fabs (/ 4.0 y_m))
         (if (or (<= x 6.2e+172) (not (<= x 8.2e+216))) t_1 t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -4.8e+259) {
		tmp = t_0;
	} else if (x <= -1.52) {
		tmp = t_1;
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 6.2e+172) || !(x <= 8.2e+216)) {
		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((z * (x / y_m)))
    t_1 = abs((x / y_m))
    if (x <= (-4.8d+259)) then
        tmp = t_0
    else if (x <= (-1.52d0)) then
        tmp = t_1
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 6.2d+172) .or. (.not. (x <= 8.2d+216))) 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((z * (x / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -4.8e+259) {
		tmp = t_0;
	} else if (x <= -1.52) {
		tmp = t_1;
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 6.2e+172) || !(x <= 8.2e+216)) {
		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((z * (x / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -4.8e+259:
		tmp = t_0
	elif x <= -1.52:
		tmp = t_1
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 6.2e+172) or not (x <= 8.2e+216):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -4.8e+259)
		tmp = t_0;
	elseif (x <= -1.52)
		tmp = t_1;
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 6.2e+172) || !(x <= 8.2e+216))
		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((z * (x / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -4.8e+259)
		tmp = t_0;
	elseif (x <= -1.52)
		tmp = t_1;
	elseif (x <= 4.0)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 6.2e+172) || ~((x <= 8.2e+216)))
		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[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4.8e+259], t$95$0, If[LessEqual[x, -1.52], t$95$1, If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 6.2e+172], N[Not[LessEqual[x, 8.2e+216]], $MachinePrecision]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.7999999999999999e259 or 6.19999999999999976e172 < x < 8.1999999999999995e216

   1. Initial program 93.6%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in z around inf 64.7%

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

      1. mul-1-neg 64.7%

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

      2. distribute-frac-neg2 64.7%

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

      3. associate-/l* 88.0%

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

   7. Simplified 88.0%

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

      1. clear-num 87.9%

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

      2. un-div-inv 88.0%

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

      3. add-sqr-sqrt 44.1%

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

      4. sqrt-unprod 75.7%

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

      5. sqr-neg 75.7%

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

      6. sqrt-unprod 43.6%

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

      7. add-sqr-sqrt 88.0%

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

   9. Applied egg-rr 88.0%

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

       1. associate-/r/ 93.9%

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

       2. *-commutative 93.9%

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

   11. Simplified 93.9%

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

   if -4.7999999999999999e259 < x < -1.52 or 4 < x < 6.19999999999999976e172 or 8.1999999999999995e216 < x

   1. Initial program 93.1%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in x around inf 94.0%

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

      1. mul-1-neg 94.0%

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

      2. *-commutative 94.0%

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

      3. associate-/l* 99.4%

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

      4. distribute-lft-neg-in 99.4%

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

      5. neg-sub0 99.4%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 99.4%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 99.4%

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

      8. +-commutative 99.4%

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

      9. unsub-neg 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in z around 0 73.4%

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

   if -1.52 < x < 4

   1. Initial program 93.2%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 75.5%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+259}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.52:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+172} \lor \neg \left(x \leq 8.2 \cdot 10^{+216}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+43}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{+141}:\\ \;\;\;\;\left|\frac{x}{y\_m} + 4 \cdot \frac{1}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -6.5e+43)
   (fabs (/ x (/ y_m z)))
   (if (<= z 3.05e+141)
     (fabs (+ (/ x y_m) (* 4.0 (/ 1.0 y_m))))
     (fabs (* x (/ (- 1.0 z) y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -6.5e+43) {
		tmp = fabs((x / (y_m / z)));
	} else if (z <= 3.05e+141) {
		tmp = fabs(((x / y_m) + (4.0 * (1.0 / y_m))));
	} else {
		tmp = fabs((x * ((1.0 - 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 (z <= (-6.5d+43)) then
        tmp = abs((x / (y_m / z)))
    else if (z <= 3.05d+141) then
        tmp = abs(((x / y_m) + (4.0d0 * (1.0d0 / y_m))))
    else
        tmp = abs((x * ((1.0d0 - 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 (z <= -6.5e+43) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (z <= 3.05e+141) {
		tmp = Math.abs(((x / y_m) + (4.0 * (1.0 / y_m))));
	} else {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -6.5e+43:
		tmp = math.fabs((x / (y_m / z)))
	elif z <= 3.05e+141:
		tmp = math.fabs(((x / y_m) + (4.0 * (1.0 / y_m))))
	else:
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -6.5e+43)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (z <= 3.05e+141)
		tmp = abs(Float64(Float64(x / y_m) + Float64(4.0 * Float64(1.0 / y_m))));
	else
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -6.5e+43)
		tmp = abs((x / (y_m / z)));
	elseif (z <= 3.05e+141)
		tmp = abs(((x / y_m) + (4.0 * (1.0 / y_m))));
	else
		tmp = abs((x * ((1.0 - z) / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -6.5e+43], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.05e+141], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] + N[(4.0 * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.4999999999999998e43

   1. Initial program 89.0%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in z around inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. distribute-frac-neg2 75.9%

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

      3. associate-/l* 79.6%

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

   7. Simplified 79.6%

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

      1. clear-num 79.5%

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

      2. un-div-inv 80.4%

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

      3. add-sqr-sqrt 37.9%

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

      4. sqrt-unprod 53.6%

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

      5. sqr-neg 53.6%

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

      6. sqrt-unprod 42.3%

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

      7. add-sqr-sqrt 80.4%

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

   9. Applied egg-rr 80.4%

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

   if -6.4999999999999998e43 < z < 3.04999999999999996e141

   1. Initial program 98.2%

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

   3. Taylor expanded in z around 0 93.2%

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

   if 3.04999999999999996e141 < z

   1. Initial program 74.5%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in x around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. associate-/l* 76.7%

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

      3. associate-*r* 76.7%

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

      4. *-commutative 76.7%

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

      5. associate-*r/ 76.7%

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

      6. mul-1-neg 76.7%

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

      7. neg-sub0 76.7%

         \[\leadsto \left|x \cdot \frac{\color{blue}{0 - \left(z - 1\right)}}{y}\right| \]

      8. associate-+l- 76.7%

         \[\leadsto \left|x \cdot \frac{\color{blue}{\left(0 - z\right) + 1}}{y}\right| \]

      9. neg-sub0 76.7%

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

      10. +-commutative 76.7%

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

      11. unsub-neg 76.7%

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

   7. Simplified 76.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+43}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{+141}:\\ \;\;\;\;\left|\frac{x}{y} + 4 \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+43}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+141}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -7.8e+43)
   (fabs (/ x (/ y_m z)))
   (if (<= z 2.4e+141)
     (fabs (/ (- -4.0 x) y_m))
     (fabs (* x (/ (- 1.0 z) y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -7.8e+43) {
		tmp = fabs((x / (y_m / z)));
	} else if (z <= 2.4e+141) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((x * ((1.0 - 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 (z <= (-7.8d+43)) then
        tmp = abs((x / (y_m / z)))
    else if (z <= 2.4d+141) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((x * ((1.0d0 - 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 (z <= -7.8e+43) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (z <= 2.4e+141) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -7.8e+43:
		tmp = math.fabs((x / (y_m / z)))
	elif z <= 2.4e+141:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -7.8e+43)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (z <= 2.4e+141)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -7.8e+43)
		tmp = abs((x / (y_m / z)));
	elseif (z <= 2.4e+141)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x * ((1.0 - z) / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -7.8e+43], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.4e+141], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.8000000000000001e43

   1. Initial program 89.0%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in z around inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. distribute-frac-neg2 75.9%

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

      3. associate-/l* 79.6%

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

   7. Simplified 79.6%

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

      1. clear-num 79.5%

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

      2. un-div-inv 80.4%

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

      3. add-sqr-sqrt 37.9%

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

      4. sqrt-unprod 53.6%

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

      5. sqr-neg 53.6%

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

      6. sqrt-unprod 42.3%

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

      7. add-sqr-sqrt 80.4%

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

   9. Applied egg-rr 80.4%

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

   if -7.8000000000000001e43 < z < 2.39999999999999997e141

   1. Initial program 98.2%

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

   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 0 93.2%

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

      1. +-commutative 93.2%

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

      2. rem-square-sqrt 48.0%

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

      3. fabs-sqr 48.0%

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

      4. rem-square-sqrt 93.2%

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

      5. fabs-neg 93.2%

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

      6. distribute-neg-frac 93.2%

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

      7. distribute-neg-in 93.2%

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

      8. metadata-eval 93.2%

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

      9. +-commutative 93.2%

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

      10. sub-neg 93.2%

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

      11. rem-square-sqrt 44.5%

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

      12. fabs-sqr 44.5%

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

      13. rem-square-sqrt 93.2%

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

   7. Simplified 93.2%

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

   if 2.39999999999999997e141 < z

   1. Initial program 74.5%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in x around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. associate-/l* 76.7%

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

      3. associate-*r* 76.7%

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

      4. *-commutative 76.7%

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

      5. associate-*r/ 76.7%

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

      6. mul-1-neg 76.7%

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

      7. neg-sub0 76.7%

         \[\leadsto \left|x \cdot \frac{\color{blue}{0 - \left(z - 1\right)}}{y}\right| \]

      8. associate-+l- 76.7%

         \[\leadsto \left|x \cdot \frac{\color{blue}{\left(0 - z\right) + 1}}{y}\right| \]

      9. neg-sub0 76.7%

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

      10. +-commutative 76.7%

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

      11. unsub-neg 76.7%

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

   7. Simplified 76.7%

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

Alternative 6: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - x \cdot \frac{z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e-47)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (* x (/ z y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2e-47) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (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 (y_m <= 2d-47) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (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 (y_m <= 2e-47) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (x * (z / y_m))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 2e-47:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (x * (z / y_m))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 2e-47)
		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(x * 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 (y_m <= 2e-47)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((((x + 4.0) / y_m) - (x * (z / 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, 2e-47], 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[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e-47

   1. Initial program 93.0%

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

      1. associate-*l/ 93.3%

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

      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 1.9999999999999999e-47 < y

   1. Initial program 93.7%

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

      1. associate-*l/ 95.3%

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

      2. sub-div 95.3%

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

   4. Applied egg-rr 95.3%

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

      1. div-sub 95.3%

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

      2. associate-/l* 99.9%

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

   6. Applied egg-rr 99.9%

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

Alternative 7: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+44}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+142}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -2.3e+44)
   (fabs (/ x (/ y_m z)))
   (if (<= z 1.15e+142) (fabs (/ (- -4.0 x) y_m)) (fabs (* x (/ z y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -2.3e+44) {
		tmp = fabs((x / (y_m / z)));
	} else if (z <= 1.15e+142) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((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 (z <= (-2.3d+44)) then
        tmp = abs((x / (y_m / z)))
    else if (z <= 1.15d+142) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((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 (z <= -2.3e+44) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (z <= 1.15e+142) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x * (z / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -2.3e+44:
		tmp = math.fabs((x / (y_m / z)))
	elif z <= 1.15e+142:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x * (z / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -2.3e+44)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (z <= 1.15e+142)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x * 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 (z <= -2.3e+44)
		tmp = abs((x / (y_m / z)));
	elseif (z <= 1.15e+142)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x * (z / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -2.3e+44], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.15e+142], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.30000000000000004e44

   1. Initial program 89.0%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in z around inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. distribute-frac-neg2 75.9%

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

      3. associate-/l* 79.6%

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

   7. Simplified 79.6%

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

      1. clear-num 79.5%

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

      2. un-div-inv 80.4%

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

      3. add-sqr-sqrt 37.9%

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

      4. sqrt-unprod 53.6%

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

      5. sqr-neg 53.6%

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

      6. sqrt-unprod 42.3%

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

      7. add-sqr-sqrt 80.4%

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

   9. Applied egg-rr 80.4%

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

   if -2.30000000000000004e44 < z < 1.15000000000000001e142

   1. Initial program 98.2%

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

   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 0 93.2%

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

      1. +-commutative 93.2%

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

      2. rem-square-sqrt 48.0%

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

      3. fabs-sqr 48.0%

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

      4. rem-square-sqrt 93.2%

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

      5. fabs-neg 93.2%

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

      6. distribute-neg-frac 93.2%

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

      7. distribute-neg-in 93.2%

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

      8. metadata-eval 93.2%

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

      9. +-commutative 93.2%

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

      10. sub-neg 93.2%

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

      11. rem-square-sqrt 44.5%

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

      12. fabs-sqr 44.5%

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

      13. rem-square-sqrt 93.2%

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

   7. Simplified 93.2%

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

   if 1.15000000000000001e142 < z

   1. Initial program 74.5%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in z around inf 72.7%

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

      1. associate-*r/ 72.7%

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

      2. neg-mul-1 72.7%

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

      3. distribute-rgt-neg-in 72.7%

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

   7. Simplified 72.7%

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

      1. distribute-rgt-neg-out 72.7%

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

      2. distribute-frac-neg 72.7%

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

      3. distribute-frac-neg2 72.7%

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

      4. associate-*r/ 76.7%

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

      5. *-commutative 76.7%

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

      6. add-sqr-sqrt 34.3%

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

      7. sqrt-unprod 53.7%

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

      8. sqr-neg 53.7%

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

      9. sqrt-unprod 42.1%

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

      10. add-sqr-sqrt 76.7%

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

   9. Applied egg-rr 76.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+44}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+142}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 6e+16)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (* (- 1.0 z) (/ x y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 6e+16) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 6d+16) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 6e+16) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 6e+16:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 6e+16)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 6e+16)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs(((1.0 - z) * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 6e+16], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6e16

   1. Initial program 92.5%

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

      1. associate-*l/ 95.8%

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

      2. sub-div 97.9%

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

   4. Applied egg-rr 97.9%

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

   if 6e16 < x

   1. Initial program 95.4%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in x around inf 91.4%

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

      1. mul-1-neg 91.4%

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

      2. *-commutative 91.4%

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

      3. associate-/l* 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. neg-sub0 100.0%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 100.0%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

Alternative 9: 68.9% 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 93.2%

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

   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 x around inf 91.7%

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

      1. mul-1-neg 91.7%

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

      2. *-commutative 91.7%

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

      3. associate-/l* 99.5%

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

      4. distribute-lft-neg-in 99.5%

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

      5. neg-sub0 99.5%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 99.5%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 99.5%

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

      8. +-commutative 99.5%

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

      9. unsub-neg 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in z around 0 66.7%

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

   if -1.52 < x < 4

   1. Initial program 93.2%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 75.5%

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

4. Final simplification 71.4%

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

Alternative 10: 40.2% 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 93.2%

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

3. Simplified 96.1%

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

5. Taylor expanded in x around 0 42.8%

   \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024092 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))