fabs fraction 1

Percentage Accurate: 91.6% → 99.9%

Time: 9.8s

Alternatives: 12

Speedup: 1.0×

• 20.7% 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: 91.6% 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

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3 \cdot 10^{+24}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - \frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 3e+24)
   (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 <= 3e+24) {
		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 <= 3d+24) 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 <= 3e+24) {
		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 <= 3e+24:
		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 <= 3e+24)
		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 <= 3e+24)
		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, 3e+24], 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 < 2.99999999999999995e24

   1. Initial program 90.0%

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

      1. associate-*l/ 91.8%

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

      2. sub-div 97.7%

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

   4. Applied egg-rr 97.7%

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

   if 2.99999999999999995e24 < y

   1. Initial program 96.6%

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

      1. associate-*l/ 92.4%

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

      2. associate-*r/ 99.7%

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

      3. clear-num 99.6%

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

      4. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+24}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ t_1 := \left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.52 \cdot 10^{+112}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-30}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* x (/ z y_m)))))
   (if (<= x -1.52e+112)
     t_0
     (if (<= x -5.5e-39)
       t_1
       (if (<= x 2.95e-30) (fabs (/ 4.0 y_m)) (if (<= x 3.4e+80) t_1 t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((x * (z / y_m)));
	double tmp;
	if (x <= -1.52e+112) {
		tmp = t_0;
	} else if (x <= -5.5e-39) {
		tmp = t_1;
	} else if (x <= 2.95e-30) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 3.4e+80) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y_m))
    t_1 = abs((x * (z / y_m)))
    if (x <= (-1.52d+112)) then
        tmp = t_0
    else if (x <= (-5.5d-39)) then
        tmp = t_1
    else if (x <= 2.95d-30) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 3.4d+80) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double t_1 = Math.abs((x * (z / y_m)));
	double tmp;
	if (x <= -1.52e+112) {
		tmp = t_0;
	} else if (x <= -5.5e-39) {
		tmp = t_1;
	} else if (x <= 2.95e-30) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 3.4e+80) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	t_1 = math.fabs((x * (z / y_m)))
	tmp = 0
	if x <= -1.52e+112:
		tmp = t_0
	elif x <= -5.5e-39:
		tmp = t_1
	elif x <= 2.95e-30:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 3.4e+80:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(x * Float64(z / y_m)))
	tmp = 0.0
	if (x <= -1.52e+112)
		tmp = t_0;
	elseif (x <= -5.5e-39)
		tmp = t_1;
	elseif (x <= 2.95e-30)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 3.4e+80)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	t_1 = abs((x * (z / y_m)));
	tmp = 0.0;
	if (x <= -1.52e+112)
		tmp = t_0;
	elseif (x <= -5.5e-39)
		tmp = t_1;
	elseif (x <= 2.95e-30)
		tmp = abs((4.0 / y_m));
	elseif (x <= 3.4e+80)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.52e+112], t$95$0, If[LessEqual[x, -5.5e-39], t$95$1, If[LessEqual[x, 2.95e-30], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.4e+80], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5199999999999999e112 or 3.39999999999999992e80 < x

   1. Initial program 82.3%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in x around inf 91.2%

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

      1. associate-/l* 99.7%

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

      2. div-sub 99.7%

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

      3. associate-*r* 99.7%

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

      4. neg-mul-1 99.7%

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

      5. *-commutative 99.7%

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

      6. div-sub 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

      9. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in z around 0 73.7%

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

   if -1.5199999999999999e112 < x < -5.50000000000000018e-39 or 2.9499999999999999e-30 < x < 3.39999999999999992e80

   1. Initial program 95.5%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in z around inf 64.6%

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

      1. associate-*r/ 64.6%

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

      2. neg-mul-1 64.6%

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

      3. distribute-lft-neg-in 64.6%

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

      4. *-commutative 64.6%

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

      5. associate-*r/ 67.2%

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

   7. Simplified 67.2%

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

      1. associate-*r/ 64.6%

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

      2. clear-num 64.5%

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

      3. *-commutative 64.5%

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

      4. add-sqr-sqrt 33.5%

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

      5. sqrt-unprod 64.5%

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

      6. sqr-neg 64.5%

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

      7. sqrt-unprod 30.8%

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

      8. add-sqr-sqrt 64.5%

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

   9. Applied egg-rr 64.5%

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

       1. associate-/r/ 64.4%

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

       2. associate-*l/ 64.6%

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

       3. *-lft-identity 64.6%

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

       4. associate-/l* 64.6%

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

   11. Simplified 64.6%

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

   if -5.50000000000000018e-39 < x < 2.9499999999999999e-30

   1. Initial program 96.0%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 84.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52 \cdot 10^{+112}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-39}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-30}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+80}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-35}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+80}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -1.75e+113)
     t_0
     (if (<= x -5.6e-35)
       (fabs (/ z (/ y_m x)))
       (if (<= x 2.15e-23)
         (fabs (/ 4.0 y_m))
         (if (<= x 1.06e+80) (fabs (* x (/ z y_m))) t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double tmp;
	if (x <= -1.75e+113) {
		tmp = t_0;
	} else if (x <= -5.6e-35) {
		tmp = fabs((z / (y_m / x)));
	} else if (x <= 2.15e-23) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 1.06e+80) {
		tmp = fabs((x * (z / y_m)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y_m))
    if (x <= (-1.75d+113)) then
        tmp = t_0
    else if (x <= (-5.6d-35)) then
        tmp = abs((z / (y_m / x)))
    else if (x <= 2.15d-23) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 1.06d+80) then
        tmp = abs((x * (z / y_m)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double tmp;
	if (x <= -1.75e+113) {
		tmp = t_0;
	} else if (x <= -5.6e-35) {
		tmp = Math.abs((z / (y_m / x)));
	} else if (x <= 2.15e-23) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 1.06e+80) {
		tmp = Math.abs((x * (z / y_m)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	tmp = 0
	if x <= -1.75e+113:
		tmp = t_0
	elif x <= -5.6e-35:
		tmp = math.fabs((z / (y_m / x)))
	elif x <= 2.15e-23:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 1.06e+80:
		tmp = math.fabs((x * (z / y_m)))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -1.75e+113)
		tmp = t_0;
	elseif (x <= -5.6e-35)
		tmp = abs(Float64(z / Float64(y_m / x)));
	elseif (x <= 2.15e-23)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 1.06e+80)
		tmp = abs(Float64(x * Float64(z / y_m)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -1.75e+113)
		tmp = t_0;
	elseif (x <= -5.6e-35)
		tmp = abs((z / (y_m / x)));
	elseif (x <= 2.15e-23)
		tmp = abs((4.0 / y_m));
	elseif (x <= 1.06e+80)
		tmp = abs((x * (z / y_m)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.75e+113], t$95$0, If[LessEqual[x, -5.6e-35], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.15e-23], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.06e+80], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.75e113 or 1.05999999999999996e80 < x

   1. Initial program 82.3%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in x around inf 91.2%

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

      1. associate-/l* 99.7%

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

      2. div-sub 99.7%

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

      3. associate-*r* 99.7%

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

      4. neg-mul-1 99.7%

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

      5. *-commutative 99.7%

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

      6. div-sub 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

      9. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in z around 0 73.7%

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

   if -1.75e113 < x < -5.5999999999999999e-35

   1. Initial program 99.8%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in z around inf 58.2%

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

      1. associate-*r/ 58.2%

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

      2. neg-mul-1 58.2%

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

      3. distribute-lft-neg-in 58.2%

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

      4. *-commutative 58.2%

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

      5. associate-*r/ 63.9%

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

   7. Simplified 63.9%

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

      1. clear-num 63.8%

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

      2. un-div-inv 63.9%

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

      3. add-sqr-sqrt 63.7%

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

      4. sqrt-unprod 63.9%

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

      5. sqr-neg 63.9%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 63.9%

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

   9. Applied egg-rr 63.9%

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

   if -5.5999999999999999e-35 < x < 2.15000000000000001e-23

   1. Initial program 96.0%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 84.0%

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

   if 2.15000000000000001e-23 < x < 1.05999999999999996e80

   1. Initial program 89.7%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in z around inf 73.4%

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

      1. associate-*r/ 73.4%

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

      2. neg-mul-1 73.4%

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

      3. distribute-lft-neg-in 73.4%

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

      4. *-commutative 73.4%

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

      5. associate-*r/ 71.7%

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

   7. Simplified 71.7%

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

      1. associate-*r/ 73.4%

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

      2. clear-num 73.3%

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

      3. *-commutative 73.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 73.3%

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

      6. sqr-neg 73.3%

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

      7. sqrt-unprod 73.2%

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

      8. add-sqr-sqrt 73.3%

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

   9. Applied egg-rr 73.3%

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

       1. associate-/r/ 73.1%

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

       2. associate-*l/ 73.4%

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

       3. *-lft-identity 73.4%

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

       4. associate-/l* 69.6%

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

   11. Simplified 69.6%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+113}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-35}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+80}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+112}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-34}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+80}:\\ \;\;\;\;\left|\frac{x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))))
   (if (<= x -1.35e+112)
     t_0
     (if (<= x -1.05e-34)
       (fabs (/ z (/ y_m x)))
       (if (<= x 2e-23)
         (fabs (/ 4.0 y_m))
         (if (<= x 1.12e+80) (fabs (/ (* x z) y_m)) t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double tmp;
	if (x <= -1.35e+112) {
		tmp = t_0;
	} else if (x <= -1.05e-34) {
		tmp = fabs((z / (y_m / x)));
	} else if (x <= 2e-23) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 1.12e+80) {
		tmp = fabs(((x * z) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y_m))
    if (x <= (-1.35d+112)) then
        tmp = t_0
    else if (x <= (-1.05d-34)) then
        tmp = abs((z / (y_m / x)))
    else if (x <= 2d-23) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 1.12d+80) then
        tmp = abs(((x * z) / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double tmp;
	if (x <= -1.35e+112) {
		tmp = t_0;
	} else if (x <= -1.05e-34) {
		tmp = Math.abs((z / (y_m / x)));
	} else if (x <= 2e-23) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 1.12e+80) {
		tmp = Math.abs(((x * z) / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	tmp = 0
	if x <= -1.35e+112:
		tmp = t_0
	elif x <= -1.05e-34:
		tmp = math.fabs((z / (y_m / x)))
	elif x <= 2e-23:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 1.12e+80:
		tmp = math.fabs(((x * z) / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -1.35e+112)
		tmp = t_0;
	elseif (x <= -1.05e-34)
		tmp = abs(Float64(z / Float64(y_m / x)));
	elseif (x <= 2e-23)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 1.12e+80)
		tmp = abs(Float64(Float64(x * z) / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -1.35e+112)
		tmp = t_0;
	elseif (x <= -1.05e-34)
		tmp = abs((z / (y_m / x)));
	elseif (x <= 2e-23)
		tmp = abs((4.0 / y_m));
	elseif (x <= 1.12e+80)
		tmp = abs(((x * z) / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.35e+112], t$95$0, If[LessEqual[x, -1.05e-34], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2e-23], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.12e+80], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.3500000000000001e112 or 1.12e80 < x

   1. Initial program 82.3%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in x around inf 91.2%

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

      1. associate-/l* 99.7%

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

      2. div-sub 99.7%

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

      3. associate-*r* 99.7%

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

      4. neg-mul-1 99.7%

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

      5. *-commutative 99.7%

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

      6. div-sub 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

      9. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in z around 0 73.7%

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

   if -1.3500000000000001e112 < x < -1.05e-34

   1. Initial program 99.8%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in z around inf 58.2%

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

      1. associate-*r/ 58.2%

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

      2. neg-mul-1 58.2%

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

      3. distribute-lft-neg-in 58.2%

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

      4. *-commutative 58.2%

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

      5. associate-*r/ 63.9%

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

   7. Simplified 63.9%

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

      1. clear-num 63.8%

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

      2. un-div-inv 63.9%

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

      3. add-sqr-sqrt 63.7%

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

      4. sqrt-unprod 63.9%

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

      5. sqr-neg 63.9%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 63.9%

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

   9. Applied egg-rr 63.9%

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

   if -1.05e-34 < x < 1.99999999999999992e-23

   1. Initial program 96.0%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 84.0%

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

   if 1.99999999999999992e-23 < x < 1.12e80

   1. Initial program 89.7%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in z around inf 73.4%

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

      1. associate-*r/ 73.4%

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

      2. neg-mul-1 73.4%

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

      3. distribute-lft-neg-in 73.4%

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

      4. *-commutative 73.4%

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

      5. associate-*r/ 71.7%

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

   7. Simplified 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*l/ 73.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 73.4%

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

      5. sqr-neg 73.4%

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

      6. sqrt-unprod 73.1%

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

      7. add-sqr-sqrt 73.4%

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

   9. Applied egg-rr 73.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+112}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-34}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+80}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.36 \cdot 10^{-39} \lor \neg \left(x \leq 3 \cdot 10^{-28}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.36e-39) (not (<= x 3e-28)))
   (fabs (* x (/ (- 1.0 z) y_m)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.36e-39) || !(x <= 3e-28)) {
		tmp = fabs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.36d-39)) .or. (.not. (x <= 3d-28))) then
        tmp = abs((x * ((1.0d0 - z) / y_m)))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.36e-39) || !(x <= 3e-28)) {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.36e-39) or not (x <= 3e-28):
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.36e-39) || !(x <= 3e-28))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1.36e-39) || ~((x <= 3e-28)))
		tmp = abs((x * ((1.0 - z) / y_m)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1.36e-39], N[Not[LessEqual[x, 3e-28]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35999999999999994e-39 or 3.00000000000000003e-28 < x

   1. Initial program 87.6%

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

      1. associate-*l/ 85.5%

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

      2. sub-div 94.0%

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

   4. Applied egg-rr 94.0%

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

   5. Taylor expanded in x around inf 89.2%

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

      1. sub-neg 89.2%

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

      2. neg-mul-1 89.2%

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

      3. associate-/l* 94.3%

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

      4. neg-mul-1 94.3%

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

      5. sub-neg 94.3%

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

   7. Simplified 94.3%

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

   if -1.35999999999999994e-39 < x < 3.00000000000000003e-28

   1. Initial program 96.0%

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

      1. fabs-sub 96.0%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 93.4%

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

      4. fma-neg 93.4%

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

      5. distribute-neg-frac 93.4%

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

      6. +-commutative 93.4%

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

      7. distribute-neg-in 93.4%

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

      8. unsub-neg 93.4%

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

      9. metadata-eval 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in z around 0 84.0%

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

      1. associate-*r/ 84.0%

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

      2. distribute-lft-in 84.0%

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

      3. metadata-eval 84.0%

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

      4. neg-mul-1 84.0%

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

      5. sub-neg 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.36 \cdot 10^{-39} \lor \neg \left(x \leq 3 \cdot 10^{-28}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-41} \lor \neg \left(x \leq 7.2 \cdot 10^{-30}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -3.1e-41) (not (<= x 7.2e-30)))
   (fabs (* (- 1.0 z) (/ x y_m)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -3.1e-41) || !(x <= 7.2e-30)) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = fabs(((-4.0 - 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 <= (-3.1d-41)) .or. (.not. (x <= 7.2d-30))) then
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    else
        tmp = abs((((-4.0d0) - 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 <= -3.1e-41) || !(x <= 7.2e-30)) {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -3.1e-41) or not (x <= 7.2e-30):
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -3.1e-41) || !(x <= 7.2e-30))
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	else
		tmp = abs(Float64(Float64(-4.0 - 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 <= -3.1e-41) || ~((x <= 7.2e-30)))
		tmp = abs(((1.0 - z) * (x / y_m)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -3.1e-41], N[Not[LessEqual[x, 7.2e-30]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.10000000000000001e-41 or 7.2000000000000006e-30 < x

   1. Initial program 87.6%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in x around -inf 89.2%

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

      1. *-commutative 89.2%

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

      2. associate-/l* 94.8%

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

      3. neg-mul-1 94.8%

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

      4. unsub-neg 94.8%

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

   7. Simplified 94.8%

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

   if -3.10000000000000001e-41 < x < 7.2000000000000006e-30

   1. Initial program 96.0%

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

      1. fabs-sub 96.0%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 93.4%

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

      4. fma-neg 93.4%

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

      5. distribute-neg-frac 93.4%

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

      6. +-commutative 93.4%

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

      7. distribute-neg-in 93.4%

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

      8. unsub-neg 93.4%

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

      9. metadata-eval 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in z around 0 84.0%

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

      1. associate-*r/ 84.0%

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

      2. distribute-lft-in 84.0%

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

      3. metadata-eval 84.0%

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

      4. neg-mul-1 84.0%

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

      5. sub-neg 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-41} \lor \neg \left(x \leq 7.2 \cdot 10^{-30}\right):\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+46} \lor \neg \left(z \leq 2.9 \cdot 10^{+85}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= z -2.3e+46) (not (<= z 2.9e+85)))
   (fabs (/ x (/ y_m z)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((z <= -2.3e+46) || !(z <= 2.9e+85)) {
		tmp = fabs((x / (y_m / z)));
	} else {
		tmp = fabs(((-4.0 - 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 ((z <= (-2.3d+46)) .or. (.not. (z <= 2.9d+85))) then
        tmp = abs((x / (y_m / z)))
    else
        tmp = abs((((-4.0d0) - 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 ((z <= -2.3e+46) || !(z <= 2.9e+85)) {
		tmp = Math.abs((x / (y_m / z)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (z <= -2.3e+46) or not (z <= 2.9e+85):
		tmp = math.fabs((x / (y_m / z)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((z <= -2.3e+46) || !(z <= 2.9e+85))
		tmp = abs(Float64(x / Float64(y_m / z)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / 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+46) || ~((z <= 2.9e+85)))
		tmp = abs((x / (y_m / z)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[z, -2.3e+46], N[Not[LessEqual[z, 2.9e+85]], $MachinePrecision]], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3000000000000001e46 or 2.89999999999999997e85 < z

   1. Initial program 87.6%

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

   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 z around inf 72.4%

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

      1. mul-1-neg 72.4%

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

      2. associate-*r/ 77.2%

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

   7. Simplified 77.2%

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

      1. clear-num 77.1%

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

      2. un-div-inv 79.6%

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

   9. Applied egg-rr 79.6%

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

   if -2.3000000000000001e46 < z < 2.89999999999999997e85

   1. Initial program 93.3%

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

      1. fabs-sub 93.3%

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

      2. associate-*l/ 95.8%

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

      3. associate-*r/ 95.3%

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

      4. fma-neg 98.3%

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

      5. distribute-neg-frac 98.3%

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

      6. +-commutative 98.3%

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

      7. distribute-neg-in 98.3%

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

      8. unsub-neg 98.3%

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

      9. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in z around 0 92.0%

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

      1. associate-*r/ 92.0%

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

      2. distribute-lft-in 92.0%

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

      3. metadata-eval 92.0%

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

      4. neg-mul-1 92.0%

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

      5. sub-neg 92.0%

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

   7. Simplified 92.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+46} \lor \neg \left(z \leq 2.9 \cdot 10^{+85}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+50}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+99}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -4.5e+50)
   (fabs (/ z (/ y_m x)))
   (if (<= z 1.95e+99) (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 <= -4.5e+50) {
		tmp = fabs((z / (y_m / x)));
	} else if (z <= 1.95e+99) {
		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 <= (-4.5d+50)) then
        tmp = abs((z / (y_m / x)))
    else if (z <= 1.95d+99) 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 <= -4.5e+50) {
		tmp = Math.abs((z / (y_m / x)));
	} else if (z <= 1.95e+99) {
		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 <= -4.5e+50:
		tmp = math.fabs((z / (y_m / x)))
	elif z <= 1.95e+99:
		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 <= -4.5e+50)
		tmp = abs(Float64(z / Float64(y_m / x)));
	elseif (z <= 1.95e+99)
		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 <= -4.5e+50)
		tmp = abs((z / (y_m / x)));
	elseif (z <= 1.95e+99)
		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, -4.5e+50], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.95e+99], 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 < -4.50000000000000014e50

   1. Initial program 97.2%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in z around inf 74.9%

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

      1. associate-*r/ 74.9%

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

      2. neg-mul-1 74.9%

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

      3. distribute-lft-neg-in 74.9%

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

      4. *-commutative 74.9%

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

      5. associate-*r/ 77.8%

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

   7. Simplified 77.8%

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

      1. clear-num 76.5%

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

      2. un-div-inv 76.6%

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

      3. add-sqr-sqrt 40.9%

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

      4. sqrt-unprod 71.6%

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

      5. sqr-neg 71.6%

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

      6. sqrt-unprod 35.6%

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

      7. add-sqr-sqrt 76.6%

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

   9. Applied egg-rr 76.6%

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

   if -4.50000000000000014e50 < z < 1.94999999999999997e99

   1. Initial program 93.4%

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

      1. fabs-sub 93.4%

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

      2. associate-*l/ 95.8%

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

      3. associate-*r/ 95.3%

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

      4. fma-neg 98.3%

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

      5. distribute-neg-frac 98.3%

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

      6. +-commutative 98.3%

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

      7. distribute-neg-in 98.3%

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

      8. unsub-neg 98.3%

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

      9. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in z around 0 91.6%

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

      1. associate-*r/ 91.6%

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

      2. distribute-lft-in 91.6%

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

      3. metadata-eval 91.6%

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

      4. neg-mul-1 91.6%

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

      5. sub-neg 91.6%

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

   7. Simplified 91.6%

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

   if 1.94999999999999997e99 < z

   1. Initial program 76.7%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in z around inf 70.6%

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

      1. associate-*r/ 70.6%

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

      2. neg-mul-1 70.6%

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

      3. distribute-lft-neg-in 70.6%

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

      4. *-commutative 70.6%

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

      5. associate-*r/ 75.4%

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

   7. Simplified 75.4%

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

      1. associate-*r/ 70.6%

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

      2. clear-num 70.4%

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

      3. *-commutative 70.4%

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

      4. add-sqr-sqrt 39.3%

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

      5. sqrt-unprod 60.6%

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

      6. sqr-neg 60.6%

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

      7. sqrt-unprod 30.9%

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

      8. add-sqr-sqrt 70.4%

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

   9. Applied egg-rr 70.4%

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

       1. associate-/r/ 70.4%

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

       2. associate-*l/ 70.6%

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

       3. *-lft-identity 70.6%

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

       4. associate-/l* 81.4%

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

   11. Simplified 81.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+50}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+99}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+47}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+85}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y\_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -3.2e+47)
   (fabs (* z (/ x y_m)))
   (if (<= z 9.2e+85) (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 <= -3.2e+47) {
		tmp = fabs((z * (x / y_m)));
	} else if (z <= 9.2e+85) {
		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 <= (-3.2d+47)) then
        tmp = abs((z * (x / y_m)))
    else if (z <= 9.2d+85) 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 <= -3.2e+47) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (z <= 9.2e+85) {
		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 <= -3.2e+47:
		tmp = math.fabs((z * (x / y_m)))
	elif z <= 9.2e+85:
		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 <= -3.2e+47)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (z <= 9.2e+85)
		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 <= -3.2e+47)
		tmp = abs((z * (x / y_m)));
	elseif (z <= 9.2e+85)
		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, -3.2e+47], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 9.2e+85], 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 < -3.2e47

   1. Initial program 97.2%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in z around inf 74.9%

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

      1. associate-*r/ 74.9%

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

      2. neg-mul-1 74.9%

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

      3. distribute-lft-neg-in 74.9%

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

      4. *-commutative 74.9%

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

      5. associate-*r/ 77.8%

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

   7. Simplified 77.8%

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

   if -3.2e47 < z < 9.1999999999999996e85

   1. Initial program 93.3%

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

      1. fabs-sub 93.3%

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

      2. associate-*l/ 95.8%

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

      3. associate-*r/ 95.3%

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

      4. fma-neg 98.3%

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

      5. distribute-neg-frac 98.3%

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

      6. +-commutative 98.3%

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

      7. distribute-neg-in 98.3%

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

      8. unsub-neg 98.3%

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

      9. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in z around 0 92.0%

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

      1. associate-*r/ 92.0%

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

      2. distribute-lft-in 92.0%

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

      3. metadata-eval 92.0%

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

      4. neg-mul-1 92.0%

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

      5. sub-neg 92.0%

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

   7. Simplified 92.0%

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

   if 9.1999999999999996e85 < z

   1. Initial program 77.7%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in z around inf 69.8%

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

      1. mul-1-neg 69.8%

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

      2. associate-*r/ 80.1%

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

   7. Simplified 80.1%

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

      1. clear-num 80.0%

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

      2. un-div-inv 82.1%

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

   9. Applied egg-rr 82.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+47}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+85}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+55}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 1e+55)
   (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 <= 1e+55) {
		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 <= 1d+55) 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 <= 1e+55) {
		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 <= 1e+55:
		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 <= 1e+55)
		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 <= 1e+55)
		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, 1e+55], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000001e55

   1. Initial program 93.3%

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

      1. associate-*l/ 94.8%

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

      2. sub-div 98.1%

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

   4. Applied egg-rr 98.1%

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

   if 1.00000000000000001e55 < x

   1. Initial program 82.5%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in x around -inf 89.8%

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

      1. *-commutative 89.8%

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

      2. associate-/l* 99.9%

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

      3. neg-mul-1 99.9%

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

      4. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+55}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.45) (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 <= -1.45) || !(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 <= (-1.45d0)) .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 <= -1.45) || !(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 <= -1.45) 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 <= -1.45) || !(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 <= -1.45) || ~((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, -1.45], 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 < -1.44999999999999996 or 4 < x

   1. Initial program 86.3%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in x around inf 91.4%

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

      1. associate-/l* 97.9%

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

      2. div-sub 97.9%

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

      3. associate-*r* 97.9%

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

      4. neg-mul-1 97.9%

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

      5. *-commutative 97.9%

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

      6. div-sub 97.9%

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

      7. sub-neg 97.9%

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

      8. metadata-eval 97.9%

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

      9. +-commutative 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in z around 0 62.0%

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

   if -1.44999999999999996 < x < 4

   1. Initial program 96.2%

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

   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 76.6%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

3. Simplified 96.4%

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

5. Taylor expanded in x around 0 41.8%

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

6. Final simplification 41.8%

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

Reproduce

herbie shell --seed 2024050 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))