fabs fraction 1

Percentage Accurate: 91.4% → 99.6%

Time: 7.1s

Alternatives: 9

Speedup: 1.0×

• 21.7% 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: 91.4% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+109} \lor \neg \left(x \leq 3.6 \cdot 10^{+16}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.85e+109) (not (<= x 3.6e+16)))
   (fabs (/ x (/ y (- 1.0 z))))
   (fabs (/ (- (+ 4.0 x) (* x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.85e+109) || !(x <= 3.6e+16)) {
		tmp = fabs((x / (y / (1.0 - z))));
	} else {
		tmp = fabs((((4.0 + x) - (x * z)) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.85d+109)) .or. (.not. (x <= 3.6d+16))) then
        tmp = abs((x / (y / (1.0d0 - z))))
    else
        tmp = abs((((4.0d0 + x) - (x * z)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.85e+109) || !(x <= 3.6e+16)) {
		tmp = Math.abs((x / (y / (1.0 - z))));
	} else {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.85e+109) or not (x <= 3.6e+16):
		tmp = math.fabs((x / (y / (1.0 - z))))
	else:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.85e+109) || !(x <= 3.6e+16))
		tmp = abs(Float64(x / Float64(y / Float64(1.0 - z))));
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.85e+109) || ~((x <= 3.6e+16)))
		tmp = abs((x / (y / (1.0 - z))));
	else
		tmp = abs((((4.0 + x) - (x * z)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.85e+109], N[Not[LessEqual[x, 3.6e+16]], $MachinePrecision]], N[Abs[N[(x / N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8500000000000001e109 or 3.6e16 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in y around 0 90.5%

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

   4. Taylor expanded in x around inf 90.5%

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

      1. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

   if -1.8500000000000001e109 < x < 3.6e16

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+109} \lor \neg \left(x \leq 3.6 \cdot 10^{+16}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-99}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.3e-99)
   (fabs (/ (- (+ 4.0 x) (* x z)) y))
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.3e-99) {
		tmp = fabs((((4.0 + x) - (x * z)) / y));
	} else {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.3e-99)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y));
	else
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.3e-99], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.30000000000000003e-99

   1. Initial program 90.4%

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

   3. Taylor expanded in y around 0 96.5%

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

   if 1.30000000000000003e-99 < y

   1. Initial program 96.6%

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

   3. Simplified 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-99}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-20}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+38} \lor \neg \left(x \leq 1.5 \cdot 10^{+103}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))) (t_1 (fabs (/ x (/ y z)))))
   (if (<= x -1.05e+21)
     t_0
     (if (<= x -5.8e-12)
       t_1
       (if (<= x 2.15e-20)
         (fabs (/ 4.0 y))
         (if (or (<= x 1.35e+38) (not (<= x 1.5e+103))) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double t_1 = fabs((x / (y / z)));
	double tmp;
	if (x <= -1.05e+21) {
		tmp = t_0;
	} else if (x <= -5.8e-12) {
		tmp = t_1;
	} else if (x <= 2.15e-20) {
		tmp = fabs((4.0 / y));
	} else if ((x <= 1.35e+38) || !(x <= 1.5e+103)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y))
    t_1 = abs((x / (y / z)))
    if (x <= (-1.05d+21)) then
        tmp = t_0
    else if (x <= (-5.8d-12)) then
        tmp = t_1
    else if (x <= 2.15d-20) then
        tmp = abs((4.0d0 / y))
    else if ((x <= 1.35d+38) .or. (.not. (x <= 1.5d+103))) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double t_1 = Math.abs((x / (y / z)));
	double tmp;
	if (x <= -1.05e+21) {
		tmp = t_0;
	} else if (x <= -5.8e-12) {
		tmp = t_1;
	} else if (x <= 2.15e-20) {
		tmp = Math.abs((4.0 / y));
	} else if ((x <= 1.35e+38) || !(x <= 1.5e+103)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	t_1 = math.fabs((x / (y / z)))
	tmp = 0
	if x <= -1.05e+21:
		tmp = t_0
	elif x <= -5.8e-12:
		tmp = t_1
	elif x <= 2.15e-20:
		tmp = math.fabs((4.0 / y))
	elif (x <= 1.35e+38) or not (x <= 1.5e+103):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	t_1 = abs(Float64(x / Float64(y / z)))
	tmp = 0.0
	if (x <= -1.05e+21)
		tmp = t_0;
	elseif (x <= -5.8e-12)
		tmp = t_1;
	elseif (x <= 2.15e-20)
		tmp = abs(Float64(4.0 / y));
	elseif ((x <= 1.35e+38) || !(x <= 1.5e+103))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	t_1 = abs((x / (y / z)));
	tmp = 0.0;
	if (x <= -1.05e+21)
		tmp = t_0;
	elseif (x <= -5.8e-12)
		tmp = t_1;
	elseif (x <= 2.15e-20)
		tmp = abs((4.0 / y));
	elseif ((x <= 1.35e+38) || ~((x <= 1.5e+103)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.05e+21], t$95$0, If[LessEqual[x, -5.8e-12], t$95$1, If[LessEqual[x, 2.15e-20], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 1.35e+38], N[Not[LessEqual[x, 1.5e+103]], $MachinePrecision]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e21 or 1.34999999999999998e38 < x < 1.5e103

   1. Initial program 88.8%

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

   3. Taylor expanded in y around 0 90.8%

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

   4. Taylor expanded in x around inf 90.9%

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

      1. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in z around 0 70.0%

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

   if -1.05e21 < x < -5.8000000000000003e-12 or 2.15000000000000006e-20 < x < 1.34999999999999998e38 or 1.5e103 < x

   1. Initial program 89.8%

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

   3. Taylor expanded in z around inf 68.4%

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

      1. mul-1-neg 68.4%

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

      2. associate-*l/ 79.1%

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

      3. distribute-rgt-neg-out 79.1%

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

   5. Simplified 79.1%

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

      1. associate-*l/ 68.4%

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

      2. associate-/l* 73.7%

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

      3. add-sqr-sqrt 41.6%

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

      4. sqrt-unprod 59.8%

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

      5. sqr-neg 59.8%

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

      6. sqrt-unprod 32.0%

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

      7. add-sqr-sqrt 73.7%

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

   7. Applied egg-rr 73.7%

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

   if -5.8000000000000003e-12 < x < 2.15000000000000006e-20

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0 80.3%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-12}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-20}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+38} \lor \neg \left(x \leq 1.5 \cdot 10^{+103}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-20}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))) (t_1 (fabs (/ x (/ y z)))))
   (if (<= x -4.4e+27)
     t_0
     (if (<= x -4.5e-13)
       t_1
       (if (<= x 1.25e-20)
         (fabs (/ 4.0 y))
         (if (<= x 1.65e+38)
           t_1
           (if (<= x 6.5e+97) t_0 (fabs (/ z (/ y x))))))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((x / y));
	double t_1 = fabs((x / (y / z)));
	double tmp;
	if (x <= -4.4e+27) {
		tmp = t_0;
	} else if (x <= -4.5e-13) {
		tmp = t_1;
	} else if (x <= 1.25e-20) {
		tmp = fabs((4.0 / y));
	} else if (x <= 1.65e+38) {
		tmp = t_1;
	} else if (x <= 6.5e+97) {
		tmp = t_0;
	} else {
		tmp = fabs((z / (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y))
    t_1 = abs((x / (y / z)))
    if (x <= (-4.4d+27)) then
        tmp = t_0
    else if (x <= (-4.5d-13)) then
        tmp = t_1
    else if (x <= 1.25d-20) then
        tmp = abs((4.0d0 / y))
    else if (x <= 1.65d+38) then
        tmp = t_1
    else if (x <= 6.5d+97) then
        tmp = t_0
    else
        tmp = abs((z / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x / y));
	double t_1 = Math.abs((x / (y / z)));
	double tmp;
	if (x <= -4.4e+27) {
		tmp = t_0;
	} else if (x <= -4.5e-13) {
		tmp = t_1;
	} else if (x <= 1.25e-20) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 1.65e+38) {
		tmp = t_1;
	} else if (x <= 6.5e+97) {
		tmp = t_0;
	} else {
		tmp = Math.abs((z / (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((x / y))
	t_1 = math.fabs((x / (y / z)))
	tmp = 0
	if x <= -4.4e+27:
		tmp = t_0
	elif x <= -4.5e-13:
		tmp = t_1
	elif x <= 1.25e-20:
		tmp = math.fabs((4.0 / y))
	elif x <= 1.65e+38:
		tmp = t_1
	elif x <= 6.5e+97:
		tmp = t_0
	else:
		tmp = math.fabs((z / (y / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(x / y))
	t_1 = abs(Float64(x / Float64(y / z)))
	tmp = 0.0
	if (x <= -4.4e+27)
		tmp = t_0;
	elseif (x <= -4.5e-13)
		tmp = t_1;
	elseif (x <= 1.25e-20)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 1.65e+38)
		tmp = t_1;
	elseif (x <= 6.5e+97)
		tmp = t_0;
	else
		tmp = abs(Float64(z / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((x / y));
	t_1 = abs((x / (y / z)));
	tmp = 0.0;
	if (x <= -4.4e+27)
		tmp = t_0;
	elseif (x <= -4.5e-13)
		tmp = t_1;
	elseif (x <= 1.25e-20)
		tmp = abs((4.0 / y));
	elseif (x <= 1.65e+38)
		tmp = t_1;
	elseif (x <= 6.5e+97)
		tmp = t_0;
	else
		tmp = abs((z / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4.4e+27], t$95$0, If[LessEqual[x, -4.5e-13], t$95$1, If[LessEqual[x, 1.25e-20], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.65e+38], t$95$1, If[LessEqual[x, 6.5e+97], t$95$0, N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.3999999999999997e27 or 1.65e38 < x < 6.4999999999999999e97

   1. Initial program 88.8%

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

   3. Taylor expanded in y around 0 90.8%

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

   4. Taylor expanded in x around inf 90.9%

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

      1. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in z around 0 70.0%

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

   if -4.3999999999999997e27 < x < -4.5e-13 or 1.25e-20 < x < 1.65e38

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf 77.4%

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

      1. mul-1-neg 77.4%

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

      2. associate-*l/ 77.3%

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

      3. distribute-rgt-neg-out 77.3%

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

   5. Simplified 77.3%

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

      1. associate-*l/ 77.4%

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

      2. associate-/l* 77.5%

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

      3. add-sqr-sqrt 48.0%

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

      4. sqrt-unprod 54.5%

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

      5. sqr-neg 54.5%

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

      6. sqrt-unprod 29.5%

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

      7. add-sqr-sqrt 77.5%

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

   7. Applied egg-rr 77.5%

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

   if -4.5e-13 < x < 1.25e-20

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0 80.3%

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

   if 6.4999999999999999e97 < x

   1. Initial program 85.6%

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

   3. Taylor expanded in z around inf 64.5%

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

      1. mul-1-neg 64.5%

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

      2. associate-*l/ 79.8%

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

      3. distribute-rgt-neg-out 79.8%

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

   5. Simplified 79.8%

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

      1. add-sqr-sqrt 44.8%

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

      2. sqrt-unprod 63.8%

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

      3. sqr-neg 63.8%

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

      4. sqrt-unprod 34.9%

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

      5. add-sqr-sqrt 79.8%

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

      6. associate-*l/ 64.5%

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

      7. *-commutative 64.5%

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

      8. associate-/l* 79.9%

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

   7. Applied egg-rr 79.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-20}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+97}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-11)
   (fabs (/ (- (+ 4.0 x) (* x z)) y))
   (fabs (- (/ (+ 4.0 x) y) (* z (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-11) {
		tmp = fabs((((4.0 + x) - (x * z)) / y));
	} else {
		tmp = fabs((((4.0 + x) / y) - (z * (x / y))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5d-11) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y))
    else
        tmp = abs((((4.0d0 + x) / y) - (z * (x / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-11) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y));
	} else {
		tmp = Math.abs((((4.0 + x) / y) - (z * (x / y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5e-11:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y))
	else:
		tmp = math.fabs((((4.0 + x) / y) - (z * (x / y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-11)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) / y) - Float64(z * Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e-11)
		tmp = abs((((4.0 + x) - (x * z)) / y));
	else
		tmp = abs((((4.0 + x) / y) - (z * (x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e-11], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision] - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.00000000000000018e-11

   1. Initial program 90.4%

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

   3. Taylor expanded in y around 0 96.9%

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

   if 5.00000000000000018e-11 < y

   1. Initial program 98.5%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-57)
   (fabs (/ (- (+ 4.0 x) (* x z)) y))
   (fabs (- (/ (+ 4.0 x) y) (/ x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-57) {
		tmp = fabs((((4.0 + x) - (x * z)) / y));
	} else {
		tmp = fabs((((4.0 + x) / y) - (x / (y / z))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5d-57) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y))
    else
        tmp = abs((((4.0d0 + x) / y) - (x / (y / z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-57) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y));
	} else {
		tmp = Math.abs((((4.0 + x) / y) - (x / (y / z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5e-57:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y))
	else:
		tmp = math.fabs((((4.0 + x) / y) - (x / (y / z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-57)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) / y) - Float64(x / Float64(y / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e-57)
		tmp = abs((((4.0 + x) - (x * z)) / y));
	else
		tmp = abs((((4.0 + x) / y) - (x / (y / z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e-57], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] / y), $MachinePrecision] - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000002e-57

   1. Initial program 89.8%

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

   3. Taylor expanded in y around 0 96.7%

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

   if 5.0000000000000002e-57 < y

   1. Initial program 98.7%

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

      1. associate-/r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+53}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.8e+53)
   (fabs (/ x (/ y z)))
   (if (<= z 4.1e+21) (fabs (/ (- -4.0 x) y)) (fabs (/ z (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e+53) {
		tmp = fabs((x / (y / z)));
	} else if (z <= 4.1e+21) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((z / (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-4.8d+53)) then
        tmp = abs((x / (y / z)))
    else if (z <= 4.1d+21) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((z / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e+53) {
		tmp = Math.abs((x / (y / z)));
	} else if (z <= 4.1e+21) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((z / (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.8e+53:
		tmp = math.fabs((x / (y / z)))
	elif z <= 4.1e+21:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((z / (y / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.8e+53)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (z <= 4.1e+21)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(z / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.8e+53)
		tmp = abs((x / (y / z)));
	elseif (z <= 4.1e+21)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((z / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.8e+53], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4.1e+21], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.8e53

   1. Initial program 93.6%

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

   3. Taylor expanded in z around inf 76.4%

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

      1. mul-1-neg 76.4%

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

      2. associate-*l/ 79.2%

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

      3. distribute-rgt-neg-out 79.2%

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

   5. Simplified 79.2%

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

      1. associate-*l/ 76.4%

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

      2. associate-/l* 84.0%

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

      3. add-sqr-sqrt 83.8%

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

      4. sqrt-unprod 51.4%

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

      5. sqr-neg 51.4%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 84.0%

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

   7. Applied egg-rr 84.0%

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

   if -4.8e53 < z < 4.1e21

   1. Initial program 93.7%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in z around 0 96.2%

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

      1. associate-*r/ 96.2%

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

      2. distribute-lft-in 96.2%

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

      3. metadata-eval 96.2%

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

      4. neg-mul-1 96.2%

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

      5. sub-neg 96.2%

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

   7. Simplified 96.2%

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

   if 4.1e21 < z

   1. Initial program 88.3%

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

   3. Taylor expanded in z around inf 71.6%

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

      1. mul-1-neg 71.6%

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

      2. associate-*l/ 80.7%

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

      3. distribute-rgt-neg-out 80.7%

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

   5. Simplified 80.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 59.9%

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

      3. sqr-neg 59.9%

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

      4. sqrt-unprod 80.5%

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

      5. add-sqr-sqrt 80.7%

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

      6. associate-*l/ 71.6%

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

      7. *-commutative 71.6%

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

      8. associate-/l* 80.7%

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

   7. Applied egg-rr 80.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+53}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = fabs((x / y));
	} else {
		tmp = fabs((4.0 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-10.5d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10.5) || !(x <= 4.0)) {
		tmp = Math.abs((x / y));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -10.5) or not (x <= 4.0):
		tmp = math.fabs((x / y))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -10.5) || !(x <= 4.0))
		tmp = abs(Float64(x / y));
	else
		tmp = abs(Float64(4.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -10.5) || ~((x <= 4.0)))
		tmp = abs((x / y));
	else
		tmp = abs((4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -10.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 88.6%

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

   3. Taylor expanded in y around 0 92.2%

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

   4. Taylor expanded in x around inf 91.4%

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

      1. associate-/l* 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in z around 0 59.0%

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

   if -10.5 < x < 4

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0 75.8%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fabs (/ 4.0 y)))

C

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

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((4.0d0 / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in x around 0 39.0%

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

4. Final simplification 39.0%

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

Reproduce

herbie shell --seed 2024011 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))