fabs fraction 1

Percentage Accurate: 91.8% → 95.9%

Time: 7.3s

Alternatives: 9

Speedup: 1.0×

• 20.8% 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: 91.8% 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: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left|\frac{x + \left(4 - x \cdot z\right)}{y}\right| \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return fabs(((x + (4.0 - (x * z))) / 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(((x + (4.0d0 - (x * z))) / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

   1. associate-*l/ 94.5%

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

   2. sub-div 98.4%

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

   3. associate--l+ 98.4%

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

3. Applied egg-rr 98.4%

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

4. Final simplification 98.4%

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

Alternative 2: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{4}{y}\right|\\ t_1 := \left|\frac{x \cdot z}{y}\right|\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-206}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.95 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ 4.0 y))) (t_1 (fabs (/ (* x z) y))))
   (if (<= x -2.5e+34)
     (fabs (/ x y))
     (if (<= x -4.4e-72)
       t_1
       (if (<= x -2.8e-206)
         t_0
         (if (<= x -2.95e-217)
           t_1
           (if (<= x 1.75e-90) t_0 (fabs (* z (/ x y))))))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((4.0 / y));
	double t_1 = fabs(((x * z) / y));
	double tmp;
	if (x <= -2.5e+34) {
		tmp = fabs((x / y));
	} else if (x <= -4.4e-72) {
		tmp = t_1;
	} else if (x <= -2.8e-206) {
		tmp = t_0;
	} else if (x <= -2.95e-217) {
		tmp = t_1;
	} else if (x <= 1.75e-90) {
		tmp = t_0;
	} else {
		tmp = fabs((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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((4.0d0 / y))
    t_1 = abs(((x * z) / y))
    if (x <= (-2.5d+34)) then
        tmp = abs((x / y))
    else if (x <= (-4.4d-72)) then
        tmp = t_1
    else if (x <= (-2.8d-206)) then
        tmp = t_0
    else if (x <= (-2.95d-217)) then
        tmp = t_1
    else if (x <= 1.75d-90) then
        tmp = t_0
    else
        tmp = abs((z * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((4.0 / y));
	double t_1 = Math.abs(((x * z) / y));
	double tmp;
	if (x <= -2.5e+34) {
		tmp = Math.abs((x / y));
	} else if (x <= -4.4e-72) {
		tmp = t_1;
	} else if (x <= -2.8e-206) {
		tmp = t_0;
	} else if (x <= -2.95e-217) {
		tmp = t_1;
	} else if (x <= 1.75e-90) {
		tmp = t_0;
	} else {
		tmp = Math.abs((z * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((4.0 / y))
	t_1 = math.fabs(((x * z) / y))
	tmp = 0
	if x <= -2.5e+34:
		tmp = math.fabs((x / y))
	elif x <= -4.4e-72:
		tmp = t_1
	elif x <= -2.8e-206:
		tmp = t_0
	elif x <= -2.95e-217:
		tmp = t_1
	elif x <= 1.75e-90:
		tmp = t_0
	else:
		tmp = math.fabs((z * (x / y)))
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(4.0 / y))
	t_1 = abs(Float64(Float64(x * z) / y))
	tmp = 0.0
	if (x <= -2.5e+34)
		tmp = abs(Float64(x / y));
	elseif (x <= -4.4e-72)
		tmp = t_1;
	elseif (x <= -2.8e-206)
		tmp = t_0;
	elseif (x <= -2.95e-217)
		tmp = t_1;
	elseif (x <= 1.75e-90)
		tmp = t_0;
	else
		tmp = abs(Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((4.0 / y));
	t_1 = abs(((x * z) / y));
	tmp = 0.0;
	if (x <= -2.5e+34)
		tmp = abs((x / y));
	elseif (x <= -4.4e-72)
		tmp = t_1;
	elseif (x <= -2.8e-206)
		tmp = t_0;
	elseif (x <= -2.95e-217)
		tmp = t_1;
	elseif (x <= 1.75e-90)
		tmp = t_0;
	else
		tmp = abs((z * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.5e+34], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -4.4e-72], t$95$1, If[LessEqual[x, -2.8e-206], t$95$0, If[LessEqual[x, -2.95e-217], t$95$1, If[LessEqual[x, 1.75e-90], t$95$0, N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.4999999999999999e34

   1. Initial program 80.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. 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 95.4%

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

      3. associate--l+ 95.4%

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

   3. Applied egg-rr 95.4%

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

   4. Taylor expanded in x around inf 95.4%

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

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

   7. Taylor expanded in z around 0 68.7%

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

   if -2.4999999999999999e34 < x < -4.40000000000000005e-72 or -2.8000000000000001e-206 < x < -2.9499999999999999e-217

   1. Initial program 90.0%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in x around inf 63.5%

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

   5. Taylor expanded in z around inf 67.6%

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

   if -4.40000000000000005e-72 < x < -2.8000000000000001e-206 or -2.9499999999999999e-217 < x < 1.7499999999999999e-90

   1. Initial program 93.9%

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

   2. Taylor expanded in x around 0 86.9%

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

   if 1.7499999999999999e-90 < x

   1. Initial program 93.9%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in x around inf 85.6%

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

   5. Taylor expanded in z around inf 56.6%

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

      1. associate-*l/ 60.0%

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

   7. Simplified 60.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-72}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-206}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq -2.95 \cdot 10^{-217}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-90}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+35}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-63} \lor \neg \left(x \leq 1.25 \cdot 10^{-78}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.6e+35)
   (fabs (/ x y))
   (if (or (<= x -4.1e-63) (not (<= x 1.25e-78)))
     (fabs (* z (/ x y)))
     (fabs (/ 4.0 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.6e+35) {
		tmp = fabs((x / y));
	} else if ((x <= -4.1e-63) || !(x <= 1.25e-78)) {
		tmp = fabs((z * (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 <= (-3.6d+35)) then
        tmp = abs((x / y))
    else if ((x <= (-4.1d-63)) .or. (.not. (x <= 1.25d-78))) then
        tmp = abs((z * (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 <= -3.6e+35) {
		tmp = Math.abs((x / y));
	} else if ((x <= -4.1e-63) || !(x <= 1.25e-78)) {
		tmp = Math.abs((z * (x / y)));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.6e+35:
		tmp = math.fabs((x / y))
	elif (x <= -4.1e-63) or not (x <= 1.25e-78):
		tmp = math.fabs((z * (x / y)))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.6e+35)
		tmp = abs(Float64(x / y));
	elseif ((x <= -4.1e-63) || !(x <= 1.25e-78))
		tmp = abs(Float64(z * 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 <= -3.6e+35)
		tmp = abs((x / y));
	elseif ((x <= -4.1e-63) || ~((x <= 1.25e-78)))
		tmp = abs((z * (x / y)));
	else
		tmp = abs((4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.6e+35], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, -4.1e-63], N[Not[LessEqual[x, 1.25e-78]], $MachinePrecision]], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.6e35

   1. Initial program 80.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. 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 95.4%

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

      3. associate--l+ 95.4%

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

   3. Applied egg-rr 95.4%

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

   4. Taylor expanded in x around inf 95.4%

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

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

   7. Taylor expanded in z around 0 68.7%

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

   if -3.6e35 < x < -4.0999999999999998e-63 or 1.2499999999999999e-78 < x

   1. Initial program 94.3%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in x around inf 81.2%

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

   5. Taylor expanded in z around inf 57.8%

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

      1. associate-*l/ 59.7%

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

   7. Simplified 59.7%

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

   if -4.0999999999999998e-63 < x < 1.2499999999999999e-78

   1. Initial program 92.2%

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

   2. Taylor expanded in x around 0 82.4%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+35}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-63} \lor \neg \left(x \leq 1.25 \cdot 10^{-78}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 4: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-63}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-90}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9e+34)
   (fabs (/ x y))
   (if (<= x -4.1e-63)
     (fabs (/ x (/ y z)))
     (if (<= x 1.75e-90) (fabs (/ 4.0 y)) (fabs (* z (/ x y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e+34) {
		tmp = fabs((x / y));
	} else if (x <= -4.1e-63) {
		tmp = fabs((x / (y / z)));
	} else if (x <= 1.75e-90) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = fabs((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 (x <= (-9d+34)) then
        tmp = abs((x / y))
    else if (x <= (-4.1d-63)) then
        tmp = abs((x / (y / z)))
    else if (x <= 1.75d-90) then
        tmp = abs((4.0d0 / y))
    else
        tmp = abs((z * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e+34) {
		tmp = Math.abs((x / y));
	} else if (x <= -4.1e-63) {
		tmp = Math.abs((x / (y / z)));
	} else if (x <= 1.75e-90) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = Math.abs((z * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9e+34:
		tmp = math.fabs((x / y))
	elif x <= -4.1e-63:
		tmp = math.fabs((x / (y / z)))
	elif x <= 1.75e-90:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = math.fabs((z * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9e+34)
		tmp = abs(Float64(x / y));
	elseif (x <= -4.1e-63)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (x <= 1.75e-90)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = abs(Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9e+34)
		tmp = abs((x / y));
	elseif (x <= -4.1e-63)
		tmp = abs((x / (y / z)));
	elseif (x <= 1.75e-90)
		tmp = abs((4.0 / y));
	else
		tmp = abs((z * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9e+34], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -4.1e-63], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.75e-90], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.0000000000000001e34

   1. Initial program 80.2%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   2. 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 95.4%

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

      3. associate--l+ 95.4%

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

   3. Applied egg-rr 95.4%

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

   4. Taylor expanded in x around inf 95.4%

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

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

   7. Taylor expanded in z around 0 68.7%

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

   if -9.0000000000000001e34 < x < -4.0999999999999998e-63

   1. Initial program 95.7%

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

   2. Taylor expanded in z around inf 62.2%

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

      1. mul-1-neg 62.2%

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

      2. associate-*l/ 58.4%

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

      3. distribute-rgt-neg-out 58.4%

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

   4. Simplified 58.4%

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

      1. associate-*l/ 62.2%

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

      2. associate-/l* 61.0%

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

      3. add-sqr-sqrt 22.1%

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

      4. sqrt-unprod 39.8%

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

      5. sqr-neg 39.8%

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

      6. sqrt-unprod 38.6%

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

      7. add-sqr-sqrt 61.0%

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

   6. Applied egg-rr 61.0%

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

   if -4.0999999999999998e-63 < x < 1.7499999999999999e-90

   1. Initial program 92.2%

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

   2. Taylor expanded in x around 0 82.4%

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

   if 1.7499999999999999e-90 < x

   1. Initial program 93.9%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in x around inf 85.6%

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

   5. Taylor expanded in z around inf 56.6%

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

      1. associate-*l/ 60.0%

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

   7. Simplified 60.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-63}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-90}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 5: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -55000000:\\ \;\;\;\;\left|\frac{x}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 10^{+49}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -55000000.0)
   (fabs (- (/ x y) (/ (* x z) y)))
   (if (<= z 1e+49) (fabs (/ (- -4.0 x) y)) (fabs (/ x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -55000000.0) {
		tmp = fabs(((x / y) - ((x * z) / y)));
	} else if (z <= 1e+49) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((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 (z <= (-55000000.0d0)) then
        tmp = abs(((x / y) - ((x * z) / y)))
    else if (z <= 1d+49) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((x / (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -55000000.0) {
		tmp = Math.abs(((x / y) - ((x * z) / y)));
	} else if (z <= 1e+49) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((x / (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -55000000.0:
		tmp = math.fabs(((x / y) - ((x * z) / y)))
	elif z <= 1e+49:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((x / (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -55000000.0)
		tmp = abs(Float64(Float64(x / y) - Float64(Float64(x * z) / y)));
	elseif (z <= 1e+49)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(x / Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -55000000.0)
		tmp = abs(((x / y) - ((x * z) / y)));
	elseif (z <= 1e+49)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((x / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -55000000.0], N[Abs[N[(N[(x / y), $MachinePrecision] - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1e+49], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5e7

   1. Initial program 91.6%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in x around inf 84.3%

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

   if -5.5e7 < z < 9.99999999999999946e48

   1. Initial program 90.2%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in z around 0 95.3%

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

      1. associate-*r/ 95.3%

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

      2. distribute-lft-in 95.3%

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

      3. metadata-eval 95.3%

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

      4. neg-mul-1 95.3%

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

      5. sub-neg 95.3%

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

   6. Simplified 95.3%

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

   if 9.99999999999999946e48 < z

   1. Initial program 88.9%

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

   2. Taylor expanded in z around inf 77.6%

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

      1. mul-1-neg 77.6%

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

      2. associate-*l/ 78.2%

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

      3. distribute-rgt-neg-out 78.2%

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

   4. Simplified 78.2%

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

      1. associate-*l/ 77.6%

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

      2. associate-/l* 81.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 53.2%

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

      5. sqr-neg 53.2%

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

      6. sqrt-unprod 81.2%

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

      7. add-sqr-sqrt 81.4%

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

   6. Applied egg-rr 81.4%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -55000000:\\ \;\;\;\;\left|\frac{x}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 10^{+49}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 6: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -126000000:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -126000000.0)
   (fabs (/ (* x z) y))
   (if (<= z 4.2e+49) (fabs (/ (- -4.0 x) y)) (fabs (/ x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -126000000.0) {
		tmp = fabs(((x * z) / y));
	} else if (z <= 4.2e+49) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((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 (z <= (-126000000.0d0)) then
        tmp = abs(((x * z) / y))
    else if (z <= 4.2d+49) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((x / (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -126000000.0) {
		tmp = Math.abs(((x * z) / y));
	} else if (z <= 4.2e+49) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((x / (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -126000000.0:
		tmp = math.fabs(((x * z) / y))
	elif z <= 4.2e+49:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((x / (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -126000000.0)
		tmp = abs(Float64(Float64(x * z) / y));
	elseif (z <= 4.2e+49)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(x / Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -126000000.0)
		tmp = abs(((x * z) / y));
	elseif (z <= 4.2e+49)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((x / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -126000000.0], N[Abs[N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 4.2e+49], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.26e8

   1. Initial program 91.6%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in x around inf 82.6%

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

   5. Taylor expanded in z around inf 83.2%

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

   if -1.26e8 < z < 4.20000000000000022e49

   1. Initial program 90.2%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in z around 0 95.3%

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

      1. associate-*r/ 95.3%

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

      2. distribute-lft-in 95.3%

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

      3. metadata-eval 95.3%

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

      4. neg-mul-1 95.3%

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

      5. sub-neg 95.3%

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

   6. Simplified 95.3%

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

   if 4.20000000000000022e49 < z

   1. Initial program 88.9%

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

   2. Taylor expanded in z around inf 77.6%

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

      1. mul-1-neg 77.6%

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

      2. associate-*l/ 78.2%

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

      3. distribute-rgt-neg-out 78.2%

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

   4. Simplified 78.2%

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

      1. associate-*l/ 77.6%

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

      2. associate-/l* 81.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 53.2%

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

      5. sqr-neg 53.2%

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

      6. sqrt-unprod 81.2%

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

      7. add-sqr-sqrt 81.4%

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

   6. Applied egg-rr 81.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -126000000:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 7: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -118000000:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x \cdot z}}\right|\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -118000000.0)
   (fabs (/ 1.0 (/ y (* x z))))
   (if (<= z 2.6e+49) (fabs (/ (- -4.0 x) y)) (fabs (/ x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -118000000.0) {
		tmp = fabs((1.0 / (y / (x * z))));
	} else if (z <= 2.6e+49) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((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 (z <= (-118000000.0d0)) then
        tmp = abs((1.0d0 / (y / (x * z))))
    else if (z <= 2.6d+49) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((x / (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -118000000.0) {
		tmp = Math.abs((1.0 / (y / (x * z))));
	} else if (z <= 2.6e+49) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((x / (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -118000000.0:
		tmp = math.fabs((1.0 / (y / (x * z))))
	elif z <= 2.6e+49:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((x / (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -118000000.0)
		tmp = abs(Float64(1.0 / Float64(y / Float64(x * z))));
	elseif (z <= 2.6e+49)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(x / Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -118000000.0)
		tmp = abs((1.0 / (y / (x * z))));
	elseif (z <= 2.6e+49)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((x / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -118000000.0], N[Abs[N[(1.0 / N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.6e+49], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.18e8

   1. Initial program 91.6%

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

   2. Taylor expanded in z around inf 83.2%

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

      1. mul-1-neg 83.2%

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

      2. associate-*l/ 76.9%

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

      3. distribute-rgt-neg-out 76.9%

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

   4. Simplified 76.9%

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

      1. associate-*l/ 83.2%

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

      2. clear-num 83.2%

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

      3. add-sqr-sqrt 83.2%

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

      4. sqrt-unprod 62.8%

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

      5. sqr-neg 62.8%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 83.2%

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

   6. Applied egg-rr 83.2%

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

   if -1.18e8 < z < 2.59999999999999989e49

   1. Initial program 90.2%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in z around 0 95.3%

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

      1. associate-*r/ 95.3%

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

      2. distribute-lft-in 95.3%

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

      3. metadata-eval 95.3%

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

      4. neg-mul-1 95.3%

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

      5. sub-neg 95.3%

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

   6. Simplified 95.3%

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

   if 2.59999999999999989e49 < z

   1. Initial program 88.9%

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

   2. Taylor expanded in z around inf 77.6%

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

      1. mul-1-neg 77.6%

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

      2. associate-*l/ 78.2%

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

      3. distribute-rgt-neg-out 78.2%

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

   4. Simplified 78.2%

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

      1. associate-*l/ 77.6%

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

      2. associate-/l* 81.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 53.2%

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

      5. sqr-neg 53.2%

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

      6. sqrt-unprod 81.2%

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

      7. add-sqr-sqrt 81.4%

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

   6. Applied egg-rr 81.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -118000000:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x \cdot z}}\right|\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 8: 69.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.55) || !(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 <= (-1.55d0)) .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 <= -1.55) || !(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 <= -1.55) 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 <= -1.55) || !(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 <= -1.55) || ~((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, -1.55], 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 < -1.55000000000000004 or 4 < x

   1. Initial program 86.7%

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

      1. associate-*l/ 89.2%

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

      2. sub-div 97.0%

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

      3. associate--l+ 97.0%

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

   3. Applied egg-rr 97.0%

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

   4. Taylor expanded in x around inf 96.4%

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

      1. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in z around 0 63.2%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 93.8%

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

   2. Taylor expanded in x around 0 70.8%

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

4. Final simplification 67.0%

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

Alternative 9: 40.3% 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 90.2%

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

2. Taylor expanded in x around 0 38.1%

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

3. Final simplification 38.1%

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

Reproduce

herbie shell --seed 2023319 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))