fabs fraction 1

Percentage Accurate: 92.3% → 96.3%

Time: 7.0s

Alternatives: 7

Speedup: 1.0×

• 20.1% 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: 92.3% 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: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + 4}{y}\\ \mathbf{if}\;t_0 - \frac{x}{y} \cdot z \leq 2 \cdot 10^{+110}:\\ \;\;\;\;\left|t_0 - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z - \left(x + 4\right)}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y)))
   (if (<= (- t_0 (* (/ x y) z)) 2e+110)
     (fabs (- t_0 (/ x (/ y z))))
     (fabs (/ (- (* x z) (+ x 4.0)) y)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double tmp;
	if ((t_0 - ((x / y) * z)) <= 2e+110) {
		tmp = fabs((t_0 - (x / (y / z))));
	} else {
		tmp = fabs((((x * z) - (x + 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) :: t_0
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y
    if ((t_0 - ((x / y) * z)) <= 2d+110) then
        tmp = abs((t_0 - (x / (y / z))))
    else
        tmp = abs((((x * z) - (x + 4.0d0)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double tmp;
	if ((t_0 - ((x / y) * z)) <= 2e+110) {
		tmp = Math.abs((t_0 - (x / (y / z))));
	} else {
		tmp = Math.abs((((x * z) - (x + 4.0)) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + 4.0) / y
	tmp = 0
	if (t_0 - ((x / y) * z)) <= 2e+110:
		tmp = math.fabs((t_0 - (x / (y / z))))
	else:
		tmp = math.fabs((((x * z) - (x + 4.0)) / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + 4.0) / y)
	tmp = 0.0
	if (Float64(t_0 - Float64(Float64(x / y) * z)) <= 2e+110)
		tmp = abs(Float64(t_0 - Float64(x / Float64(y / z))));
	else
		tmp = abs(Float64(Float64(Float64(x * z) - Float64(x + 4.0)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + 4.0) / y;
	tmp = 0.0;
	if ((t_0 - ((x / y) * z)) <= 2e+110)
		tmp = abs((t_0 - (x / (y / z))));
	else
		tmp = abs((((x * z) - (x + 4.0)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], 2e+110], N[Abs[N[(t$95$0 - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x * z), $MachinePrecision] - N[(x + 4.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < 2e110

   1. Initial program 96.6%

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

      1. associate-*l/ 95.3%

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

      2. associate-/l* 99.4%

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

   3. Applied egg-rr 99.4%

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

   if 2e110 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))

   1. Initial program 73.3%

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

      1. associate-*l/ 79.6%

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

      2. sub-div 97.2%

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

   3. Applied egg-rr 97.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 2 \cdot 10^{+110}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot z - \left(x + 4\right)}{y}\right|\\ \end{array} \]

Alternative 2: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+120} \lor \neg \left(z \leq 3.2 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{\left|x\right|}{\left|\frac{y}{z}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.8e+120) (not (<= z 3.2e+51)))
   (/ (fabs x) (fabs (/ y z)))
   (fabs (/ (- -4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.8e+120) || !(z <= 3.2e+51)) {
		tmp = fabs(x) / fabs((y / z));
	} else {
		tmp = fabs(((-4.0 - 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 ((z <= (-3.8d+120)) .or. (.not. (z <= 3.2d+51))) then
        tmp = abs(x) / abs((y / z))
    else
        tmp = abs((((-4.0d0) - x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.8e+120) || !(z <= 3.2e+51)) {
		tmp = Math.abs(x) / Math.abs((y / z));
	} else {
		tmp = Math.abs(((-4.0 - x) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.8e+120) or not (z <= 3.2e+51):
		tmp = math.fabs(x) / math.fabs((y / z))
	else:
		tmp = math.fabs(((-4.0 - x) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.8e+120) || !(z <= 3.2e+51))
		tmp = Float64(abs(x) / abs(Float64(y / z)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.8e+120) || ~((z <= 3.2e+51)))
		tmp = abs(x) / abs((y / z));
	else
		tmp = abs(((-4.0 - x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.8e+120], N[Not[LessEqual[z, 3.2e+51]], $MachinePrecision]], N[(N[Abs[x], $MachinePrecision] / N[Abs[N[(y / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7999999999999998e120 or 3.2000000000000002e51 < z

   1. Initial program 85.6%

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

   2. Taylor expanded in z around inf 70.4%

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

      1. *-commutative 70.4%

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

      2. metadata-eval 70.4%

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

      3. times-frac 70.4%

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

      4. *-lft-identity 70.4%

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

      5. neg-mul-1 70.4%

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

      6. associate-/l* 79.4%

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

      7. distribute-frac-neg 79.4%

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

   4. Simplified 79.4%

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

   if -3.7999999999999998e120 < z < 3.2000000000000002e51

   1. Initial program 93.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in z around 0 92.6%

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

      1. associate-*r/ 92.6%

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

      2. distribute-lft-in 92.6%

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

      3. metadata-eval 92.6%

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

      4. neg-mul-1 92.6%

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

      5. sub-neg 92.6%

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

   6. Simplified 92.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+120} \lor \neg \left(z \leq 3.2 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{\left|x\right|}{\left|\frac{y}{z}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 3: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{4}{y}\right|\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-8}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-97}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (/ 4.0 y))))
   (if (<= x -4.4e-8)
     (fabs (* (/ x y) z))
     (if (<= x -2.4e-76)
       t_0
       (if (<= x -7.6e-97)
         (fabs (* x (/ z y)))
         (if (<= x 4.0) t_0 (fabs (/ x y))))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((4.0 / y));
	double tmp;
	if (x <= -4.4e-8) {
		tmp = fabs(((x / y) * z));
	} else if (x <= -2.4e-76) {
		tmp = t_0;
	} else if (x <= -7.6e-97) {
		tmp = fabs((x * (z / y)));
	} else if (x <= 4.0) {
		tmp = t_0;
	} else {
		tmp = fabs((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) :: tmp
    t_0 = abs((4.0d0 / y))
    if (x <= (-4.4d-8)) then
        tmp = abs(((x / y) * z))
    else if (x <= (-2.4d-76)) then
        tmp = t_0
    else if (x <= (-7.6d-97)) then
        tmp = abs((x * (z / y)))
    else if (x <= 4.0d0) then
        tmp = t_0
    else
        tmp = abs((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 tmp;
	if (x <= -4.4e-8) {
		tmp = Math.abs(((x / y) * z));
	} else if (x <= -2.4e-76) {
		tmp = t_0;
	} else if (x <= -7.6e-97) {
		tmp = Math.abs((x * (z / y)));
	} else if (x <= 4.0) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((4.0 / y))
	tmp = 0
	if x <= -4.4e-8:
		tmp = math.fabs(((x / y) * z))
	elif x <= -2.4e-76:
		tmp = t_0
	elif x <= -7.6e-97:
		tmp = math.fabs((x * (z / y)))
	elif x <= 4.0:
		tmp = t_0
	else:
		tmp = math.fabs((x / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(4.0 / y))
	tmp = 0.0
	if (x <= -4.4e-8)
		tmp = abs(Float64(Float64(x / y) * z));
	elseif (x <= -2.4e-76)
		tmp = t_0;
	elseif (x <= -7.6e-97)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (x <= 4.0)
		tmp = t_0;
	else
		tmp = abs(Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((4.0 / y));
	tmp = 0.0;
	if (x <= -4.4e-8)
		tmp = abs(((x / y) * z));
	elseif (x <= -2.4e-76)
		tmp = t_0;
	elseif (x <= -7.6e-97)
		tmp = abs((x * (z / y)));
	elseif (x <= 4.0)
		tmp = t_0;
	else
		tmp = abs((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4.4e-8], N[Abs[N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -2.4e-76], t$95$0, If[LessEqual[x, -7.6e-97], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], t$95$0, N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.3999999999999997e-8

   1. Initial program 85.5%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in z around inf 47.4%

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

      1. *-commutative 47.4%

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

      2. associate-*l/ 62.3%

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

      3. *-commutative 62.3%

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

   6. Simplified 62.3%

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

   if -4.3999999999999997e-8 < x < -2.40000000000000013e-76 or -7.6000000000000001e-97 < x < 4

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0 78.4%

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

   if -2.40000000000000013e-76 < x < -7.6000000000000001e-97

   1. Initial program 81.2%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around inf 99.4%

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

      1. associate-*l/ 99.4%

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

      2. *-commutative 99.4%

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

   6. Simplified 99.4%

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

   if 4 < x

   1. Initial program 86.2%

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

      1. associate-*l/ 80.9%

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

      2. sub-div 90.0%

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

   3. Applied egg-rr 90.0%

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

   4. Taylor expanded in x around inf 89.1%

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

      1. associate-/l* 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in z around 0 66.1%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-8}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-76}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-97}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+43}:\\ \;\;\;\;\left|\frac{x \cdot z - \left(x + 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e+43)
   (fabs (/ (- (* x z) (+ x 4.0)) y))
   (fabs (/ (+ z -1.0) (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e+43) {
		tmp = fabs((((x * z) - (x + 4.0)) / y));
	} else {
		tmp = fabs(((z + -1.0) / (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 (x <= 2d+43) then
        tmp = abs((((x * z) - (x + 4.0d0)) / y))
    else
        tmp = abs(((z + (-1.0d0)) / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e+43) {
		tmp = Math.abs((((x * z) - (x + 4.0)) / y));
	} else {
		tmp = Math.abs(((z + -1.0) / (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2e+43:
		tmp = math.fabs((((x * z) - (x + 4.0)) / y))
	else:
		tmp = math.fabs(((z + -1.0) / (y / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e+43)
		tmp = abs(Float64(Float64(Float64(x * z) - Float64(x + 4.0)) / y));
	else
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2e+43)
		tmp = abs((((x * z) - (x + 4.0)) / y));
	else
		tmp = abs(((z + -1.0) / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e+43], N[Abs[N[(N[(N[(x * z), $MachinePrecision] - N[(x + 4.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000003e43

   1. Initial program 92.3%

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

      1. associate-*l/ 94.9%

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

      2. sub-div 98.0%

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

   3. Applied egg-rr 98.0%

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

   if 2.00000000000000003e43 < x

   1. Initial program 84.1%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in x around inf 88.4%

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

      1. associate-/l* 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+43}:\\ \;\;\;\;\left|\frac{x \cdot z - \left(x + 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 5: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+121} \lor \neg \left(z \leq 2.2 \cdot 10^{+52}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.9e+121) (not (<= z 2.2e+52)))
   (fabs (* x (/ z y)))
   (fabs (/ (- -4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.9e+121) || !(z <= 2.2e+52)) {
		tmp = fabs((x * (z / y)));
	} else {
		tmp = fabs(((-4.0 - 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 ((z <= (-1.9d+121)) .or. (.not. (z <= 2.2d+52))) then
        tmp = abs((x * (z / y)))
    else
        tmp = abs((((-4.0d0) - x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.9e+121) || !(z <= 2.2e+52)) {
		tmp = Math.abs((x * (z / y)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.9e+121) or not (z <= 2.2e+52):
		tmp = math.fabs((x * (z / y)))
	else:
		tmp = math.fabs(((-4.0 - x) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.9e+121) || !(z <= 2.2e+52))
		tmp = abs(Float64(x * Float64(z / y)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.9e+121) || ~((z <= 2.2e+52)))
		tmp = abs((x * (z / y)));
	else
		tmp = abs(((-4.0 - x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.9e+121], N[Not[LessEqual[z, 2.2e+52]], $MachinePrecision]], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9e121 or 2.2e52 < z

   1. Initial program 85.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in z around inf 70.4%

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

      1. associate-*l/ 79.4%

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

      2. *-commutative 79.4%

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

   6. Simplified 79.4%

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

   if -1.9e121 < z < 2.2e52

   1. Initial program 93.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in z around 0 92.6%

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

      1. associate-*r/ 92.6%

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

      2. distribute-lft-in 92.6%

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

      3. metadata-eval 92.6%

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

      4. neg-mul-1 92.6%

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

      5. sub-neg 92.6%

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

   6. Simplified 92.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+121} \lor \neg \left(z \leq 2.2 \cdot 10^{+52}\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 6: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -11 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -11.0) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -11.0) || !(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 <= (-11.0d0)) .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 <= -11.0) || !(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 <= -11.0) 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 <= -11.0) || !(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 <= -11.0) || ~((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, -11.0], 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 < -11 or 4 < x

   1. Initial program 85.5%

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

      1. associate-*l/ 82.2%

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

      2. sub-div 91.7%

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

   3. Applied egg-rr 91.7%

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

   4. Taylor expanded in x around inf 91.2%

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

      1. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in z around 0 64.3%

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

   if -11 < x < 4

   1. Initial program 95.2%

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

   2. Taylor expanded in x around 0 74.4%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 7: 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 90.4%

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

2. Taylor expanded in x around 0 40.2%

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

3. Final simplification 40.2%

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

Reproduce

herbie shell --seed 2023196 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))