fabs fraction 1

Average Accuracy: 97.4% → 99.7%

Time: 10.3s

Precision: binary64

Cost: 8648

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ t_1 := t_0 - \frac{x}{y} \cdot z\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\left|t_0 - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;t_1 \leq 10^{+46}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1\right|\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y)) (t_1 (- t_0 (* (/ x y) z))))
   (if (<= t_1 -2e+35)
     (fabs (- t_0 (/ z (/ y x))))
     (if (<= t_1 1e+46) (fabs (/ (- (+ x 4.0) (* x z)) y)) (fabs t_1)))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double t_1 = t_0 - ((x / y) * z);
	double tmp;
	if (t_1 <= -2e+35) {
		tmp = fabs((t_0 - (z / (y / x))));
	} else if (t_1 <= 1e+46) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(t_1);
	}
	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
    code = abs((((x + 4.0d0) / y) - ((x / y) * z)))
end function

↓

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 = (x + 4.0d0) / y
    t_1 = t_0 - ((x / y) * z)
    if (t_1 <= (-2d+35)) then
        tmp = abs((t_0 - (z / (y / x))))
    else if (t_1 <= 1d+46) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs(t_1)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double t_1 = t_0 - ((x / y) * z);
	double tmp;
	if (t_1 <= -2e+35) {
		tmp = Math.abs((t_0 - (z / (y / x))));
	} else if (t_1 <= 1e+46) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x + 4.0) / y
	t_1 = t_0 - ((x / y) * z)
	tmp = 0
	if t_1 <= -2e+35:
		tmp = math.fabs((t_0 - (z / (y / x))))
	elif t_1 <= 1e+46:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(x + 4.0) / y)
	t_1 = Float64(t_0 - Float64(Float64(x / y) * z))
	tmp = 0.0
	if (t_1 <= -2e+35)
		tmp = abs(Float64(t_0 - Float64(z / Float64(y / x))));
	elseif (t_1 <= 1e+46)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = (x + 4.0) / y;
	t_1 = t_0 - ((x / y) * z);
	tmp = 0.0;
	if (t_1 <= -2e+35)
		tmp = abs((t_0 - (z / (y / x))));
	elseif (t_1 <= 1e+46)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs(t_1);
	end
	tmp_2 = tmp;
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]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+35], N[Abs[N[(t$95$0 - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+46], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < -1.9999999999999999e35

   1. Initial program 99.8%

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

   2. Simplified 99.8%

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

      [Start] 99.8	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      *-lft-identity [<=] 99.8	\[ \color{blue}{1 \cdot \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]
      metadata-eval [<=] 99.8	\[ \color{blue}{\left|-1\right|} \cdot \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      fabs-sub [=>] 99.8	\[ \left|-1\right| \cdot \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]
      fabs-mul [<=] 99.8	\[ \color{blue}{\left|-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right|} \]
      neg-mul-1 [<=] 99.8	\[ \left|\color{blue}{-\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)}\right| \]
      sub0-neg [<=] 99.8	\[ \left|\color{blue}{0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)}\right| \]
      associate-+l- [<=] 99.8	\[ \left|\color{blue}{\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}}\right| \]
      neg-sub0 [<=] 99.8	\[ \left|\color{blue}{\left(-\frac{x}{y} \cdot z\right)} + \frac{x + 4}{y}\right| \]
      +-commutative [<=] 99.8	\[ \left|\color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)}\right| \]
      sub-neg [<=] 99.8	\[ \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]
      associate-*l/ [=>] 90.6	\[ \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right| \]
      *-commutative [=>] 90.6	\[ \left|\frac{x + 4}{y} - \frac{\color{blue}{z \cdot x}}{y}\right| \]
      associate-/l* [=>] 99.8	\[ \left|\frac{x + 4}{y} - \color{blue}{\frac{z}{\frac{y}{x}}}\right| \]

   if -1.9999999999999999e35 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < 9.9999999999999999e45

   1. Initial program 94.6%

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

   2. Applied egg-rr 99.6%

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

      [Start] 94.6	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      associate-*l/ [=>] 99.6	\[ \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right| \]
      sub-div [=>] 99.6	\[ \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]

   if 9.9999999999999999e45 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))

   1. Initial program 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 10^{+46}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+102}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-84}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array} \]

Alternative 2
Accuracy	81.1%
Cost	7249

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+151}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{+84} \lor \neg \left(z \leq -7.3 \cdot 10^{+46}\right) \land z \leq 1.55 \cdot 10^{+62}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 3
Accuracy	86.0%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-18} \lor \neg \left(x \leq 80000000\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \]

Alternative 4
Accuracy	96.7%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 5
Accuracy	70.6%
Cost	6988

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-19}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	70.6%
Cost	6988

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-19}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	70.5%
Cost	6988

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -9.4 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-20}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	70.4%
Cost	6857

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

Alternative 9
Accuracy	49.1%
Cost	6592

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

Reproduce

herbie shell --seed 2023136 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))