fabs fraction 1

Average Accuracy: 97.8% → 99.7%

Time: 13.6s

Precision: binary64

Cost: 14276

Math

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

↓

\[\begin{array}{l} t_0 := \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z)))))
   (if (<= t_0 5e+36) (fabs (/ (- (+ x 4.0) (* x z)) y)) t_0)))

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 = fabs((((x + 4.0) / y) - ((x / y) * z)));
	double tmp;
	if (t_0 <= 5e+36) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} 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
    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) :: tmp
    t_0 = abs((((x + 4.0d0) / y) - ((x / y) * z)))
    if (t_0 <= 5d+36) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = t_0
    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 = Math.abs((((x + 4.0) / y) - ((x / y) * z)));
	double tmp;
	if (t_0 <= 5e+36) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = t_0;
	}
	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 = math.fabs((((x + 4.0) / y) - ((x / y) * z)))
	tmp = 0
	if t_0 <= 5e+36:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = t_0
	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 = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z)))
	tmp = 0.0
	if (t_0 <= 5e+36)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = t_0;
	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 = abs((((x + 4.0) / y) - ((x / y) * z)));
	tmp = 0.0;
	if (t_0 <= 5e+36)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = t_0;
	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[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 5e+36], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 4.99999999999999977e36

   1. Initial program 95.4%

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

   2. Taylor expanded in x around 0 99.8%

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

   3. Simplified 99.6%

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

      [Start] 99.8	\[ \left|4 \cdot \frac{1}{y} + \left(\frac{1}{y} - \frac{z}{y}\right) \cdot x\right| \]
      *-commutative [=>] 99.8	\[ \left|4 \cdot \frac{1}{y} + \color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]
      sub-neg [=>] 99.8	\[ \left|4 \cdot \frac{1}{y} + x \cdot \color{blue}{\left(\frac{1}{y} + \left(-\frac{z}{y}\right)\right)}\right| \]
      mul-1-neg [<=] 99.8	\[ \left|4 \cdot \frac{1}{y} + x \cdot \left(\frac{1}{y} + \color{blue}{-1 \cdot \frac{z}{y}}\right)\right| \]
      distribute-rgt-in [=>] 99.8	\[ \left|4 \cdot \frac{1}{y} + \color{blue}{\left(\frac{1}{y} \cdot x + \left(-1 \cdot \frac{z}{y}\right) \cdot x\right)}\right| \]
      *-commutative [<=] 99.8	\[ \left|4 \cdot \frac{1}{y} + \left(\color{blue}{x \cdot \frac{1}{y}} + \left(-1 \cdot \frac{z}{y}\right) \cdot x\right)\right| \]
      associate-*r* [<=] 99.8	\[ \left|4 \cdot \frac{1}{y} + \left(x \cdot \frac{1}{y} + \color{blue}{-1 \cdot \left(\frac{z}{y} \cdot x\right)}\right)\right| \]
      associate-*l/ [=>] 99.6	\[ \left|4 \cdot \frac{1}{y} + \left(x \cdot \frac{1}{y} + -1 \cdot \color{blue}{\frac{z \cdot x}{y}}\right)\right| \]
      mul-1-neg [=>] 99.6	\[ \left|4 \cdot \frac{1}{y} + \left(x \cdot \frac{1}{y} + \color{blue}{\left(-\frac{z \cdot x}{y}\right)}\right)\right| \]
      *-commutative [=>] 99.6	\[ \left|4 \cdot \frac{1}{y} + \left(x \cdot \frac{1}{y} + \left(-\frac{\color{blue}{x \cdot z}}{y}\right)\right)\right| \]
      associate-*l/ [<=] 95.3	\[ \left|4 \cdot \frac{1}{y} + \left(x \cdot \frac{1}{y} + \left(-\color{blue}{\frac{x}{y} \cdot z}\right)\right)\right| \]
      distribute-rgt-neg-out [<=] 95.3	\[ \left|4 \cdot \frac{1}{y} + \left(x \cdot \frac{1}{y} + \color{blue}{\frac{x}{y} \cdot \left(-z\right)}\right)\right| \]
      associate-+r+ [=>] 95.3	\[ \left|\color{blue}{\left(4 \cdot \frac{1}{y} + x \cdot \frac{1}{y}\right) + \frac{x}{y} \cdot \left(-z\right)}\right| \]
      +-commutative [<=] 95.3	\[ \left|\color{blue}{\left(x \cdot \frac{1}{y} + 4 \cdot \frac{1}{y}\right)} + \frac{x}{y} \cdot \left(-z\right)\right| \]
      distribute-rgt-in [<=] 95.3	\[ \left|\color{blue}{\frac{1}{y} \cdot \left(x + 4\right)} + \frac{x}{y} \cdot \left(-z\right)\right| \]
      associate-*l/ [=>] 95.4	\[ \left|\color{blue}{\frac{1 \cdot \left(x + 4\right)}{y}} + \frac{x}{y} \cdot \left(-z\right)\right| \]
      *-lft-identity [=>] 95.4	\[ \left|\frac{\color{blue}{x + 4}}{y} + \frac{x}{y} \cdot \left(-z\right)\right| \]
      distribute-rgt-neg-out [=>] 95.4	\[ \left|\frac{x + 4}{y} + \color{blue}{\left(-\frac{x}{y} \cdot z\right)}\right| \]

   if 4.99999999999999977e36 < (fabs.f64 (-.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 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.9%
Cost	7376

\[\begin{array}{l} t_0 := \left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{if}\;x \leq -165:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-141}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-112}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 9500000:\\ \;\;\;\;\frac{1}{\left|\frac{y}{x + 4}\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	96.2%
Cost	7372

\[\begin{array}{l} t_0 := \left|\frac{x}{y} \cdot z - \frac{4}{y}\right|\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+224}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+33} \lor \neg \left(x \leq 8.5 \cdot 10^{+16}\right):\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 4
Accuracy	70.0%
Cost	7120

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \frac{4}{\left|y\right|}\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-119}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	69.1%
Cost	7120

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \frac{4}{\left|y\right|}\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-112}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	69.6%
Cost	7120

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \frac{4}{\left|y\right|}\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-112}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	69.3%
Cost	7120

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \frac{4}{\left|y\right|}\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-112}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	98.4%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 3.6\right):\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 9
Accuracy	83.1%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+32}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+52}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 10
Accuracy	70.6%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \end{array} \]

Alternative 11
Accuracy	49.0%
Cost	6592

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

Reproduce

herbie shell --seed 2023140 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))