fabs fraction 1

Average Accuracy: 97.4% → 99.5%

Time: 11.2s

Precision: binary64

Cost: 7368

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+31}:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-103}:\\ \;\;\;\;\left|t_0 - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_0 - z \cdot \frac{x}{y}\right|\\ \end{array} \]

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)))
   (if (<= x -2e+31)
     (fabs (/ (- 1.0 z) (/ y x)))
     (if (<= x 3e-103)
       (fabs (- t_0 (/ (* x z) y)))
       (fabs (- t_0 (* z (/ x y))))))))

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 tmp;
	if (x <= -2e+31) {
		tmp = fabs(((1.0 - z) / (y / x)));
	} else if (x <= 3e-103) {
		tmp = fabs((t_0 - ((x * z) / y)));
	} else {
		tmp = fabs((t_0 - (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
    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 = (x + 4.0d0) / y
    if (x <= (-2d+31)) then
        tmp = abs(((1.0d0 - z) / (y / x)))
    else if (x <= 3d-103) then
        tmp = abs((t_0 - ((x * z) / y)))
    else
        tmp = abs((t_0 - (z * (x / y))))
    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 tmp;
	if (x <= -2e+31) {
		tmp = Math.abs(((1.0 - z) / (y / x)));
	} else if (x <= 3e-103) {
		tmp = Math.abs((t_0 - ((x * z) / y)));
	} else {
		tmp = Math.abs((t_0 - (z * (x / y))));
	}
	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
	tmp = 0
	if x <= -2e+31:
		tmp = math.fabs(((1.0 - z) / (y / x)))
	elif x <= 3e-103:
		tmp = math.fabs((t_0 - ((x * z) / y)))
	else:
		tmp = math.fabs((t_0 - (z * (x / y))))
	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)
	tmp = 0.0
	if (x <= -2e+31)
		tmp = abs(Float64(Float64(1.0 - z) / Float64(y / x)));
	elseif (x <= 3e-103)
		tmp = abs(Float64(t_0 - Float64(Float64(x * z) / y)));
	else
		tmp = abs(Float64(t_0 - Float64(z * Float64(x / y))));
	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;
	tmp = 0.0;
	if (x <= -2e+31)
		tmp = abs(((1.0 - z) / (y / x)));
	elseif (x <= 3e-103)
		tmp = abs((t_0 - ((x * z) / y)));
	else
		tmp = abs((t_0 - (z * (x / y))));
	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]}, If[LessEqual[x, -2e+31], N[Abs[N[(N[(1.0 - z), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3e-103], N[Abs[N[(t$95$0 - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$0 - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9999999999999999e31

   1. Initial program 99.8%

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

   2. Simplified 84.4%

      \[\leadsto \color{blue}{\left|\frac{x - \mathsf{fma}\left(x, z, -4\right)}{y}\right|} \]
      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/ [=>] 84.4	\[ \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right| \]
      div-sub [<=] 84.4	\[ \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]
      /-rgt-identity [<=] 84.4	\[ \left|\frac{\left(x + 4\right) - x \cdot z}{\color{blue}{\frac{y}{1}}}\right| \]
      metadata-eval [<=] 84.4	\[ \left|\frac{\left(x + 4\right) - x \cdot z}{\frac{y}{\color{blue}{--1}}}\right| \]
      associate-/l* [<=] 84.4	\[ \left|\color{blue}{\frac{\left(\left(x + 4\right) - x \cdot z\right) \cdot \left(--1\right)}{y}}\right| \]
      *-commutative [=>] 84.4	\[ \left|\frac{\color{blue}{\left(--1\right) \cdot \left(\left(x + 4\right) - x \cdot z\right)}}{y}\right| \]

   3. Taylor expanded in x around inf 84.3%

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

   4. Simplified 99.6%

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

      [Start] 84.3	\[ \left|\frac{\left(1 - z\right) \cdot x}{y}\right| \]
      associate-/l* [=>] 99.6	\[ \left|\color{blue}{\frac{1 - z}{\frac{y}{x}}}\right| \]

   if -1.9999999999999999e31 < x < 3e-103

   1. Initial program 95.9%

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

   2. Taylor expanded in x around 0 99.8%

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

   if 3e-103 < x

   1. Initial program 99.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+31}:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-103}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	68.0%
Cost	7512

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{+182}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -220000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -0.0095:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	67.9%
Cost	7512

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{+181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3200000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -0.012:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-22}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	99.5%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+29}:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-103}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 4
Accuracy	85.8%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 5
Accuracy	99.8%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+32}:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 6
Accuracy	80.5%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.52 \lor \neg \left(x \leq 1.8 \cdot 10^{-22}\right):\\ \;\;\;\;\left|\frac{x \cdot \left(1 - z\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \end{array} \]

Alternative 7
Accuracy	85.8%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.75 \lor \neg \left(x \leq 2.5 \cdot 10^{-22}\right):\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \end{array} \]

Alternative 8
Accuracy	82.5%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+103}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+74}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 9
Accuracy	70.0%
Cost	6857

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

Alternative 10
Accuracy	48.7%
Cost	6592

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

Reproduce

herbie shell --seed 2023133 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))