fabs fraction 1

Average Error: 1.7 → 0.9

Time: 10.9s

Precision: binary64

Cost: 7368

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+201}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z + -1}}\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.4e+201)
   (fabs (/ x (/ y (+ z -1.0))))
   (if (<= x 2e+29)
     (fabs (- (/ (+ x 4.0) y) (/ (* x z) y)))
     (fabs (* (/ x y) (+ z -1.0))))))

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 tmp;
	if (x <= -2.4e+201) {
		tmp = fabs((x / (y / (z + -1.0))));
	} else if (x <= 2e+29) {
		tmp = fabs((((x + 4.0) / y) - ((x * z) / y)));
	} else {
		tmp = fabs(((x / y) * (z + -1.0)));
	}
	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) :: tmp
    if (x <= (-2.4d+201)) then
        tmp = abs((x / (y / (z + (-1.0d0)))))
    else if (x <= 2d+29) then
        tmp = abs((((x + 4.0d0) / y) - ((x * z) / y)))
    else
        tmp = abs(((x / y) * (z + (-1.0d0))))
    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 tmp;
	if (x <= -2.4e+201) {
		tmp = Math.abs((x / (y / (z + -1.0))));
	} else if (x <= 2e+29) {
		tmp = Math.abs((((x + 4.0) / y) - ((x * z) / y)));
	} else {
		tmp = Math.abs(((x / y) * (z + -1.0)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if x <= -2.4e+201:
		tmp = math.fabs((x / (y / (z + -1.0))))
	elif x <= 2e+29:
		tmp = math.fabs((((x + 4.0) / y) - ((x * z) / y)))
	else:
		tmp = math.fabs(((x / y) * (z + -1.0)))
	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)
	tmp = 0.0
	if (x <= -2.4e+201)
		tmp = abs(Float64(x / Float64(y / Float64(z + -1.0))));
	elseif (x <= 2e+29)
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x * z) / y)));
	else
		tmp = abs(Float64(Float64(x / y) * Float64(z + -1.0)));
	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)
	tmp = 0.0;
	if (x <= -2.4e+201)
		tmp = abs((x / (y / (z + -1.0))));
	elseif (x <= 2e+29)
		tmp = abs((((x + 4.0) / y) - ((x * z) / y)));
	else
		tmp = abs(((x / y) * (z + -1.0)));
	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_] := If[LessEqual[x, -2.4e+201], N[Abs[N[(x / N[(y / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2e+29], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.39999999999999993e201

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      fabs-sub [=>] 0.1	\[ \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]
      associate-*l/ [=>] 17.0	\[ \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]
      associate-*r/ [<=] 0.1	\[ \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]
      *-commutative [<=] 0.1	\[ \left|\color{blue}{\frac{z}{y} \cdot x} - \frac{x + 4}{y}\right| \]
      *-commutative [=>] 0.1	\[ \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]
      fma-neg [=>] 0.1	\[ \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]
      distribute-neg-frac [=>] 0.1	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]
      neg-sub0 [=>] 0.1	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{0 - \left(x + 4\right)}}{y}\right)\right| \]
      +-commutative [=>] 0.1	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{0 - \color{blue}{\left(4 + x\right)}}{y}\right)\right| \]
      associate--r+ [=>] 0.1	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(0 - 4\right) - x}}{y}\right)\right| \]
      metadata-eval [=>] 0.1	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Taylor expanded in x around inf 0.3

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

   4. Simplified 0.1

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

      [Start] 0.3	\[ \left|\left(\frac{z}{y} - \frac{1}{y}\right) \cdot x\right| \]
      *-commutative [=>] 0.3	\[ \left|\color{blue}{x \cdot \left(\frac{z}{y} - \frac{1}{y}\right)}\right| \]
      sub-neg [=>] 0.3	\[ \left|x \cdot \color{blue}{\left(\frac{z}{y} + \left(-\frac{1}{y}\right)\right)}\right| \]
      distribute-lft-in [=>] 0.3	\[ \left|\color{blue}{x \cdot \frac{z}{y} + x \cdot \left(-\frac{1}{y}\right)}\right| \]
      distribute-neg-frac [=>] 0.3	\[ \left|x \cdot \frac{z}{y} + x \cdot \color{blue}{\frac{-1}{y}}\right| \]
      metadata-eval [=>] 0.3	\[ \left|x \cdot \frac{z}{y} + x \cdot \frac{\color{blue}{-1}}{y}\right| \]
      associate-*r/ [=>] 0.1	\[ \left|x \cdot \frac{z}{y} + \color{blue}{\frac{x \cdot -1}{y}}\right| \]
      *-commutative [<=] 0.1	\[ \left|x \cdot \frac{z}{y} + \frac{\color{blue}{-1 \cdot x}}{y}\right| \]
      associate-*r/ [<=] 0.1	\[ \left|x \cdot \frac{z}{y} + \color{blue}{-1 \cdot \frac{x}{y}}\right| \]
      mul-1-neg [=>] 0.1	\[ \left|x \cdot \frac{z}{y} + \color{blue}{\left(-\frac{x}{y}\right)}\right| \]
      sub-neg [<=] 0.1	\[ \left|\color{blue}{x \cdot \frac{z}{y} - \frac{x}{y}}\right| \]
      *-commutative [=>] 0.1	\[ \left|\color{blue}{\frac{z}{y} \cdot x} - \frac{x}{y}\right| \]
      associate-/r/ [<=] 0.1	\[ \left|\color{blue}{\frac{z}{\frac{y}{x}}} - \frac{x}{y}\right| \]

   5. Applied egg-rr 0.3

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

   6. Applied egg-rr 0.1

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

   if -2.39999999999999993e201 < x < 1.99999999999999983e29

   1. Initial program 2.2

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

   2. Taylor expanded in x around 0 1.2

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

   if 1.99999999999999983e29 < x

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      fabs-sub [=>] 0.1	\[ \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]
      associate-*l/ [=>] 9.1	\[ \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]
      associate-*r/ [<=] 0.1	\[ \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]
      *-commutative [<=] 0.1	\[ \left|\color{blue}{\frac{z}{y} \cdot x} - \frac{x + 4}{y}\right| \]
      *-commutative [=>] 0.1	\[ \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]
      fma-neg [=>] 0.1	\[ \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]
      distribute-neg-frac [=>] 0.1	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]
      neg-sub0 [=>] 0.1	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{0 - \left(x + 4\right)}}{y}\right)\right| \]
      +-commutative [=>] 0.1	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{0 - \color{blue}{\left(4 + x\right)}}{y}\right)\right| \]
      associate--r+ [=>] 0.1	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(0 - 4\right) - x}}{y}\right)\right| \]
      metadata-eval [=>] 0.1	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   3. Taylor expanded in x around inf 0.3

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

   4. Simplified 0.1

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

      [Start] 0.3	\[ \left|\left(\frac{z}{y} - \frac{1}{y}\right) \cdot x\right| \]
      *-commutative [=>] 0.3	\[ \left|\color{blue}{x \cdot \left(\frac{z}{y} - \frac{1}{y}\right)}\right| \]
      sub-neg [=>] 0.3	\[ \left|x \cdot \color{blue}{\left(\frac{z}{y} + \left(-\frac{1}{y}\right)\right)}\right| \]
      distribute-lft-in [=>] 0.3	\[ \left|\color{blue}{x \cdot \frac{z}{y} + x \cdot \left(-\frac{1}{y}\right)}\right| \]
      distribute-neg-frac [=>] 0.3	\[ \left|x \cdot \frac{z}{y} + x \cdot \color{blue}{\frac{-1}{y}}\right| \]
      metadata-eval [=>] 0.3	\[ \left|x \cdot \frac{z}{y} + x \cdot \frac{\color{blue}{-1}}{y}\right| \]
      associate-*r/ [=>] 0.1	\[ \left|x \cdot \frac{z}{y} + \color{blue}{\frac{x \cdot -1}{y}}\right| \]
      *-commutative [<=] 0.1	\[ \left|x \cdot \frac{z}{y} + \frac{\color{blue}{-1 \cdot x}}{y}\right| \]
      associate-*r/ [<=] 0.1	\[ \left|x \cdot \frac{z}{y} + \color{blue}{-1 \cdot \frac{x}{y}}\right| \]
      mul-1-neg [=>] 0.1	\[ \left|x \cdot \frac{z}{y} + \color{blue}{\left(-\frac{x}{y}\right)}\right| \]
      sub-neg [<=] 0.1	\[ \left|\color{blue}{x \cdot \frac{z}{y} - \frac{x}{y}}\right| \]
      *-commutative [=>] 0.1	\[ \left|\color{blue}{\frac{z}{y} \cdot x} - \frac{x}{y}\right| \]
      associate-/r/ [<=] 0.1	\[ \left|\color{blue}{\frac{z}{\frac{y}{x}}} - \frac{x}{y}\right| \]

   5. Applied egg-rr 0.1

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+201}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z + -1}}\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	12.3
Cost	7248

\[\begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{-4 - x}{y}\right|\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 15000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 2
Error	0.9
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -95000:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z + -1}}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \end{array} \]

Alternative 3
Error	0.9
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -95000:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}} - \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 3.9:\\ \;\;\;\;\left|\frac{4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \end{array} \]

Alternative 4
Error	0.4
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+119}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z + -1}}\right|\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \end{array} \]

Alternative 5
Error	8.8
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -5500 \lor \neg \left(x \leq 14500000\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 6
Error	8.8
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00076 \lor \neg \left(x \leq 9000000\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \end{array} \]

Alternative 7
Error	8.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -15500:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z + -1}}\right|\\ \mathbf{elif}\;x \leq 7800000:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \end{array} \]

Alternative 8
Error	0.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -95000:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z + -1}}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \end{array} \]

Alternative 9
Error	18.9
Cost	6857

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

Alternative 10
Error	32.4
Cost	6592

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

Reproduce

herbie shell --seed 2023027 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))