fabs fraction 1

Average Error: 1.5 → 0.5

Time: 9.9s

Precision: binary64

Cost: 13641

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+84} \lor \neg \left(y \leq 10^{-89}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x \cdot \left(1 - z\right)}{y}\right|\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4e+84) (not (<= y 1e-89)))
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))
   (fabs (/ (+ 4.0 (* x (- 1.0 z))) 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 tmp;
	if ((y <= -4e+84) || !(y <= 1e-89)) {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	} else {
		tmp = fabs(((4.0 + (x * (1.0 - z))) / 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)
	tmp = 0.0
	if ((y <= -4e+84) || !(y <= 1e-89))
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	else
		tmp = abs(Float64(Float64(4.0 + Float64(x * Float64(1.0 - z))) / y));
	end
	return 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[Or[LessEqual[y, -4e+84], N[Not[LessEqual[y, 1e-89]], $MachinePrecision]], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 + N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.00000000000000023e84 or 1.00000000000000004e-89 < y

   1. Initial program 2.5

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

   2. Simplified 0.4

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

      [Start] 2.5	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      fabs-sub [=>] 2.5	\[ \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]
      associate-*l/ [=>] 5.4	\[ \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]
      associate-*r/ [<=] 0.4	\[ \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]
      *-commutative [<=] 0.4	\[ \left|\color{blue}{\frac{z}{y} \cdot x} - \frac{x + 4}{y}\right| \]
      *-commutative [=>] 0.4	\[ \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]
      fma-neg [=>] 0.4	\[ \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]
      distribute-neg-frac [=>] 0.4	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]
      neg-sub0 [=>] 0.4	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{0 - \left(x + 4\right)}}{y}\right)\right| \]
      +-commutative [=>] 0.4	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{0 - \color{blue}{\left(4 + x\right)}}{y}\right)\right| \]
      associate--r+ [=>] 0.4	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(0 - 4\right) - x}}{y}\right)\right| \]
      metadata-eval [=>] 0.4	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   if -4.00000000000000023e84 < y < 1.00000000000000004e-89

   1. Initial program 0.1

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

   2. Simplified 0.6

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

      [Start] 0.1	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      *-lft-identity [<=] 0.1	\[ \color{blue}{1 \cdot \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]
      metadata-eval [<=] 0.1	\[ \color{blue}{\left|-1\right|} \cdot \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      fabs-sub [=>] 0.1	\[ \left|-1\right| \cdot \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]
      fabs-mul [<=] 0.1	\[ \color{blue}{\left|-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right|} \]
      neg-mul-1 [<=] 0.1	\[ \left|\color{blue}{-\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)}\right| \]
      sub0-neg [<=] 0.1	\[ \left|\color{blue}{0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)}\right| \]
      associate-+l- [<=] 0.1	\[ \left|\color{blue}{\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}}\right| \]
      neg-sub0 [<=] 0.1	\[ \left|\color{blue}{\left(-\frac{x}{y} \cdot z\right)} + \frac{x + 4}{y}\right| \]
      +-commutative [<=] 0.1	\[ \left|\color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)}\right| \]
      sub-neg [<=] 0.1	\[ \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]
      associate-*l/ [=>] 0.6	\[ \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right| \]
      div-sub [<=] 0.6	\[ \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]
      /-rgt-identity [<=] 0.6	\[ \left|\frac{\left(x + 4\right) - x \cdot z}{\color{blue}{\frac{y}{1}}}\right| \]
      metadata-eval [<=] 0.6	\[ \left|\frac{\left(x + 4\right) - x \cdot z}{\frac{y}{\color{blue}{--1}}}\right| \]
      associate-/l* [<=] 0.6	\[ \left|\color{blue}{\frac{\left(\left(x + 4\right) - x \cdot z\right) \cdot \left(--1\right)}{y}}\right| \]
      *-commutative [=>] 0.6	\[ \left|\frac{\color{blue}{\left(--1\right) \cdot \left(\left(x + 4\right) - x \cdot z\right)}}{y}\right| \]

   3. Taylor expanded in x around 0 0.6

      \[\leadsto \left|\frac{\color{blue}{4 + \left(1 - z\right) \cdot x}}{y}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+84} \lor \neg \left(y \leq 10^{-89}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x \cdot \left(1 - z\right)}{y}\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	8649

\[\begin{array}{l} t_0 := \frac{x + 4}{y} - z \cdot \frac{x}{y}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+33} \lor \neg \left(t_0 \leq 2 \cdot 10^{-63}\right):\\ \;\;\;\;\left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x \cdot \left(1 - z\right)}{y}\right|\\ \end{array} \]

Alternative 2
Error	20.9
Cost	7381

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{+145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+122} \lor \neg \left(x \leq 4.5 \cdot 10^{+269}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	20.8
Cost	7380

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+120}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+267}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7369

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+45} \lor \neg \left(y \leq 4.5 \cdot 10^{-91}\right):\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}} + \frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x \cdot \left(1 - z\right)}{y}\right|\\ \end{array} \]

Alternative 5
Error	12.3
Cost	7248

\[\begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x + 4}{y}\right|\\ \mathbf{if}\;z \leq -1.18 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+189}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 6
Error	8.7
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -2700000 \lor \neg \left(x \leq 22500000000\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \]

Alternative 7
Error	1.0
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 4.6\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 8
Error	1.9
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+44}:\\ \;\;\;\;\left|\frac{4 + x \cdot \left(1 - z\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \end{array} \]

Alternative 9
Error	19.0
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 10
Error	32.6
Cost	6592

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

Reproduce

herbie shell --seed 2023010 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))