Data.Colour.Matrix:inverse from colour-2.3.3, B

Average Error: 7.7 → 0.9

Time: 15.1s

Precision: binary64

Cost: 8136

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

\[\frac{x \cdot y - z \cdot t}{a} \]

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+249}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (/ (- (* x y) (* z t)) a))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 -1e+249)
     (- (/ x (/ a y)) (/ z (/ a t)))
     (if (<= t_1 2e+194) (/ t_1 a) (fma -1.0 (/ t (/ a z)) (/ y (/ a x)))))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (z * t)) / a;
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -1e+249) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if (t_1 <= 2e+194) {
		tmp = t_1 / a;
	} else {
		tmp = fma(-1.0, (t / (a / z)), (y / (a / x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= -1e+249)
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	elseif (t_1 <= 2e+194)
		tmp = Float64(t_1 / a);
	else
		tmp = fma(-1.0, Float64(t / Float64(a / z)), Float64(y / Float64(a / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+249], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+194], N[(t$95$1 / a), $MachinePrecision], N[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.7
Target	6.4
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -9.9999999999999992e248

   1. Initial program 37.5

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Applied egg-rr 0.8

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}} \]

   if -9.9999999999999992e248 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.99999999999999989e194

   1. Initial program 0.8

      \[\frac{x \cdot y - z \cdot t}{a} \]

   if 1.99999999999999989e194 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 27.3

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in a around 0 27.3

      \[\leadsto \color{blue}{\frac{y \cdot x - t \cdot z}{a}} \]

   3. Simplified 1.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)} \]
      Proof
      (fma.f64 -1 (/.f64 t (/.f64 a z)) (/.f64 y (/.f64 a x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t z) a)) (/.f64 y (/.f64 a x))): 24 points increase in error, 30 points decrease in error
      (fma.f64 -1 (/.f64 (*.f64 t z) a) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y x) a))): 35 points increase in error, 32 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 t z) a)) (/.f64 (*.f64 y x) a))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 (*.f64 y x) a) (*.f64 -1 (/.f64 (*.f64 t z) a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y x) a) (Rewrite=> mul-1-neg_binary64 (neg.f64 (/.f64 (*.f64 t z) a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> unsub-neg_binary64 (-.f64 (/.f64 (*.f64 y x) a) (/.f64 (*.f64 t z) a))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 y x) (*.f64 t z)) a)): 2 points increase in error, 5 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+249}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	1736

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+249}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	4.2
Cost	1608

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \]

Alternative 3
Error	25.2
Cost	1440

\[\begin{array}{l} t_1 := t \cdot \frac{-z}{a}\\ t_2 := \frac{-z}{\frac{a}{t}}\\ t_3 := y \cdot \frac{x}{a}\\ \mathbf{if}\;t \leq -2.318350207130133 \cdot 10^{-78}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.0885942739481147 \cdot 10^{-85}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;t \leq 2.0454223489668182 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 68711.01731258573:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.1952633833528638 \cdot 10^{+35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+71}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+153}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	25.2
Cost	1440

\[\begin{array}{l} t_1 := t \cdot \frac{-z}{a}\\ t_2 := \frac{-z}{\frac{a}{t}}\\ t_3 := y \cdot \frac{x}{a}\\ \mathbf{if}\;t \leq -2.318350207130133 \cdot 10^{-78}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.0885942739481147 \cdot 10^{-85}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;t \leq 2.0454223489668182 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 68711.01731258573:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.1952633833528638 \cdot 10^{+35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+71}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+99}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+153}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	25.3
Cost	1440

\[\begin{array}{l} t_1 := \frac{-z}{\frac{a}{t}}\\ t_2 := t \cdot \frac{-z}{a}\\ t_3 := y \cdot \frac{x}{a}\\ \mathbf{if}\;t \leq -2.318350207130133 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.0885942739481147 \cdot 10^{-85}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y}}\\ \mathbf{elif}\;t \leq 2.0454223489668182 \cdot 10^{-47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 68711.01731258573:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.1952633833528638 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+71}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+99}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+153}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	25.8
Cost	1176

\[\begin{array}{l} t_1 := \frac{-z}{\frac{a}{t}}\\ t_2 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;y \leq -3.7621125855454825 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 9.838452958616032 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+111}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	25.8
Cost	1176

\[\begin{array}{l} t_1 := \frac{-z}{\frac{a}{t}}\\ t_2 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;y \leq -3.7621125855454825 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 9.838452958616032 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+66}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+111}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	32.1
Cost	716

\[\begin{array}{l} \mathbf{if}\;t \leq -2.5521299367014494 \cdot 10^{-247}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+237}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 9
Error	32.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -4.791424674612273 \cdot 10^{-263}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 10
Error	31.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{+78}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 11
Error	32.4
Cost	320

\[\frac{y}{\frac{a}{x}} \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

  (/ (- (* x y) (* z t)) a))