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

Average Error: 11.77% → 1.22%

Time: 10.4s

Precision: binary64

Cost: 7945

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+276} \lor \neg \left(t_1 \leq 2 \cdot 10^{+210}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \end{array} \]

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 (or (<= t_1 -5e+276) (not (<= t_1 2e+210)))
     (- (/ x (/ a y)) (/ z (/ a t)))
     (/ (fma x y (* z (- t))) a))))

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 <= -5e+276) || !(t_1 <= 2e+210)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = fma(x, y, (z * -t)) / a;
	}
	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 <= -5e+276) || !(t_1 <= 2e+210))
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	else
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a);
	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[Or[LessEqual[t$95$1, -5e+276], N[Not[LessEqual[t$95$1, 2e+210]], $MachinePrecision]], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Target

Original	11.77%
Target	8.85%
Herbie	1.22%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -5.00000000000000001e276 or 1.99999999999999985e210 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 57.48

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

   2. Applied egg-rr 1.46

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

   if -5.00000000000000001e276 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.99999999999999985e210

   1. Initial program 1.17

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

   2. Simplified 1.16

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

      [Start] 1.17	\[ \frac{x \cdot y - z \cdot t}{a} \]
      fma-neg [=>] 1.16	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y, -z \cdot t\right)}}{a} \]
      distribute-rgt-neg-in [=>] 1.16	\[ \frac{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-t\right)}\right)}{a} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.22

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+276} \lor \neg \left(x \cdot y - z \cdot t \leq 2 \cdot 10^{+210}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.43%
Cost	1864

\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+290}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \end{array} \]

Alternative 2
Error	1.22%
Cost	1737

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+276} \lor \neg \left(t_1 \leq 2 \cdot 10^{+210}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]

Alternative 3
Error	29.96%
Cost	1684

\[\begin{array}{l} t_1 := \frac{-z}{\frac{a}{t}}\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 10^{-68}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+262}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	37.06%
Cost	914

\[\begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+28} \lor \neg \left(z \leq -1.12 \cdot 10^{-27} \lor \neg \left(z \leq -6.2 \cdot 10^{-51}\right) \land z \leq 6.8 \cdot 10^{-136}\right):\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 5
Error	38.07%
Cost	913

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-27} \lor \neg \left(z \leq -1.7 \cdot 10^{-69}\right) \land z \leq 2.9 \cdot 10^{-207}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \]

Alternative 6
Error	37.86%
Cost	913

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+26}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-24} \lor \neg \left(z \leq -1.65 \cdot 10^{-69}\right) \land z \leq 2.9 \cdot 10^{-207}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \]

Alternative 7
Error	37.78%
Cost	913

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+27}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-23} \lor \neg \left(z \leq -7 \cdot 10^{-73}\right) \land z \leq 2.8 \cdot 10^{-207}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \end{array} \]

Alternative 8
Error	51.96%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-306} \lor \neg \left(x \leq 2.65 \cdot 10^{-152}\right):\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 9
Error	50.31%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-104} \lor \neg \left(x \leq 8.5 \cdot 10^{-135}\right):\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]

Alternative 10
Error	52.11%
Cost	320

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

Reproduce

herbie shell --seed 2023121 
(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))