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

Average Error: 7.9 → 0.8

Time: 13.7s

Precision: binary64

Cost: 8072

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+266}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{+214}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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))) (t_2 (fma x (/ y a) (/ (- z) (/ a t)))))
   (if (<= t_1 -2e+266) t_2 (if (<= t_1 1e+214) (/ t_1 a) t_2))))

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 t_2 = fma(x, (y / a), (-z / (a / t)));
	double tmp;
	if (t_1 <= -2e+266) {
		tmp = t_2;
	} else if (t_1 <= 1e+214) {
		tmp = t_1 / a;
	} else {
		tmp = t_2;
	}
	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))
	t_2 = fma(x, Float64(y / a), Float64(Float64(-z) / Float64(a / t)))
	tmp = 0.0
	if (t_1 <= -2e+266)
		tmp = t_2;
	elseif (t_1 <= 1e+214)
		tmp = Float64(t_1 / a);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(x * N[(y / a), $MachinePrecision] + N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+266], t$95$2, If[LessEqual[t$95$1, 1e+214], N[(t$95$1 / a), $MachinePrecision], t$95$2]]]]

Target

Original	7.9
Target	5.9
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -2.0000000000000001e266 or 9.9999999999999995e213 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 37.8

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

   2. Applied egg-rr 0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\frac{z}{\frac{a}{t}}\right)} \]

   if -2.0000000000000001e266 < (-.f64 (*.f64 x y) (*.f64 z t)) < 9.9999999999999995e213

   1. Initial program 0.9

      \[\frac{x \cdot y - z \cdot t}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -2 \cdot 10^{+266}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+214}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.0
Cost	1864

\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{elif}\;t_1 \leq 10^{+281}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 2
Error	0.9
Cost	1736

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+238}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{+214}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	23.4
Cost	912

\[\begin{array}{l} t_1 := z \cdot \frac{t}{-a}\\ \mathbf{if}\;z \leq -9.6 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.8453643417257675 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -5.8115841226768406 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.418167049913747 \cdot 10^{-151}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	24.7
Cost	912

\[\begin{array}{l} \mathbf{if}\;x \leq -0.011050384226630655:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;x \leq -4.7849893519551545 \cdot 10^{-60}:\\ \;\;\;\;\frac{t}{\frac{a}{-z}}\\ \mathbf{elif}\;x \leq -4.0716769304200983 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 2.8174570250262018 \cdot 10^{-151}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 5
Error	24.7
Cost	912

\[\begin{array}{l} \mathbf{if}\;x \leq -0.011050384226630655:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;x \leq -4.7849893519551545 \cdot 10^{-60}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;x \leq -4.0716769304200983 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 2.8174570250262018 \cdot 10^{-151}:\\ \;\;\;\;z \cdot \frac{t}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 6
Error	32.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 1.3887580629653318 \cdot 10^{-210}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 7
Error	32.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -7.144789600729026 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]

Alternative 8
Error	32.3
Cost	320

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

Reproduce

herbie shell --seed 2022311 
(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))