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

Average Error: 7.7 → 0.7

Time: 10.9s

Precision: binary64

Cost: 8072

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+264}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \end{array} \]

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 (- INFINITY))
     (- (/ x (/ a y)) (/ z (/ a t)))
     (if (<= t_1 5e+264) (/ t_1 a) (fma x (/ y a) (/ (- z) (/ a t)))))))

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 <= -((double) INFINITY)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if (t_1 <= 5e+264) {
		tmp = t_1 / a;
	} else {
		tmp = fma(x, (y / a), (-z / (a / t)));
	}
	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 <= Float64(-Inf))
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	elseif (t_1 <= 5e+264)
		tmp = Float64(t_1 / a);
	else
		tmp = fma(x, Float64(y / a), Float64(Float64(-z) / Float64(a / t)));
	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, (-Infinity)], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+264], N[(t$95$1 / a), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision] + N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.7
Target	5.5
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0

   1. Initial program 64.0

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

   2. Applied egg-rr 0.2

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.00000000000000033e264

   1. Initial program 0.8

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

   if 5.00000000000000033e264 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 45.9

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

   2. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\frac{z}{\frac{a}{t}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+264}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	31.6
Cost	2364

\[\begin{array}{l} t_1 := -\frac{z \cdot t}{a}\\ t_2 := \left(-z\right) \cdot \frac{t}{a}\\ t_3 := \frac{x}{\frac{a}{y}}\\ t_4 := \frac{x \cdot y}{a}\\ \mathbf{if}\;a \leq -1.2730870475754085 \cdot 10^{+129}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -3.274912530705219 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -5.4 \cdot 10^{-23}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.15 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-155}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-193}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.42 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-89}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;a \leq 3.890129449987877 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.8147640579063267 \cdot 10^{+172}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 5.684383120183049 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.0086502889763638 \cdot 10^{+233}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.6694950571182055 \cdot 10^{+279}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	0.7
Cost	1736

\[\begin{array}{l} t_1 := \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	4.3
Cost	1608

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;t_1 \leq 10^{+285}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \end{array} \]

Alternative 4
Error	27.4
Cost	1176

\[\begin{array}{l} t_1 := \left(-z\right) \cdot \frac{t}{a}\\ t_2 := \frac{x}{\frac{a}{y}}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+168}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;z \leq -3.745526105770942 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -2.198231315383873 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.504351234452438 \cdot 10^{-283}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.0913829374307953 \cdot 10^{-224}:\\ \;\;\;\;-\frac{z \cdot t}{a}\\ \mathbf{elif}\;z \leq 1.405940892000433 \cdot 10^{-155}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	23.5
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -5.709523417542095 \cdot 10^{-56}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;y \leq 170413.23978528462:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 6
Error	31.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+48}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 7
Error	32.3
Cost	320

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

Alternative 8
Error	32.3
Cost	320

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

Reproduce

herbie shell --seed 2022316 
(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))