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

Average Error: 7.7 → 1.0

Time: 8.7s

Precision: binary64

Cost: 8136

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+279}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+170}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

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 -5e+279)
     (- (/ x (/ a y)) (/ z (/ a t)))
     (if (<= t_1 2e+170) (/ 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 <= -5e+279) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if (t_1 <= 2e+170) {
		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 <= -5e+279)
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	elseif (t_1 <= 2e+170)
		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, -5e+279], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+170], 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	5.8
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -5.0000000000000002e279

   1. Initial program 48.1

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

   2. Applied egg-rr 0.3

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

   if -5.0000000000000002e279 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2.00000000000000007e170

   1. Initial program 0.9

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

   if 2.00000000000000007e170 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 25.3

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

   2. Taylor expanded in x around 0 25.3

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a}} \]

   3. Simplified 1.7

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

      [Start] 25.3	\[ -1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a} \]
      fma-def [=>] 25.3	\[ \color{blue}{\mathsf{fma}\left(-1, \frac{t \cdot z}{a}, \frac{y \cdot x}{a}\right)} \]
      associate-/l* [=>] 14.5	\[ \mathsf{fma}\left(-1, \color{blue}{\frac{t}{\frac{a}{z}}}, \frac{y \cdot x}{a}\right) \]
      associate-/l* [=>] 1.7	\[ \mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \color{blue}{\frac{y}{\frac{a}{x}}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+279}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+170}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	4.5
Cost	1864

\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \]

Alternative 2
Error	0.8
Cost	1864

\[\begin{array}{l} t_1 := \frac{x}{\frac{a}{y}}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+279}:\\ \;\;\;\;t_1 - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_2 \leq 10^{+281}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \frac{\frac{z}{a}}{\frac{1}{t}}\\ \end{array} \]

Alternative 3
Error	0.8
Cost	1737

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+279} \lor \neg \left(t_1 \leq 10^{+281}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]

Alternative 4
Error	0.8
Cost	1736

\[\begin{array}{l} t_1 := \frac{z}{\frac{a}{t}}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+279}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - t_1\\ \mathbf{elif}\;t_2 \leq 10^{+281}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - t_1\\ \end{array} \]

Alternative 5
Error	18.5
Cost	1164

\[\begin{array}{l} t_1 := t \cdot \frac{-z}{a}\\ \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq -4 \cdot 10^{-147}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-70}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	24.3
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-134}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 7
Error	31.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 8
Error	32.7
Cost	320

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

Alternative 9
Error	32.7
Cost	320

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

Reproduce

herbie shell --seed 2023012 
(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))