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

Average Error: 7.5 → 0.9

Time: 9.6s

Precision: binary64

Cost: 7748

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+291}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

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)) a)))
   (if (<= t_1 -5e+296)
     (fma -1.0 (/ t (/ a z)) (/ y (/ a x)))
     (if (<= t_1 2e+291) t_1 (- (/ x (/ a y)) (/ 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)) / a;
	double tmp;
	if (t_1 <= -5e+296) {
		tmp = fma(-1.0, (t / (a / z)), (y / (a / x)));
	} else if (t_1 <= 2e+291) {
		tmp = t_1;
	} else {
		tmp = (x / (a / y)) - (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(Float64(x * y) - Float64(z * t)) / a)
	tmp = 0.0
	if (t_1 <= -5e+296)
		tmp = fma(-1.0, Float64(t / Float64(a / z)), Float64(y / Float64(a / x)));
	elseif (t_1 <= 2e+291)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / Float64(a / y)) - 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[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+296], N[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+291], t$95$1, N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.5
Target	5.7
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 (*.f64 x y) (*.f64 z t)) a) < -5.0000000000000001e296

   1. Initial program 56.7

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

   2. Taylor expanded in x around 0 56.7

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

   3. Simplified 3.1

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

      [Start] 56.7	\[ -1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a} \]
      fma-def [=>] 56.7	\[ \color{blue}{\mathsf{fma}\left(-1, \frac{t \cdot z}{a}, \frac{y \cdot x}{a}\right)} \]
      associate-/l* [=>] 31.1	\[ \mathsf{fma}\left(-1, \color{blue}{\frac{t}{\frac{a}{z}}}, \frac{y \cdot x}{a}\right) \]
      associate-/l* [=>] 3.1	\[ \mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \color{blue}{\frac{y}{\frac{a}{x}}}\right) \]

   if -5.0000000000000001e296 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 1.9999999999999999e291

   1. Initial program 0.6

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

   if 1.9999999999999999e291 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 54.4

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

   2. Applied egg-rr 2.9

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} \leq -5 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{elif}\;\frac{x \cdot y - z \cdot t}{a} \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	1993

\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+299} \lor \neg \left(t_1 \leq 2 \cdot 10^{+291}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	4.3
Cost	1864

\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;t_1 \leq 10^{+297}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \]

Alternative 3
Error	23.6
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00018 \lor \neg \left(z \leq 1.15 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{t}{a} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]

Alternative 4
Error	23.3
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00019 \lor \neg \left(z \leq 6 \cdot 10^{-66}\right):\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]

Alternative 5
Error	23.1
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-5} \lor \neg \left(z \leq 1.85 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]

Alternative 6
Error	32.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-48} \lor \neg \left(z \leq -2 \cdot 10^{-88}\right):\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 7
Error	32.5
Cost	320

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

Alternative 8
Error	33.1
Cost	320

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

Reproduce

herbie shell --seed 2023039 
(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))