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

Average Error: 7.1 → 0.9

Time: 5.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ t_2 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 1.197418276934376 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, -z, x \cdot y\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ a y)) (/ z (/ a t)))) (t_2 (/ (- (* x y) (* z t)) a)))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 1.197418276934376e+292) (/ (fma t (- z) (* x y)) a) t_1))))

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 / (a / y)) - (z / (a / t));
	double t_2 = ((x * y) - (z * t)) / a;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1.197418276934376e+292) {
		tmp = fma(t, -z, (x * y)) / a;
	} else {
		tmp = t_1;
	}
	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 / Float64(a / y)) - Float64(z / Float64(a / t)))
	t_2 = Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1.197418276934376e+292)
		tmp = Float64(fma(t, Float64(-z), Float64(x * y)) / a);
	else
		tmp = t_1;
	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 / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1.197418276934376e+292], N[(N[(t * (-z) + N[(x * y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.1
Target	5.2
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 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < -inf.0 or 1.1974182769343759e292 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 58.4

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

   2. Applied egg-rr 2.5

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 1.1974182769343759e292

   1. Initial program 0.7

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

   2. Taylor expanded in x around 0 0.7

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

   3. Simplified 0.7

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, -z, y \cdot x\right)}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;\frac{x \cdot y - z \cdot t}{a} \leq 1.197418276934376 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, -z, x \cdot y\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))