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

Average Error: 7.3 → 0.7

Time: 5.5s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -2.4568731715777092 \cdot 10^{+275}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{a}, z, x \cdot \frac{y}{a}\right)\\ \mathbf{elif}\;t_1 \leq 3.264168360365097 \cdot 10^{+290}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t \cdot \left(-z\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \end{array} \]

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 -2.4568731715777092e+275)
     (fma (/ (- t) a) z (* x (/ y a)))
     (if (<= t_1 3.264168360365097e+290)
       (/ (fma x y (* t (- z))) 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 <= -2.4568731715777092e+275) {
		tmp = fma((-t / a), z, (x * (y / a)));
	} else if (t_1 <= 3.264168360365097e+290) {
		tmp = fma(x, y, (t * -z)) / 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 <= -2.4568731715777092e+275)
		tmp = fma(Float64(Float64(-t) / a), z, Float64(x * Float64(y / a)));
	elseif (t_1 <= 3.264168360365097e+290)
		tmp = Float64(fma(x, y, Float64(t * Float64(-z))) / 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, -2.4568731715777092e+275], N[(N[((-t) / a), $MachinePrecision] * z + N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 3.264168360365097e+290], N[(N[(x * y + N[(t * (-z)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision] + N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.3
Target	5.8
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)) < -2.4568731715777092e275

   1. Initial program 46.7

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

   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. Applied egg-rr 0.4

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

   4. Applied egg-rr 0.4

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

   if -2.4568731715777092e275 < (-.f64 (*.f64 x y) (*.f64 z t)) < 3.2641683603650968e290

   1. Initial program 0.7

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

   2. Applied egg-rr 0.7

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

   if 3.2641683603650968e290 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 53.5

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -2.4568731715777092 \cdot 10^{+275}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{a}, z, x \cdot \frac{y}{a}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 3.264168360365097 \cdot 10^{+290}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t \cdot \left(-z\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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))