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

Average Error: 7.3 → 1.3

Time: 5.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+180}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -t, x \cdot y\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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))) (t_2 (fma x (/ y a) (/ (- z) (/ a t)))))
   (if (<= t_1 -5e+180)
     t_2
     (if (<= t_1 2e+152) (/ (fma z (- t) (* x y)) a) t_2))))

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 t_2 = fma(x, (y / a), (-z / (a / t)));
	double tmp;
	if (t_1 <= -5e+180) {
		tmp = t_2;
	} else if (t_1 <= 2e+152) {
		tmp = fma(z, -t, (x * y)) / a;
	} else {
		tmp = t_2;
	}
	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))
	t_2 = fma(x, Float64(y / a), Float64(Float64(-z) / Float64(a / t)))
	tmp = 0.0
	if (t_1 <= -5e+180)
		tmp = t_2;
	elseif (t_1 <= 2e+152)
		tmp = Float64(fma(z, Float64(-t), Float64(x * y)) / a);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(x * N[(y / a), $MachinePrecision] + N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+180], t$95$2, If[LessEqual[t$95$1, 2e+152], N[(N[(z * (-t) + N[(x * y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$2]]]]

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	5.3
Herbie	1.3

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -4.9999999999999996e180 or 2.0000000000000001e152 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 22.3

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

   2. Applied egg-rr 2.3

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

   if -4.9999999999999996e180 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2.0000000000000001e152

   1. Initial program 0.9

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

   2. Applied egg-rr 1.4

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot {\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y}, z \cdot \left(-t\right)\right)}}{a} \]

   3. Taylor expanded in x around 0 0.9

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

   4. Simplified 0.8

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+180}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -t, x \cdot y\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))