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

Average Error: 7.5 → 1.4

Time: 6.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+181}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - t \cdot \frac{z}{a}\\ \mathbf{elif}\;t_1 \leq 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a}, t_1, \frac{1}{a} \cdot \mathsf{fma}\left(z, -t, z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, z \cdot \left(-\frac{t}{a}\right)\right)\\ \end{array} \]

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+181)
     (- (/ x (/ a y)) (* t (/ z a)))
     (if (<= t_1 1e+151)
       (fma (/ 1.0 a) t_1 (* (/ 1.0 a) (fma z (- t) (* z t))))
       (fma x (/ y a) (* z (- (/ t a))))))))

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+181) {
		tmp = (x / (a / y)) - (t * (z / a));
	} else if (t_1 <= 1e+151) {
		tmp = fma((1.0 / a), t_1, ((1.0 / a) * fma(z, -t, (z * t))));
	} else {
		tmp = fma(x, (y / a), (z * -(t / a)));
	}
	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+181)
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(t * Float64(z / a)));
	elseif (t_1 <= 1e+151)
		tmp = fma(Float64(1.0 / a), t_1, Float64(Float64(1.0 / a) * fma(z, Float64(-t), Float64(z * t))));
	else
		tmp = fma(x, Float64(y / a), Float64(z * Float64(-Float64(t / a))));
	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+181], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+151], N[(N[(1.0 / a), $MachinePrecision] * t$95$1 + N[(N[(1.0 / a), $MachinePrecision] * N[(z * (-t) + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision] + N[(z * (-N[(t / a), $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	7.5
Target	5.7
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -5.0000000000000003e181

   1. Initial program 25.1

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

   2. Applied egg-rr 1.3

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

   3. Applied egg-rr 1.6

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

   if -5.0000000000000003e181 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.00000000000000002e151

   1. Initial program 1.0

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

   2. Applied egg-rr 1.0

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

   3. Applied egg-rr 1.0

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

   if 1.00000000000000002e151 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 20.7

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

   2. Applied egg-rr 3.0

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

   3. Applied egg-rr 2.9

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+181}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - t \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a}, x \cdot y - z \cdot t, \frac{1}{a} \cdot \mathsf{fma}\left(z, -t, z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, z \cdot \left(-\frac{t}{a}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(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))