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

Average Error: 7.2 → 0.8

Time: 11.9s

Precision: binary64

Cost: 8392

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

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ t_2 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (fma -1.0 (/ t (/ a z)) (/ y (/ a x))))
        (t_2 (/ (- (* x y) (* z t)) a)))
   (if (<= t_2 -5e+304) t_1 (if (<= t_2 5e+303) t_2 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 = fma(-1.0, (t / (a / z)), (y / (a / x)));
	double t_2 = ((x * y) - (z * t)) / a;
	double tmp;
	if (t_2 <= -5e+304) {
		tmp = t_1;
	} else if (t_2 <= 5e+303) {
		tmp = t_2;
	} 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 = fma(-1.0, Float64(t / Float64(a / z)), Float64(y / Float64(a / x)))
	t_2 = Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
	tmp = 0.0
	if (t_2 <= -5e+304)
		tmp = t_1;
	elseif (t_2 <= 5e+303)
		tmp = t_2;
	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[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(a / x), $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, -5e+304], t$95$1, If[LessEqual[t$95$2, 5e+303], t$95$2, t$95$1]]]]

Target

Original	7.2
Target	5.6
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < -4.9999999999999997e304 or 4.9999999999999997e303 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 61.7

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

   2. Taylor expanded in a around 0 61.7

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

   3. Simplified 1.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)} \]
      Proof
      (fma.f64 -1 (/.f64 t (/.f64 a z)) (/.f64 y (/.f64 a x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t z) a)) (/.f64 y (/.f64 a x))): 25 points increase in error, 39 points decrease in error
      (fma.f64 -1 (/.f64 (*.f64 t z) a) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y x) a))): 33 points increase in error, 32 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 t z) a)) (/.f64 (*.f64 y x) a))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 (*.f64 y x) a) (*.f64 -1 (/.f64 (*.f64 t z) a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y x) a) (Rewrite=> mul-1-neg_binary64 (neg.f64 (/.f64 (*.f64 t z) a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> unsub-neg_binary64 (-.f64 (/.f64 (*.f64 y x) a) (/.f64 (*.f64 t z) a))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 y x) (*.f64 t z)) a)): 0 points increase in error, 5 points decrease in error

   if -4.9999999999999997e304 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 4.9999999999999997e303

   1. Initial program 0.8

      \[\frac{x \cdot y - z \cdot t}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} \leq -5 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{elif}\;\frac{x \cdot y - z \cdot t}{a} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	1992

\[\begin{array}{l} t_1 := \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ t_2 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{+292}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	3.9
Cost	1608

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;t_1 \leq 10^{+308}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \]

Alternative 3
Error	16.6
Cost	1424

\[\begin{array}{l} t_1 := \frac{z \cdot \left(-t\right)}{a}\\ t_2 := z \cdot \frac{-t}{a}\\ \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-55}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+207}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	24.7
Cost	1176

\[\begin{array}{l} t_1 := z \cdot \frac{-t}{a}\\ \mathbf{if}\;t \leq -9.01781693283319 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.78287283961909 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;t \leq -4.386380058600963 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.165401728485936 \cdot 10^{-20}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;t \leq 2.003576418661335 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.426440366550298 \cdot 10^{+97}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	32.3
Cost	716

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;z \leq -4.429522976598987 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.502268182838024 \cdot 10^{-120}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \leq -3.544090985272632 \cdot 10^{-148}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	32.3
Cost	716

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;z \leq -4.429522976598987 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.621225455397464 \cdot 10^{-119}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \leq -3.544090985272632 \cdot 10^{-148}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	32.3
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -4.429522976598987 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;z \leq -7.621225455397464 \cdot 10^{-119}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \leq -3.544090985272632 \cdot 10^{-148}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 8
Error	32.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -3.544090985272632 \cdot 10^{-148}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 9
Error	32.7
Cost	320

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

Reproduce

herbie shell --seed 2022294 
(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))