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

Average Error: 7.8 → 0.9

Time: 10.2s

Precision: binary64

Cost: 8136

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+210}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+185}:\\ \;\;\;\;\frac{x \cdot y}{a} - \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

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 -2e+210)
     (- (/ x (/ a y)) (/ z (/ a t)))
     (if (<= t_1 5e+185)
       (- (/ (* x y) a) (/ (* z t) a))
       (fma -1.0 (/ t (/ a z)) (/ y (/ a x)))))))

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 <= -2e+210) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if (t_1 <= 5e+185) {
		tmp = ((x * y) / a) - ((z * t) / a);
	} else {
		tmp = fma(-1.0, (t / (a / z)), (y / (a / x)));
	}
	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 <= -2e+210)
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	elseif (t_1 <= 5e+185)
		tmp = Float64(Float64(Float64(x * y) / a) - Float64(Float64(z * t) / a));
	else
		tmp = fma(-1.0, Float64(t / Float64(a / z)), Float64(y / Float64(a / x)));
	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, -2e+210], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+185], N[(N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision] - N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.8
Target	5.8
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 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -1.99999999999999985e210

   1. Initial program 28.8

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

   2. Applied egg-rr 0.9

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

   if -1.99999999999999985e210 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999999e185

   1. Initial program 0.8

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

   2. Applied egg-rr 0.9

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

   3. Applied egg-rr 0.8

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

   if 4.9999999999999999e185 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 25.7

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

   2. Taylor expanded in a around 0 25.7

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

   3. Simplified 1.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)} \]
      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))): 39 points increase in error, 31 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, 25 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)): 2 points increase in error, 2 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -2 \cdot 10^{+210}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+185}:\\ \;\;\;\;\frac{x \cdot y}{a} - \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	1736

\[\begin{array}{l} t_1 := x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.7
Cost	1736

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+210}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\frac{x \cdot y}{a} - \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 3
Error	4.2
Cost	1608

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 10^{+294}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \]

Alternative 4
Error	24.5
Cost	912

\[\begin{array}{l} t_1 := z \cdot \frac{-t}{a}\\ \mathbf{if}\;t \leq -1.906795633342993 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.8016239397804074 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;t \leq 4.6895457389654756 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 426638915198.51843:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	24.4
Cost	912

\[\begin{array}{l} t_1 := z \cdot \frac{-t}{a}\\ \mathbf{if}\;t \leq -1.906795633342993 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.8016239397804074 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;t \leq 4.6895457389654756 \cdot 10^{-39}:\\ \;\;\;\;\frac{t \cdot \left(-z\right)}{a}\\ \mathbf{elif}\;t \leq 426638915198.51843:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	24.2
Cost	912

\[\begin{array}{l} \mathbf{if}\;t \leq -1.906795633342993 \cdot 10^{-156}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;t \leq 1.8016239397804074 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;t \leq 4.6895457389654756 \cdot 10^{-39}:\\ \;\;\;\;\frac{t \cdot \left(-z\right)}{a}\\ \mathbf{elif}\;t \leq 426638915198.51843:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \]

Alternative 7
Error	32.8
Cost	320

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

Alternative 8
Error	32.0
Cost	320

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

Reproduce

herbie shell --seed 2022308 
(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))