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

Average Error: 7.9 → 0.5

Time: 9.3s

Precision: binary64

Cost: 7556

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{t_1}{a}\\ t_3 := \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-127}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-322}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 10^{+252}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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 (/ t_1 a))
        (t_3 (- (/ x (/ a y)) (/ z (/ a t)))))
   (if (<= t_1 -2e+285)
     (fma y (/ x a) (/ (- z) (/ a t)))
     (if (<= t_1 -2e-127)
       t_2
       (if (<= t_1 1e-322) t_3 (if (<= t_1 1e+252) t_2 t_3))))))

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 = t_1 / a;
	double t_3 = (x / (a / y)) - (z / (a / t));
	double tmp;
	if (t_1 <= -2e+285) {
		tmp = fma(y, (x / a), (-z / (a / t)));
	} else if (t_1 <= -2e-127) {
		tmp = t_2;
	} else if (t_1 <= 1e-322) {
		tmp = t_3;
	} else if (t_1 <= 1e+252) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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 = Float64(t_1 / a)
	t_3 = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)))
	tmp = 0.0
	if (t_1 <= -2e+285)
		tmp = fma(y, Float64(x / a), Float64(Float64(-z) / Float64(a / t)));
	elseif (t_1 <= -2e-127)
		tmp = t_2;
	elseif (t_1 <= 1e-322)
		tmp = t_3;
	elseif (t_1 <= 1e+252)
		tmp = t_2;
	else
		tmp = t_3;
	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[(t$95$1 / a), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+285], N[(y * N[(x / a), $MachinePrecision] + N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-127], t$95$2, If[LessEqual[t$95$1, 1e-322], t$95$3, If[LessEqual[t$95$1, 1e+252], t$95$2, t$95$3]]]]]]]

Target

Original	7.9
Target	5.8
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -2e285

   1. Initial program 51.4

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

   2. Applied egg-rr 0.3

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

   if -2e285 < (-.f64 (*.f64 x y) (*.f64 z t)) < -2.0000000000000001e-127 or 9.88131e-323 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.0000000000000001e252

   1. Initial program 0.3

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

   if -2.0000000000000001e-127 < (-.f64 (*.f64 x y) (*.f64 z t)) < 9.88131e-323 or 1.0000000000000001e252 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 24.6

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

   2. Applied egg-rr 1.6

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -2 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq -2 \cdot 10^{-127}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{-322}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+252}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	2768

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{t_1}{a}\\ t_3 := x \cdot \frac{y}{a} - \frac{z}{\frac{a}{t}}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+293}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-231}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-322}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 10^{+252}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	0.5
Cost	2768

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{t_1}{a}\\ t_3 := \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+293}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-127}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-322}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 10^{+252}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	18.7
Cost	1684

\[\begin{array}{l} t_1 := \frac{x \cdot y}{a}\\ t_2 := y \cdot \frac{x}{a}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+285}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;x \cdot y \leq 100000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+53}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	5.2
Cost	1608

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := y \cdot \frac{x}{a}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+297}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	23.3
Cost	1176

\[\begin{array}{l} t_1 := z \cdot \frac{-t}{a}\\ \mathbf{if}\;z \leq -6 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{+80}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-89}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	23.0
Cost	1176

\[\begin{array}{l} t_1 := z \cdot \frac{-t}{a}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.88 \cdot 10^{+53}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+93}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \end{array} \]

Alternative 7
Error	32.8
Cost	320

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

Alternative 8
Error	33.0
Cost	320

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

Reproduce

herbie shell --seed 2022329 
(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))