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

Average Error: 7.7 → 0.9

Time: 11.7s

Precision: binary64

Cost: 1736

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\frac{x \cdot y}{a} - \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ \end{array} \]

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 (- INFINITY))
     (- (/ x (/ a y)) (/ z (/ a t)))
     (if (<= t_1 5e+184)
       (- (/ (* x y) a) (/ (* z t) a))
       (- (* x (/ y a)) (/ t (/ a z)))))))

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 <= -((double) INFINITY)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if (t_1 <= 5e+184) {
		tmp = ((x * y) / a) - ((z * t) / a);
	} else {
		tmp = (x * (y / a)) - (t / (a / z));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (z * t)) / a;
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if (t_1 <= 5e+184) {
		tmp = ((x * y) / a) - ((z * t) / a);
	} else {
		tmp = (x * (y / a)) - (t / (a / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	return ((x * y) - (z * t)) / a

↓

def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x / (a / y)) - (z / (a / t))
	elif t_1 <= 5e+184:
		tmp = ((x * y) / a) - ((z * t) / a)
	else:
		tmp = (x * (y / a)) - (t / (a / z))
	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 <= Float64(-Inf))
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	elseif (t_1 <= 5e+184)
		tmp = Float64(Float64(Float64(x * y) / a) - Float64(Float64(z * t) / a));
	else
		tmp = Float64(Float64(x * Float64(y / a)) - Float64(t / Float64(a / z)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - (z * t)) / a;
end

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x / (a / y)) - (z / (a / t));
	elseif (t_1 <= 5e+184)
		tmp = ((x * y) / a) - ((z * t) / a);
	else
		tmp = (x * (y / a)) - (t / (a / z));
	end
	tmp_2 = 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, (-Infinity)], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+184], N[(N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision] - N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.7
Target	5.9
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)) < -inf.0

   1. Initial program 64.0

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

   2. Applied egg-rr 0.3

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999999e184

   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 4.9999999999999999e184 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 25.8

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

   2. Taylor expanded in x around 0 25.8

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a}} \]

   3. Applied egg-rr 1.4

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

   4. Applied egg-rr 1.2

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\frac{x \cdot y}{a} - \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	1736

\[\begin{array}{l} t_1 := x \cdot \frac{y}{a} - \frac{z}{\frac{a}{t}}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.9
Cost	1736

\[\begin{array}{l} t_1 := x \cdot \frac{y}{a}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1 - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 3
Error	0.9
Cost	1736

\[\begin{array}{l} t_1 := x \cdot \frac{y}{a}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1 - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\frac{x \cdot y}{a} - \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 4
Error	4.4
Cost	1608

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 5
Error	27.0
Cost	980

\[\begin{array}{l} \mathbf{if}\;y \leq -3.9400926017181465 \cdot 10^{-219}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+58}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+98}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;y \leq 10^{+225}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]

Alternative 6
Error	25.7
Cost	912

\[\begin{array}{l} \mathbf{if}\;t \leq -7.798837149938507 \cdot 10^{-100}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+261}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+294}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \]

Alternative 7
Error	24.2
Cost	648

\[\begin{array}{l} t_1 := z \cdot \frac{-t}{a}\\ \mathbf{if}\;t \leq -7.798837149938507 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	32.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{+200}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 10^{+290}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 9
Error	32.0
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{\frac{a}{y}}\\ \mathbf{if}\;a \leq -2.403447983811911 \cdot 10^{+179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-82}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	33.0
Cost	320

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

Reproduce

herbie shell --seed 2022295 
(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))