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

Average Error: 7.5 → 1.2

Time: 24.7s

Precision: binary64

Cost: 1736

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

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \frac{x}{a} - z \cdot \frac{t}{a}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\frac{x \cdot y}{a} - \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (- (* y (/ x a)) (* z (/ t a)))) (t_2 (- (* x y) (* z t))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 2e+133) (- (/ (* x y) a) (/ (* z t) a)) 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 = (y * (x / a)) - (z * (t / a));
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+133) {
		tmp = ((x * y) / a) - ((z * t) / a);
	} else {
		tmp = t_1;
	}
	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 = (y * (x / a)) - (z * (t / a));
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 2e+133) {
		tmp = ((x * y) / a) - ((z * t) / a);
	} else {
		tmp = t_1;
	}
	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 = (y * (x / a)) - (z * (t / a))
	t_2 = (x * y) - (z * t)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 2e+133:
		tmp = ((x * y) / a) - ((z * t) / a)
	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 = Float64(Float64(y * Float64(x / a)) - Float64(z * Float64(t / a)))
	t_2 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+133)
		tmp = Float64(Float64(Float64(x * y) / a) - Float64(Float64(z * t) / a));
	else
		tmp = t_1;
	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 = (y * (x / a)) - (z * (t / a));
	t_2 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 2e+133)
		tmp = ((x * y) / a) - ((z * t) / a);
	else
		tmp = t_1;
	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[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 2e+133], N[(N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision] - N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	7.5
Target	5.7
Herbie	1.2

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 2e133 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 28.5

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

   2. Applied egg-rr 2.4

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

   3. Applied egg-rr 2.5

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2e133

   1. Initial program 0.8

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

   2. Taylor expanded in x around 0 0.8

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{a} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\frac{x \cdot y}{a} - \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - z \cdot \frac{t}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.2
Cost	1736

\[\begin{array}{l} t_1 := y \cdot \frac{x}{a} - z \cdot \frac{t}{a}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	4.7
Cost	1608

\[\begin{array}{l} t_1 := z \cdot \frac{-t}{a}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{+275}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	17.9
Cost	1292

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-16}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-106}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+246}:\\ \;\;\;\;\left(z \cdot t\right) \cdot \frac{1}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a} \cdot \left(-t\right)\\ \end{array} \]

Alternative 4
Error	24.9
Cost	1176

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{x}}\\ t_2 := \frac{-t}{\frac{a}{z}}\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+157}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2 \cdot 10^{+78}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-205}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	24.3
Cost	1176

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{x}}\\ t_2 := z \cdot \frac{-t}{a}\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+130}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2 \cdot 10^{+78}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-216}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \]

Alternative 6
Error	24.0
Cost	1044

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{x}}\\ t_2 := \frac{-z}{\frac{a}{t}}\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{+130}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.6 \cdot 10^{+57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-210}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	33.4
Cost	584

\[\begin{array}{l} t_1 := y \cdot \frac{x}{a}\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-245}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	33.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-245}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 9
Error	33.9
Cost	584

\[\begin{array}{l} t_1 := \frac{x \cdot y}{a}\\ \mathbf{if}\;a \leq 1.8 \cdot 10^{-214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+92}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	33.3
Cost	320

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

Reproduce

herbie shell --seed 2022325 
(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))