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

Average Error: 7.5 → 0.8

Time: 35.4s

Precision: binary64

Cost: 1800

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

Math

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

↓

\[\begin{array}{l} t_1 := z \cdot \left(-\frac{t}{a}\right)\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1 + \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+205}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1 + y \cdot \frac{x}{a}\\ \end{array} \]

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 (* z (- (/ t a)))) (t_2 (- (* x y) (* z t))))
   (if (<= t_2 (- INFINITY))
     (+ t_1 (/ x (/ a y)))
     (if (<= t_2 2e+205) (/ t_2 a) (+ t_1 (* y (/ x a)))))))

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 = z * -(t / a);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1 + (x / (a / y));
	} else if (t_2 <= 2e+205) {
		tmp = t_2 / a;
	} else {
		tmp = t_1 + (y * (x / a));
	}
	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 = z * -(t / a);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1 + (x / (a / y));
	} else if (t_2 <= 2e+205) {
		tmp = t_2 / a;
	} else {
		tmp = t_1 + (y * (x / a));
	}
	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 = z * -(t / a)
	t_2 = (x * y) - (z * t)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1 + (x / (a / y))
	elif t_2 <= 2e+205:
		tmp = t_2 / a
	else:
		tmp = t_1 + (y * (x / a))
	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(z * Float64(-Float64(t / a)))
	t_2 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(t_1 + Float64(x / Float64(a / y)));
	elseif (t_2 <= 2e+205)
		tmp = Float64(t_2 / a);
	else
		tmp = Float64(t_1 + Float64(y * Float64(x / a)));
	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 = z * -(t / a);
	t_2 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1 + (x / (a / y));
	elseif (t_2 <= 2e+205)
		tmp = t_2 / a;
	else
		tmp = t_1 + (y * (x / a));
	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[(z * (-N[(t / a), $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(t$95$1 + N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+205], N[(t$95$2 / a), $MachinePrecision], N[(t$95$1 + N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	7.5
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 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. Taylor expanded in x around 0 64.0

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

   3. Simplified 0.2

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

      [Start] 64.0	\[ -1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a} \]
      rational.json-simplify-49 [=>] 31.0	\[ -1 \cdot \color{blue}{\left(z \cdot \frac{t}{a}\right)} + \frac{y \cdot x}{a} \]
      rational.json-simplify-43 [=>] 31.0	\[ \color{blue}{z \cdot \left(\frac{t}{a} \cdot -1\right)} + \frac{y \cdot x}{a} \]
      rational.json-simplify-9 [=>] 31.0	\[ z \cdot \color{blue}{\left(-\frac{t}{a}\right)} + \frac{y \cdot x}{a} \]
      rational.json-simplify-2 [=>] 31.0	\[ z \cdot \left(-\frac{t}{a}\right) + \frac{\color{blue}{x \cdot y}}{a} \]
      rational.json-simplify-49 [=>] 0.2	\[ z \cdot \left(-\frac{t}{a}\right) + \color{blue}{y \cdot \frac{x}{a}} \]

   4. Applied egg-rr 0.3

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2.00000000000000003e205

   1. Initial program 0.7

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

   if 2.00000000000000003e205 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 29.2

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

   2. Taylor expanded in x around 0 29.2

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

   3. Simplified 1.6

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

      [Start] 29.2	\[ -1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a} \]
      rational.json-simplify-49 [=>] 15.9	\[ -1 \cdot \color{blue}{\left(z \cdot \frac{t}{a}\right)} + \frac{y \cdot x}{a} \]
      rational.json-simplify-43 [=>] 15.9	\[ \color{blue}{z \cdot \left(\frac{t}{a} \cdot -1\right)} + \frac{y \cdot x}{a} \]
      rational.json-simplify-9 [=>] 15.9	\[ z \cdot \color{blue}{\left(-\frac{t}{a}\right)} + \frac{y \cdot x}{a} \]
      rational.json-simplify-2 [=>] 15.9	\[ z \cdot \left(-\frac{t}{a}\right) + \frac{\color{blue}{x \cdot y}}{a} \]
      rational.json-simplify-49 [=>] 1.6	\[ z \cdot \left(-\frac{t}{a}\right) + \color{blue}{y \cdot \frac{x}{a}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right) + \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+205}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right) + y \cdot \frac{x}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	25.4
Cost	1836

\[\begin{array}{l} t_1 := -\frac{t}{\frac{a}{z}}\\ t_2 := -\frac{z}{\frac{a}{t}}\\ t_3 := \frac{y}{\frac{a}{x}}\\ t_4 := \frac{y \cdot x}{a}\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{+142}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+72}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{+38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-121}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-169}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-291}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	25.5
Cost	1836

\[\begin{array}{l} t_1 := -\frac{t}{\frac{a}{z}}\\ t_2 := \frac{y}{\frac{a}{x}}\\ t_3 := -\frac{z}{\frac{a}{t}}\\ t_4 := \frac{y \cdot x}{a}\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{+38}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-85}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-121}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-169}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq 10^{-292}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	25.5
Cost	1836

\[\begin{array}{l} t_1 := \frac{t \cdot \left(-z\right)}{a}\\ t_2 := \frac{y \cdot x}{a}\\ t_3 := \frac{y}{\frac{a}{x}}\\ t_4 := -\frac{z}{\frac{a}{t}}\\ t_5 := -\frac{t}{\frac{a}{z}}\\ \mathbf{if}\;x \leq -8 \cdot 10^{+142}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{+35}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq -1.26 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-121}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-158}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-169}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-292}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-121}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	25.5
Cost	1836

\[\begin{array}{l} t_1 := \frac{t \cdot \left(-z\right)}{a}\\ t_2 := \frac{y \cdot x}{a}\\ t_3 := \frac{y}{\frac{a}{x}}\\ t_4 := -\frac{t}{\frac{a}{z}}\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{+142}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{+72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{t}{a}}{\frac{-1}{z}}\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-121}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-158}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-169}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-288}:\\ \;\;\;\;-\frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-122}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	0.8
Cost	1800

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := z \cdot \left(-\frac{t}{a}\right) + y \cdot \frac{x}{a}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+251}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+205}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	4.2
Cost	1608

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 7
Error	25.3
Cost	1440

\[\begin{array}{l} t_1 := -t \cdot \frac{z}{a}\\ t_2 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq -3.05 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	25.3
Cost	1440

\[\begin{array}{l} t_1 := -\frac{t}{\frac{a}{z}}\\ t_2 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{+72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-62}:\\ \;\;\;\;-t \cdot \frac{z}{a}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-121}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	33.2
Cost	584

\[\begin{array}{l} t_1 := y \cdot \frac{x}{a}\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 8.6 \cdot 10^{-194}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	33.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+171}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 11
Error	31.8
Cost	584

\[\begin{array}{l} t_1 := y \cdot \frac{x}{a}\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	33.4
Cost	320

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

Reproduce

herbie shell --seed 2023068 
(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))