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

Average Error: 7.3 → 4.5

Time: 22.4s

Precision: binary64

Cost: 1224

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

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+259}:\\ \;\;\;\;y \cdot \frac{x}{a} - \frac{t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* t (- (/ z a)))))
   (if (<= (* z t) (- INFINITY))
     t_1
     (if (<= (* z t) 4e+259) (- (* y (/ x a)) (/ (* t z) 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 = t * -(z / a);
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((z * t) <= 4e+259) {
		tmp = (y * (x / a)) - ((t * z) / 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 = t * -(z / a);
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if ((z * t) <= 4e+259) {
		tmp = (y * (x / a)) - ((t * z) / 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 = t * -(z / a)
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = t_1
	elif (z * t) <= 4e+259:
		tmp = (y * (x / a)) - ((t * z) / 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(t * Float64(-Float64(z / a)))
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(z * t) <= 4e+259)
		tmp = Float64(Float64(y * Float64(x / a)) - Float64(Float64(t * z) / 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 = t * -(z / a);
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = t_1;
	elseif ((z * t) <= 4e+259)
		tmp = (y * (x / a)) - ((t * z) / 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[(t * (-N[(z / a), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 4e+259], N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] - N[(N[(t * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	7.3
Target	5.6
Herbie	4.5

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 4e259 < (*.f64 z t)

   1. Initial program 51.0

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

   2. Applied egg-rr 64.0

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

   3. Taylor expanded in x around inf 51.0

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a}\right) - -1 \cdot \frac{t \cdot z}{a}} \]

   4. Simplified 51.0

      \[\leadsto \color{blue}{\left(\frac{y \cdot x}{a} + -2 \cdot \frac{t \cdot z}{a}\right) - \left(-\frac{t \cdot z}{a}\right)} \]
      Proof

      [Start] 51.0	\[ \left(-2 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a}\right) - -1 \cdot \frac{t \cdot z}{a} \]
      rational.json-simplify-1 [=>] 51.0	\[ \color{blue}{\left(\frac{y \cdot x}{a} + -2 \cdot \frac{t \cdot z}{a}\right)} - -1 \cdot \frac{t \cdot z}{a} \]
      rational.json-simplify-2 [=>] 51.0	\[ \left(\frac{y \cdot x}{a} + -2 \cdot \frac{t \cdot z}{a}\right) - \color{blue}{\frac{t \cdot z}{a} \cdot -1} \]
      rational.json-simplify-9 [=>] 51.0	\[ \left(\frac{y \cdot x}{a} + -2 \cdot \frac{t \cdot z}{a}\right) - \color{blue}{\left(-\frac{t \cdot z}{a}\right)} \]

   5. Taylor expanded in t around -inf 4.1

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \left(2 \cdot \frac{z}{a} + -1 \cdot \frac{z}{a}\right)\right)} \]

   6. Simplified 4.1

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

      [Start] 4.1	\[ -1 \cdot \left(t \cdot \left(2 \cdot \frac{z}{a} + -1 \cdot \frac{z}{a}\right)\right) \]
      rational.json-simplify-43 [=>] 4.1	\[ \color{blue}{t \cdot \left(\left(2 \cdot \frac{z}{a} + -1 \cdot \frac{z}{a}\right) \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 4.1	\[ t \cdot \color{blue}{\left(-\left(2 \cdot \frac{z}{a} + -1 \cdot \frac{z}{a}\right)\right)} \]
      rational.json-simplify-2 [=>] 4.1	\[ t \cdot \left(-\left(\color{blue}{\frac{z}{a} \cdot 2} + -1 \cdot \frac{z}{a}\right)\right) \]
      rational.json-simplify-47 [=>] 4.1	\[ t \cdot \left(-\color{blue}{\frac{z}{a} \cdot \left(-1 + 2\right)}\right) \]
      metadata-eval [=>] 4.1	\[ t \cdot \left(-\frac{z}{a} \cdot \color{blue}{1}\right) \]
      rational.json-simplify-2 [=>] 4.1	\[ t \cdot \left(-\color{blue}{1 \cdot \frac{z}{a}}\right) \]
      rational.json-simplify-6 [=>] 4.1	\[ t \cdot \left(-\color{blue}{\frac{z}{a}}\right) \]

   if -inf.0 < (*.f64 z t) < 4e259

   1. Initial program 4.0

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

   2. Applied egg-rr 23.2

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

   3. Taylor expanded in y around 0 4.5

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + y \cdot \left(2 \cdot \frac{x}{a} - \frac{x}{a}\right)} \]

   4. Simplified 4.5

      \[\leadsto \color{blue}{\left(-\frac{t \cdot z}{a}\right) + y \cdot \left(2 \cdot \frac{x}{a} - \frac{x}{a}\right)} \]
      Proof

      [Start] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + y \cdot \left(2 \cdot \frac{x}{a} - \frac{x}{a}\right) \]
      rational.json-simplify-2 [=>] 4.5	\[ \color{blue}{\frac{t \cdot z}{a} \cdot -1} + y \cdot \left(2 \cdot \frac{x}{a} - \frac{x}{a}\right) \]
      rational.json-simplify-9 [=>] 4.5	\[ \color{blue}{\left(-\frac{t \cdot z}{a}\right)} + y \cdot \left(2 \cdot \frac{x}{a} - \frac{x}{a}\right) \]

   5. Taylor expanded in t around 0 4.5

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + y \cdot \left(2 \cdot \frac{x}{a} - \frac{x}{a}\right)} \]

   6. Simplified 4.5

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

      [Start] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + y \cdot \left(2 \cdot \frac{x}{a} - \frac{x}{a}\right) \]
      rational.json-simplify-2 [=>] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + y \cdot \left(\color{blue}{\frac{x}{a} \cdot 2} - \frac{x}{a}\right) \]
      rational.json-simplify-6 [<=] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + y \cdot \left(\frac{x}{a} \cdot 2 - \color{blue}{1 \cdot \frac{x}{a}}\right) \]
      rational.json-simplify-48 [=>] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + y \cdot \color{blue}{\left(\frac{x}{a} \cdot \left(2 - 1\right)\right)} \]
      metadata-eval [=>] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + y \cdot \left(\frac{x}{a} \cdot \color{blue}{1}\right) \]
      rational.json-simplify-2 [=>] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + y \cdot \color{blue}{\left(1 \cdot \frac{x}{a}\right)} \]
      rational.json-simplify-6 [=>] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + y \cdot \color{blue}{\frac{x}{a}} \]
      rational.json-simplify-5 [<=] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + \color{blue}{\left(y \cdot \frac{x}{a} - 0\right)} \]
      metadata-eval [<=] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + \left(y \cdot \frac{x}{a} - \color{blue}{\left(0 - 0\right)}\right) \]
      rational.json-simplify-44 [<=] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + \color{blue}{\left(0 - \left(0 - y \cdot \frac{x}{a}\right)\right)} \]
      rational.json-simplify-12 [<=] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + \left(0 - \color{blue}{\left(-y \cdot \frac{x}{a}\right)}\right) \]
      rational.json-simplify-12 [<=] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + \color{blue}{\left(-\left(-y \cdot \frac{x}{a}\right)\right)} \]
      rational.json-simplify-8 [=>] 4.5	\[ -1 \cdot \frac{t \cdot z}{a} + \color{blue}{\left(-y \cdot \frac{x}{a}\right) \cdot -1} \]
      rational.json-simplify-47 [=>] 4.5	\[ \color{blue}{-1 \cdot \left(\left(-y \cdot \frac{x}{a}\right) + \frac{t \cdot z}{a}\right)} \]
      rational.json-simplify-1 [<=] 4.5	\[ -1 \cdot \color{blue}{\left(\frac{t \cdot z}{a} + \left(-y \cdot \frac{x}{a}\right)\right)} \]
      rational.json-simplify-2 [<=] 4.5	\[ \color{blue}{\left(\frac{t \cdot z}{a} + \left(-y \cdot \frac{x}{a}\right)\right) \cdot -1} \]
      rational.json-simplify-8 [<=] 4.5	\[ \color{blue}{-\left(\frac{t \cdot z}{a} + \left(-y \cdot \frac{x}{a}\right)\right)} \]
      rational.json-simplify-13 [<=] 4.5	\[ \color{blue}{0 - \left(\frac{t \cdot z}{a} + \left(-y \cdot \frac{x}{a}\right)\right)} \]
      rational.json-simplify-46 [=>] 4.5	\[ \color{blue}{\left(0 - \frac{t \cdot z}{a}\right) - \left(-y \cdot \frac{x}{a}\right)} \]
      rational.json-simplify-42 [=>] 4.5	\[ \color{blue}{\left(0 - \left(-y \cdot \frac{x}{a}\right)\right) - \frac{t \cdot z}{a}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 4.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+259}:\\ \;\;\;\;y \cdot \frac{x}{a} - \frac{t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	3.9
Cost	1864

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

Alternative 2
Error	24.9
Cost	912

\[\begin{array}{l} t_1 := t \cdot \left(-\frac{z}{a}\right)\\ t_2 := y \cdot \frac{x}{a}\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 650000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	24.9
Cost	912

\[\begin{array}{l} t_1 := y \cdot \frac{x}{a}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-63}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1100000000:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	30.8
Cost	584

\[\begin{array}{l} t_1 := y \cdot \frac{x}{a}\\ \mathbf{if}\;a \leq -3.3 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	32.2
Cost	320

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

Reproduce

herbie shell --seed 2023077 
(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))