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

Average Error: 7.8 → 1.1

Time: 10.2s

Precision: binary64

Cost: 1737

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+158} \lor \neg \left(t_1 \leq 5 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a} - \frac{z \cdot t}{a}\\ \end{array} \]

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 (- (* x y) (* z t))))
   (if (or (<= t_1 -2e+158) (not (<= t_1 5e+167)))
     (- (/ x (/ a y)) (/ z (/ a t)))
     (- (/ (* x y) a) (/ (* z t) 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 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -2e+158) || !(t_1 <= 5e+167)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = ((x * y) / a) - ((z * t) / a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) - (z * t)) / a
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) - (z * t)
    if ((t_1 <= (-2d+158)) .or. (.not. (t_1 <= 5d+167))) then
        tmp = (x / (a / y)) - (z / (a / t))
    else
        tmp = ((x * y) / a) - ((z * t) / a)
    end if
    code = tmp
end function

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 <= -2e+158) || !(t_1 <= 5e+167)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = ((x * y) / a) - ((z * t) / 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 = (x * y) - (z * t)
	tmp = 0
	if (t_1 <= -2e+158) or not (t_1 <= 5e+167):
		tmp = (x / (a / y)) - (z / (a / t))
	else:
		tmp = ((x * y) / a) - ((z * t) / 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(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= -2e+158) || !(t_1 <= 5e+167))
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	else
		tmp = Float64(Float64(Float64(x * y) / a) - Float64(Float64(z * t) / 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 = (x * y) - (z * t);
	tmp = 0.0;
	if ((t_1 <= -2e+158) || ~((t_1 <= 5e+167)))
		tmp = (x / (a / y)) - (z / (a / t));
	else
		tmp = ((x * y) / a) - ((z * t) / 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[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+158], N[Not[LessEqual[t$95$1, 5e+167]], $MachinePrecision]], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision] - N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.8
Target	5.5
Herbie	1.1

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -1.99999999999999991e158 or 4.9999999999999997e167 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 23.1

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

   2. Applied egg-rr 1.7

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

   if -1.99999999999999991e158 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999997e167

   1. Initial program 0.9

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

   2. Taylor expanded in x around 0 0.9

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -2 \cdot 10^{+158} \lor \neg \left(x \cdot y - z \cdot t \leq 5 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a} - \frac{z \cdot t}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.2
Cost	1864

\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{z}{\frac{a}{-t}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 2
Error	1.1
Cost	1737

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+158} \lor \neg \left(t_1 \leq 5 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]

Alternative 3
Error	24.8
Cost	1176

\[\begin{array}{l} t_1 := t \cdot \frac{-z}{a}\\ t_2 := z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{+108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{+56}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -12200000000000:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \leq -820:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-191}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	24.9
Cost	1176

\[\begin{array}{l} t_1 := t \cdot \frac{-z}{a}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+107}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{+56}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;z \leq -3 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1060000000000:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \leq -2000:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-191}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	24.0
Cost	913

\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-94}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-188} \lor \neg \left(y \leq 2.1 \cdot 10^{-161}\right) \land y \leq 7 \cdot 10^{+53}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 6
Error	32.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+145}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-176}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 7
Error	33.4
Cost	320

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

Alternative 8
Error	32.9
Cost	320

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

Alternative 9
Error	32.8
Cost	320

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

Reproduce

herbie shell --seed 2023031 
(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))