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

Average Error: 7.7 → 0.9

Time: 6.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+166}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \end{array} \]

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 (<= t_1 -1e+286)
     (- (/ x (/ a y)) (/ z (/ a t)))
     (if (<= t_1 5e+166) (/ t_1 a) (- (* y (/ x a)) (* t (/ z 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 <= -1e+286) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if (t_1 <= 5e+166) {
		tmp = t_1 / a;
	} else {
		tmp = (y * (x / a)) - (t * (z / 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 <= (-1d+286)) then
        tmp = (x / (a / y)) - (z / (a / t))
    else if (t_1 <= 5d+166) then
        tmp = t_1 / a
    else
        tmp = (y * (x / a)) - (t * (z / 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 <= -1e+286) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if (t_1 <= 5e+166) {
		tmp = t_1 / a;
	} else {
		tmp = (y * (x / a)) - (t * (z / 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 <= -1e+286:
		tmp = (x / (a / y)) - (z / (a / t))
	elif t_1 <= 5e+166:
		tmp = t_1 / a
	else:
		tmp = (y * (x / a)) - (t * (z / 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 <= -1e+286)
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	elseif (t_1 <= 5e+166)
		tmp = Float64(t_1 / a);
	else
		tmp = Float64(Float64(y * Float64(x / a)) - Float64(t * Float64(z / 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 <= -1e+286)
		tmp = (x / (a / y)) - (z / (a / t));
	elseif (t_1 <= 5e+166)
		tmp = t_1 / a;
	else
		tmp = (y * (x / a)) - (t * (z / 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[LessEqual[t$95$1, -1e+286], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+166], N[(t$95$1 / a), $MachinePrecision], N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] - N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.7
Target	5.6
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)) < -1.00000000000000003e286

   1. Initial program 52.6

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

   2. Applied egg-rr 0.3

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

   if -1.00000000000000003e286 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.0000000000000002e166

   1. Initial program 0.8

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

   if 5.0000000000000002e166 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 23.3

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

   2. Taylor expanded in a around 0 23.3

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

   3. Simplified 1.9

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

   4. Applied egg-rr 2.0

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+166}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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))