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

Average Error: 8.1 → 1.3

Time: 8.6s

Precision: binary64

Cost: 1736

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

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \frac{x}{a}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+134}:\\ \;\;\;\;t_1 - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1 - t \cdot \frac{z}{a}\\ \end{array} \]

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 (* y (/ x a))) (t_2 (- (* x y) (* z t))))
   (if (<= t_2 -5e+134)
     (- t_1 (/ z (/ a t)))
     (if (<= t_2 5e+284) (/ t_2 a) (- t_1 (* 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 = y * (x / a);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -5e+134) {
		tmp = t_1 - (z / (a / t));
	} else if (t_2 <= 5e+284) {
		tmp = t_2 / a;
	} else {
		tmp = t_1 - (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) :: t_2
    real(8) :: tmp
    t_1 = y * (x / a)
    t_2 = (x * y) - (z * t)
    if (t_2 <= (-5d+134)) then
        tmp = t_1 - (z / (a / t))
    else if (t_2 <= 5d+284) then
        tmp = t_2 / a
    else
        tmp = t_1 - (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 = y * (x / a);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -5e+134) {
		tmp = t_1 - (z / (a / t));
	} else if (t_2 <= 5e+284) {
		tmp = t_2 / a;
	} else {
		tmp = t_1 - (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 = y * (x / a)
	t_2 = (x * y) - (z * t)
	tmp = 0
	if t_2 <= -5e+134:
		tmp = t_1 - (z / (a / t))
	elif t_2 <= 5e+284:
		tmp = t_2 / a
	else:
		tmp = t_1 - (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(y * Float64(x / a))
	t_2 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_2 <= -5e+134)
		tmp = Float64(t_1 - Float64(z / Float64(a / t)));
	elseif (t_2 <= 5e+284)
		tmp = Float64(t_2 / a);
	else
		tmp = Float64(t_1 - 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 = y * (x / a);
	t_2 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_2 <= -5e+134)
		tmp = t_1 - (z / (a / t));
	elseif (t_2 <= 5e+284)
		tmp = t_2 / a;
	else
		tmp = t_1 - (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[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+134], N[(t$95$1 - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+284], N[(t$95$2 / a), $MachinePrecision], N[(t$95$1 - N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	8.1
Target	5.8
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -4.99999999999999981e134

   1. Initial program 19.3

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

   2. Applied egg-rr 3.3

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

   3. Taylor expanded in x around 0 11.8

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

   4. Simplified 3.4

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

   if -4.99999999999999981e134 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999999e284

   1. Initial program 0.8

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

   if 4.9999999999999999e284 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 53.5

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

   2. Applied egg-rr 0.3

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

   3. Taylor expanded in x around 0 27.1

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

   4. Simplified 0.3

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

   5. Taylor expanded in z around 0 30.3

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

   6. Simplified 0.3

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \frac{x}{a} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.7
Cost	1864

\[\begin{array}{l} t_1 := x \cdot \frac{y}{a}\\ t_2 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.7
Cost	1736

\[\begin{array}{l} t_1 := y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	19.4
Cost	1164

\[\begin{array}{l} t_1 := \frac{z \cdot t}{-a}\\ \mathbf{if}\;z \cdot t \leq -500000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 10^{-105}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{a}{-t}}\\ \end{array} \]

Alternative 4
Error	23.7
Cost	912

\[\begin{array}{l} t_1 := \left(-z\right) \cdot \frac{t}{a}\\ t_2 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;y \leq -1.5238519301656607 \cdot 10^{-54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.257638781444251 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.852676500649716 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 7.969511945945763 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	23.7
Cost	912

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;y \leq -1.5238519301656607 \cdot 10^{-54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.257638781444251 \cdot 10^{-62}:\\ \;\;\;\;\frac{z}{\frac{a}{-t}}\\ \mathbf{elif}\;y \leq 5.852676500649716 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 7.969511945945763 \cdot 10^{-14}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	24.2
Cost	912

\[\begin{array}{l} t_1 := t \cdot \frac{-z}{a}\\ t_2 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;y \leq -1.5238519301656607 \cdot 10^{-54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.257638781444251 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.852676500649716 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 7.969511945945763 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	31.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;a \leq -5.867049561934949 \cdot 10^{+46}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 8
Error	33.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3398701131690427 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 9
Error	33.2
Cost	320

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

Reproduce

herbie shell --seed 2022228 
(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))