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

Average Accuracy: 88.4% → 98.6%

Time: 10.1s

Precision: binary64

Cost: 1737

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+241} \lor \neg \left(t_1 \leq 5 \cdot 10^{+207}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]

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 -1e+241) (not (<= t_1 5e+207)))
     (- (/ x (/ a y)) (/ z (/ a t)))
     (/ t_1 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+241) || !(t_1 <= 5e+207)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = t_1 / 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+241)) .or. (.not. (t_1 <= 5d+207))) then
        tmp = (x / (a / y)) - (z / (a / t))
    else
        tmp = t_1 / 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+241) || !(t_1 <= 5e+207)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = t_1 / 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+241) or not (t_1 <= 5e+207):
		tmp = (x / (a / y)) - (z / (a / t))
	else:
		tmp = t_1 / 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+241) || !(t_1 <= 5e+207))
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	else
		tmp = Float64(t_1 / 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+241) || ~((t_1 <= 5e+207)))
		tmp = (x / (a / y)) - (z / (a / t));
	else
		tmp = t_1 / 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, -1e+241], N[Not[LessEqual[t$95$1, 5e+207]], $MachinePrecision]], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Target

Original	88.4%
Target	91.6%
Herbie	98.6%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -1.0000000000000001e241 or 4.9999999999999999e207 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 49.7%

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

   2. Applied egg-rr 98.7%

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

      [Start] 49.7	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-sub [=>] 49.7	\[ \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]
      associate-/l* [=>] 71.0	\[ \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{z \cdot t}{a} \]
      associate-/l* [=>] 98.7	\[ \frac{x}{\frac{a}{y}} - \color{blue}{\frac{z}{\frac{a}{t}}} \]

   if -1.0000000000000001e241 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999999e207

   1. Initial program 98.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+241} \lor \neg \left(x \cdot y - z \cdot t \leq 5 \cdot 10^{+207}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	93.6%
Cost	1864

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

Alternative 2
Accuracy	61.2%
Cost	1440

\[\begin{array}{l} t_1 := \frac{z}{\frac{-a}{t}}\\ t_2 := \frac{x \cdot y}{a}\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+179}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+74}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{+24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-10}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-181}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-141}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-112}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	64.5%
Cost	1308

\[\begin{array}{l} t_1 := t \cdot \frac{-z}{a}\\ t_2 := z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{-129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-197}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-36}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;t \leq 0.0074:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+247}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	72.9%
Cost	1292

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-18}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\left(z \cdot t\right) \cdot \frac{1}{-a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \]

Alternative 5
Accuracy	61.5%
Cost	1178

\[\begin{array}{l} t_1 := z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{+74}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+28} \lor \neg \left(z \leq -1.45 \cdot 10^{+24} \lor \neg \left(z \leq -1.1 \cdot 10^{-9}\right) \land z \leq 2.1 \cdot 10^{-181}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]

Alternative 6
Accuracy	61.2%
Cost	1176

\[\begin{array}{l} t_1 := t \cdot \frac{-z}{a}\\ t_2 := \frac{x \cdot y}{a}\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{+74}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -6 \cdot 10^{+29}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-9}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-181}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	61.3%
Cost	1176

\[\begin{array}{l} t_1 := \frac{x \cdot y}{a}\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+179}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{+74}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-10}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \]

Alternative 8
Accuracy	50.0%
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+26}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-306}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-267}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 9
Accuracy	49.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{-291} \lor \neg \left(a \leq 2.8 \cdot 10^{-73}\right):\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 10
Accuracy	49.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-303} \lor \neg \left(a \leq 4.9 \cdot 10^{+84}\right):\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 11
Accuracy	49.3%
Cost	320

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

Reproduce

herbie shell --seed 2023138 
(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))