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

Average Accuracy: 88.9% → 98.6%

Time: 11.3s

Precision: binary64

Cost: 1736

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{z}{\frac{a}{t}}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+245}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - t_1\\ \end{array} \]

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 (/ z (/ a t))) (t_2 (- (* x y) (* z t))))
   (if (<= t_2 -2e+245)
     (- (/ x (/ a y)) t_1)
     (if (<= t_2 5e+174) (/ t_2 a) (- (* x (/ y 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 = z / (a / t);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -2e+245) {
		tmp = (x / (a / y)) - t_1;
	} else if (t_2 <= 5e+174) {
		tmp = t_2 / a;
	} else {
		tmp = (x * (y / a)) - t_1;
	}
	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 = z / (a / t)
    t_2 = (x * y) - (z * t)
    if (t_2 <= (-2d+245)) then
        tmp = (x / (a / y)) - t_1
    else if (t_2 <= 5d+174) then
        tmp = t_2 / a
    else
        tmp = (x * (y / a)) - t_1
    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 = z / (a / t);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -2e+245) {
		tmp = (x / (a / y)) - t_1;
	} else if (t_2 <= 5e+174) {
		tmp = t_2 / a;
	} else {
		tmp = (x * (y / a)) - 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 = z / (a / t)
	t_2 = (x * y) - (z * t)
	tmp = 0
	if t_2 <= -2e+245:
		tmp = (x / (a / y)) - t_1
	elif t_2 <= 5e+174:
		tmp = t_2 / a
	else:
		tmp = (x * (y / a)) - 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(z / Float64(a / t))
	t_2 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_2 <= -2e+245)
		tmp = Float64(Float64(x / Float64(a / y)) - t_1);
	elseif (t_2 <= 5e+174)
		tmp = Float64(t_2 / a);
	else
		tmp = Float64(Float64(x * Float64(y / a)) - 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 = z / (a / t);
	t_2 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_2 <= -2e+245)
		tmp = (x / (a / y)) - t_1;
	elseif (t_2 <= 5e+174)
		tmp = t_2 / a;
	else
		tmp = (x * (y / a)) - 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[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+245], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 5e+174], N[(t$95$2 / a), $MachinePrecision], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Target

Original	88.9%
Target	91.9%
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 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -2.00000000000000009e245

   1. Initial program 41.5%

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

   2. Applied egg-rr 99.3%

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

      [Start] 41.5	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-sub [=>] 41.5	\[ \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]
      associate-/l* [=>] 68.1	\[ \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{z \cdot t}{a} \]
      associate-/l* [=>] 99.3	\[ \frac{x}{\frac{a}{y}} - \color{blue}{\frac{z}{\frac{a}{t}}} \]

   if -2.00000000000000009e245 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999997e174

   1. Initial program 98.7%

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

   if 4.9999999999999997e174 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 65.2%

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

   2. Applied egg-rr 97.3%

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

      [Start] 65.2	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-sub [=>] 65.2	\[ \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]
      associate-/l* [=>] 80.5	\[ \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{z \cdot t}{a} \]
      associate-/l* [=>] 97.3	\[ \frac{x}{\frac{a}{y}} - \color{blue}{\frac{z}{\frac{a}{t}}} \]

   3. Applied egg-rr 97.4%

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

      [Start] 97.3	\[ \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}} \]
      div-inv [=>] 97.2	\[ \color{blue}{x \cdot \frac{1}{\frac{a}{y}}} - \frac{z}{\frac{a}{t}} \]
      *-commutative [=>] 97.2	\[ \color{blue}{\frac{1}{\frac{a}{y}} \cdot x} - \frac{z}{\frac{a}{t}} \]
      clear-num [<=] 97.4	\[ \color{blue}{\frac{y}{a}} \cdot x - \frac{z}{\frac{a}{t}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -2 \cdot 10^{+245}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+174}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	93.7%
Cost	1864

\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\frac{t}{a}\right)\\ \end{array} \]

Alternative 2
Accuracy	98.6%
Cost	1737

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+246} \lor \neg \left(t_1 \leq 5 \cdot 10^{+174}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]

Alternative 3
Accuracy	62.3%
Cost	912

\[\begin{array}{l} t_1 := z \cdot \left(-\frac{t}{a}\right)\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 4
Accuracy	62.6%
Cost	912

\[\begin{array}{l} t_1 := t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+152}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.82 \cdot 10^{-22}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 5
Accuracy	62.6%
Cost	912

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+152}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{+118}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-111}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 6
Accuracy	48.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -1.26 \cdot 10^{+222}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 7
Accuracy	48.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+77}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]

Alternative 8
Accuracy	47.8%
Cost	320

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

Reproduce

herbie shell --seed 2023137 
(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))