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

Average Accuracy: 88.5% → 98.8%

Time: 10.7s

Precision: binary64

Cost: 1993

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+288} \lor \neg \left(t_1 \leq 10^{+253}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot y + z \cdot t\right) + \left(z \cdot t\right) \cdot -2}{a}\\ \end{array} \]

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+288) (not (<= t_1 1e+253)))
     (- (/ x (/ a y)) (/ z (/ a t)))
     (/ (+ (+ (* x y) (* z t)) (* (* z t) -2.0)) 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+288) || !(t_1 <= 1e+253)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = (((x * y) + (z * t)) + ((z * t) * -2.0)) / 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+288)) .or. (.not. (t_1 <= 1d+253))) then
        tmp = (x / (a / y)) - (z / (a / t))
    else
        tmp = (((x * y) + (z * t)) + ((z * t) * (-2.0d0))) / 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+288) || !(t_1 <= 1e+253)) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else {
		tmp = (((x * y) + (z * t)) + ((z * t) * -2.0)) / 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+288) or not (t_1 <= 1e+253):
		tmp = (x / (a / y)) - (z / (a / t))
	else:
		tmp = (((x * y) + (z * t)) + ((z * t) * -2.0)) / 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+288) || !(t_1 <= 1e+253))
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	else
		tmp = Float64(Float64(Float64(Float64(x * y) + Float64(z * t)) + Float64(Float64(z * t) * -2.0)) / 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+288) || ~((t_1 <= 1e+253)))
		tmp = (x / (a / y)) - (z / (a / t));
	else
		tmp = (((x * y) + (z * t)) + ((z * t) * -2.0)) / 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+288], N[Not[LessEqual[t$95$1, 1e+253]], $MachinePrecision]], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Target

Original	88.5%
Target	91.4%
Herbie	98.8%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -2e288 or 9.9999999999999994e252 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 27.1%

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

   2. Applied egg-rr 99.5%

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

      [Start] 27.1	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-sub [=>] 27.1	\[ \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]
      associate-/l* [=>] 61.0	\[ \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{z \cdot t}{a} \]
      associate-/l* [=>] 99.5	\[ \frac{x}{\frac{a}{y}} - \color{blue}{\frac{z}{\frac{a}{t}}} \]

   if -2e288 < (-.f64 (*.f64 x y) (*.f64 z t)) < 9.9999999999999994e252

   1. Initial program 98.6%

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

   2. Applied egg-rr 98.5%

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

      [Start] 98.6	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-inv [=>] 98.5	\[ \color{blue}{\left(x \cdot y - z \cdot t\right) \cdot \frac{1}{a}} \]
      *-commutative [=>] 98.5	\[ \color{blue}{\frac{1}{a} \cdot \left(x \cdot y - z \cdot t\right)} \]

   3. Applied egg-rr 98.5%

      \[\leadsto \frac{1}{a} \cdot \color{blue}{\left(\left(\left(x \cdot y - z \cdot t\right) + z \cdot \left(-t\right)\right) + z \cdot t\right)} \]
      Proof

      [Start] 98.5	\[ \frac{1}{a} \cdot \left(x \cdot y - z \cdot t\right) \]
      prod-diff [=>] 98.5	\[ \frac{1}{a} \cdot \color{blue}{\left(\mathsf{fma}\left(x, y, -t \cdot z\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)\right)} \]
      *-commutative [<=] 98.5	\[ \frac{1}{a} \cdot \left(\mathsf{fma}\left(x, y, -\color{blue}{z \cdot t}\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)\right) \]
      fma-neg [<=] 98.5	\[ \frac{1}{a} \cdot \left(\color{blue}{\left(x \cdot y - z \cdot t\right)} + \mathsf{fma}\left(-t, z, t \cdot z\right)\right) \]
      *-commutative [<=] 98.5	\[ \frac{1}{a} \cdot \left(\left(x \cdot y - z \cdot t\right) + \mathsf{fma}\left(-t, z, \color{blue}{z \cdot t}\right)\right) \]
      fma-udef [=>] 98.5	\[ \frac{1}{a} \cdot \left(\left(x \cdot y - z \cdot t\right) + \color{blue}{\left(\left(-t\right) \cdot z + z \cdot t\right)}\right) \]
      distribute-lft-neg-in [<=] 98.5	\[ \frac{1}{a} \cdot \left(\left(x \cdot y - z \cdot t\right) + \left(\color{blue}{\left(-t \cdot z\right)} + z \cdot t\right)\right) \]
      *-commutative [<=] 98.5	\[ \frac{1}{a} \cdot \left(\left(x \cdot y - z \cdot t\right) + \left(\left(-\color{blue}{z \cdot t}\right) + z \cdot t\right)\right) \]
      associate-+r+ [=>] 98.5	\[ \frac{1}{a} \cdot \color{blue}{\left(\left(\left(x \cdot y - z \cdot t\right) + \left(-z \cdot t\right)\right) + z \cdot t\right)} \]
      distribute-rgt-neg-in [=>] 98.5	\[ \frac{1}{a} \cdot \left(\left(\left(x \cdot y - z \cdot t\right) + \color{blue}{z \cdot \left(-t\right)}\right) + z \cdot t\right) \]

   4. Applied egg-rr 98.4%

      \[\leadsto \frac{1}{a} \cdot \color{blue}{\left(z \cdot t - \left(z \cdot \left(t + t\right) - x \cdot y\right)\right)} \]
      Proof

      [Start] 98.5	\[ \frac{1}{a} \cdot \left(\left(\left(x \cdot y - z \cdot t\right) + z \cdot \left(-t\right)\right) + z \cdot t\right) \]
      associate-+l+ [=>] 98.5	\[ \frac{1}{a} \cdot \color{blue}{\left(\left(x \cdot y - z \cdot t\right) + \left(z \cdot \left(-t\right) + z \cdot t\right)\right)} \]
      +-commutative [=>] 98.5	\[ \frac{1}{a} \cdot \color{blue}{\left(\left(z \cdot \left(-t\right) + z \cdot t\right) + \left(x \cdot y - z \cdot t\right)\right)} \]
      +-commutative [=>] 98.5	\[ \frac{1}{a} \cdot \left(\color{blue}{\left(z \cdot t + z \cdot \left(-t\right)\right)} + \left(x \cdot y - z \cdot t\right)\right) \]
      distribute-rgt-neg-out [=>] 98.5	\[ \frac{1}{a} \cdot \left(\left(z \cdot t + \color{blue}{\left(-z \cdot t\right)}\right) + \left(x \cdot y - z \cdot t\right)\right) \]
      distribute-lft-neg-in [=>] 98.5	\[ \frac{1}{a} \cdot \left(\left(z \cdot t + \color{blue}{\left(-z\right) \cdot t}\right) + \left(x \cdot y - z \cdot t\right)\right) \]
      add-sqr-sqrt [=>] 73.2	\[ \frac{1}{a} \cdot \left(\left(z \cdot t + \left(-z\right) \cdot \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}\right) + \left(x \cdot y - z \cdot t\right)\right) \]
      sqrt-unprod [=>] 67.5	\[ \frac{1}{a} \cdot \left(\left(z \cdot t + \left(-z\right) \cdot \color{blue}{\sqrt{t \cdot t}}\right) + \left(x \cdot y - z \cdot t\right)\right) \]
      sqr-neg [<=] 67.5	\[ \frac{1}{a} \cdot \left(\left(z \cdot t + \left(-z\right) \cdot \sqrt{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) + \left(x \cdot y - z \cdot t\right)\right) \]
      sqrt-unprod [<=] 13.1	\[ \frac{1}{a} \cdot \left(\left(z \cdot t + \left(-z\right) \cdot \color{blue}{\left(\sqrt{-t} \cdot \sqrt{-t}\right)}\right) + \left(x \cdot y - z \cdot t\right)\right) \]
      add-sqr-sqrt [<=] 52.3	\[ \frac{1}{a} \cdot \left(\left(z \cdot t + \left(-z\right) \cdot \color{blue}{\left(-t\right)}\right) + \left(x \cdot y - z \cdot t\right)\right) \]
      cancel-sign-sub-inv [<=] 52.3	\[ \frac{1}{a} \cdot \left(\color{blue}{\left(z \cdot t - z \cdot \left(-t\right)\right)} + \left(x \cdot y - z \cdot t\right)\right) \]
      add-sqr-sqrt [=>] 32.9	\[ \frac{1}{a} \cdot \left(\left(\color{blue}{\sqrt{z \cdot t} \cdot \sqrt{z \cdot t}} - z \cdot \left(-t\right)\right) + \left(x \cdot y - z \cdot t\right)\right) \]
      sqrt-unprod [=>] 67.7	\[ \frac{1}{a} \cdot \left(\left(\color{blue}{\sqrt{\left(z \cdot t\right) \cdot \left(z \cdot t\right)}} - z \cdot \left(-t\right)\right) + \left(x \cdot y - z \cdot t\right)\right) \]
      sqr-neg [<=] 67.7	\[ \frac{1}{a} \cdot \left(\left(\sqrt{\color{blue}{\left(-z \cdot t\right) \cdot \left(-z \cdot t\right)}} - z \cdot \left(-t\right)\right) + \left(x \cdot y - z \cdot t\right)\right) \]
      distribute-rgt-neg-out [<=] 67.7	\[ \frac{1}{a} \cdot \left(\left(\sqrt{\color{blue}{\left(z \cdot \left(-t\right)\right)} \cdot \left(-z \cdot t\right)} - z \cdot \left(-t\right)\right) + \left(x \cdot y - z \cdot t\right)\right) \]
      distribute-rgt-neg-out [<=] 67.7	\[ \frac{1}{a} \cdot \left(\left(\sqrt{\left(z \cdot \left(-t\right)\right) \cdot \color{blue}{\left(z \cdot \left(-t\right)\right)}} - z \cdot \left(-t\right)\right) + \left(x \cdot y - z \cdot t\right)\right) \]
      sqrt-unprod [<=] 55.0	\[ \frac{1}{a} \cdot \left(\left(\color{blue}{\sqrt{z \cdot \left(-t\right)} \cdot \sqrt{z \cdot \left(-t\right)}} - z \cdot \left(-t\right)\right) + \left(x \cdot y - z \cdot t\right)\right) \]
      add-sqr-sqrt [<=] 98.5	\[ \frac{1}{a} \cdot \left(\left(\color{blue}{z \cdot \left(-t\right)} - z \cdot \left(-t\right)\right) + \left(x \cdot y - z \cdot t\right)\right) \]

   5. Taylor expanded in a around 0 98.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -2 \cdot 10^{+288} \lor \neg \left(x \cdot y - z \cdot t \leq 10^{+253}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot y + z \cdot t\right) + \left(z \cdot t\right) \cdot -2}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.8%
Cost	1737

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+288} \lor \neg \left(t_1 \leq 10^{+253}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]

Alternative 2
Accuracy	93.4%
Cost	1608

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 3
Accuracy	62.1%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-41}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 4
Accuracy	63.8%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-123}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 5
Accuracy	50.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-91} \lor \neg \left(a \leq 2.75 \cdot 10^{+97}\right):\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]

Alternative 6
Accuracy	48.4%
Cost	320

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

Reproduce

herbie shell --seed 2023133 
(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))