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

Average Accuracy: 91.2% → 96.5%

Time: 10.3s

Precision: binary64

Cost: 2768

• Unsound rule application detected in e-graph. Results may not be sound.

• 28.5% of points produce a very large (infinite) output. You may want to add a precondition.

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{t_1}{a}\\ t_3 := x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+198}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-136}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-178}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

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)))
        (t_2 (/ t_1 a))
        (t_3 (- (* x (/ y a)) (* z (/ t a)))))
   (if (<= t_1 -1e+198)
     t_3
     (if (<= t_1 -5e-136)
       t_2
       (if (<= t_1 5e-178)
         t_3
         (if (<= t_1 2e+301) t_2 (- (/ x (/ a y)) (/ z (/ a t)))))))))

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 t_2 = t_1 / a;
	double t_3 = (x * (y / a)) - (z * (t / a));
	double tmp;
	if (t_1 <= -1e+198) {
		tmp = t_3;
	} else if (t_1 <= -5e-136) {
		tmp = t_2;
	} else if (t_1 <= 5e-178) {
		tmp = t_3;
	} else if (t_1 <= 2e+301) {
		tmp = t_2;
	} else {
		tmp = (x / (a / y)) - (z / (a / t));
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = (x * y) - (z * t)
    t_2 = t_1 / a
    t_3 = (x * (y / a)) - (z * (t / a))
    if (t_1 <= (-1d+198)) then
        tmp = t_3
    else if (t_1 <= (-5d-136)) then
        tmp = t_2
    else if (t_1 <= 5d-178) then
        tmp = t_3
    else if (t_1 <= 2d+301) then
        tmp = t_2
    else
        tmp = (x / (a / y)) - (z / (a / t))
    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 t_2 = t_1 / a;
	double t_3 = (x * (y / a)) - (z * (t / a));
	double tmp;
	if (t_1 <= -1e+198) {
		tmp = t_3;
	} else if (t_1 <= -5e-136) {
		tmp = t_2;
	} else if (t_1 <= 5e-178) {
		tmp = t_3;
	} else if (t_1 <= 2e+301) {
		tmp = t_2;
	} else {
		tmp = (x / (a / y)) - (z / (a / t));
	}
	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)
	t_2 = t_1 / a
	t_3 = (x * (y / a)) - (z * (t / a))
	tmp = 0
	if t_1 <= -1e+198:
		tmp = t_3
	elif t_1 <= -5e-136:
		tmp = t_2
	elif t_1 <= 5e-178:
		tmp = t_3
	elif t_1 <= 2e+301:
		tmp = t_2
	else:
		tmp = (x / (a / y)) - (z / (a / t))
	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))
	t_2 = Float64(t_1 / a)
	t_3 = Float64(Float64(x * Float64(y / a)) - Float64(z * Float64(t / a)))
	tmp = 0.0
	if (t_1 <= -1e+198)
		tmp = t_3;
	elseif (t_1 <= -5e-136)
		tmp = t_2;
	elseif (t_1 <= 5e-178)
		tmp = t_3;
	elseif (t_1 <= 2e+301)
		tmp = t_2;
	else
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	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);
	t_2 = t_1 / a;
	t_3 = (x * (y / a)) - (z * (t / a));
	tmp = 0.0;
	if (t_1 <= -1e+198)
		tmp = t_3;
	elseif (t_1 <= -5e-136)
		tmp = t_2;
	elseif (t_1 <= 5e-178)
		tmp = t_3;
	elseif (t_1 <= 2e+301)
		tmp = t_2;
	else
		tmp = (x / (a / y)) - (z / (a / t));
	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]}, Block[{t$95$2 = N[(t$95$1 / a), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+198], t$95$3, If[LessEqual[t$95$1, -5e-136], t$95$2, If[LessEqual[t$95$1, 5e-178], t$95$3, If[LessEqual[t$95$1, 2e+301], t$95$2, N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	91.2%
Target	91.8%
Herbie	96.5%

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -1.00000000000000002e198 or -5.0000000000000002e-136 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.99999999999999976e-178

   1. Initial program 76.3%

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

   2. Taylor expanded in x around 0 73.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a}} \]

   3. Applied egg-rr 94.8%

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

      [Start] 73.7	\[ -1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a} \]
      +-commutative [=>] 73.7	\[ \color{blue}{\frac{y \cdot x}{a} + -1 \cdot \frac{t \cdot z}{a}} \]
      mul-1-neg [=>] 73.7	\[ \frac{y \cdot x}{a} + \color{blue}{\left(-\frac{t \cdot z}{a}\right)} \]
      unsub-neg [=>] 73.7	\[ \color{blue}{\frac{y \cdot x}{a} - \frac{t \cdot z}{a}} \]
      associate-/l* [=>] 84.1	\[ \color{blue}{\frac{y}{\frac{a}{x}}} - \frac{t \cdot z}{a} \]
      associate-/r/ [=>] 84.2	\[ \color{blue}{\frac{y}{a} \cdot x} - \frac{t \cdot z}{a} \]
      associate-/l* [=>] 94.7	\[ \frac{y}{a} \cdot x - \color{blue}{\frac{t}{\frac{a}{z}}} \]
      associate-/r/ [=>] 94.8	\[ \frac{y}{a} \cdot x - \color{blue}{\frac{t}{a} \cdot z} \]

   if -1.00000000000000002e198 < (-.f64 (*.f64 x y) (*.f64 z t)) < -5.0000000000000002e-136 or 4.99999999999999976e-178 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2.00000000000000011e301

   1. Initial program 99.7%

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

   if 2.00000000000000011e301 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 78.6%

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

   2. Applied egg-rr 94.2%

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

      [Start] 78.6	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-sub [=>] 72.9	\[ \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]
      associate-/l* [=>] 83.6	\[ \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{z \cdot t}{a} \]
      associate-/l* [=>] 94.2	\[ \frac{x}{\frac{a}{y}} - \color{blue}{\frac{z}{\frac{a}{t}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+198}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{-136}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{-178}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.5%
Cost	2770

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+198} \lor \neg \left(t_1 \leq -5 \cdot 10^{-136} \lor \neg \left(t_1 \leq 5 \cdot 10^{-178}\right) \land t_1 \leq 2 \cdot 10^{+301}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]

Alternative 2
Accuracy	92.9%
Cost	1357

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+287}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-321} \lor \neg \left(x \cdot y \leq 10^{-322}\right):\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \]

Alternative 3
Accuracy	66.0%
Cost	912

\[\begin{array}{l} t_1 := z \cdot \frac{-t}{a}\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+107}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-81}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \leq 115:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 4
Accuracy	65.9%
Cost	912

\[\begin{array}{l} t_1 := \frac{z}{a} \cdot \left(-t\right)\\ t_2 := y \cdot \frac{x}{a}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+107}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 450:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 5
Accuracy	65.9%
Cost	912

\[\begin{array}{l} t_1 := y \cdot \frac{x}{a}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{z}{a} \cdot \left(-t\right)\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 210:\\ \;\;\;\;\frac{t}{\frac{-a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 6
Accuracy	51.4%
Cost	320

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

Alternative 7
Accuracy	51.4%
Cost	320

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

Reproduce

herbie shell --seed 2023160 
(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))