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

Percentage Accurate: 91.6% → 97.3%

Time: 13.3s

Precision: binary64

Cost: 7556

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

• 28.4% 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

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

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))))
   (if (<= t_1 (- INFINITY))
     (fma x (/ y a) (/ (- z) (/ a t)))
     (if (<= t_1 2e+285) (/ t_1 a) (- (/ 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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(x, (y / a), (-z / (a / t)));
	} else if (t_1 <= 2e+285) {
		tmp = t_1 / a;
	} 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))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(x, Float64(y / a), Float64(Float64(-z) / Float64(a / t)));
	elseif (t_1 <= 2e+285)
		tmp = Float64(t_1 / a);
	else
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	end
	return 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[LessEqual[t$95$1, (-Infinity)], N[(x * N[(y / a), $MachinePrecision] + N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+285], N[(t$95$1 / a), $MachinePrecision], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	91.6%
Target	92.2%
Herbie	97.3%

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0

   1. Initial program 70.5%

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

   2. Applied egg-rr 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\frac{z}{\frac{a}{t}}\right)} \]
      Step-by-step derivation

      [Start] 70.5%	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-sub [=>] 67.6%	\[ \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]
      *-un-lft-identity [=>] 67.6%	\[ \frac{x \cdot y}{\color{blue}{1 \cdot a}} - \frac{z \cdot t}{a} \]
      times-frac [=>] 80.9%	\[ \color{blue}{\frac{x}{1} \cdot \frac{y}{a}} - \frac{z \cdot t}{a} \]
      fma-neg [=>] 80.9%	\[ \color{blue}{\mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\frac{z \cdot t}{a}\right)} \]
      associate-/l* [=>] 97.1%	\[ \mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\color{blue}{\frac{z}{\frac{a}{t}}}\right) \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 2e285

   1. Initial program 98.7%

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

   if 2e285 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 77.4%

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

   2. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}} \]
      Step-by-step derivation

      [Start] 77.4%	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-sub [=>] 77.4%	\[ \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]
      associate-/l* [=>] 89.0%	\[ \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{z \cdot t}{a} \]
      associate-/l* [=>] 99.9%	\[ \frac{x}{\frac{a}{y}} - \color{blue}{\frac{z}{\frac{a}{t}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.3%
Cost	7556

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternative 2
Accuracy	97.2%
Cost	7620

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternative 3
Accuracy	97.2%
Cost	1737

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{+285}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]

Alternative 4
Accuracy	72.0%
Cost	1424

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-82}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{-32}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 5
Accuracy	68.2%
Cost	912

\[\begin{array}{l} t_1 := z \cdot \frac{-t}{a}\\ t_2 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1900000000000:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Accuracy	68.2%
Cost	912

\[\begin{array}{l} t_1 := \frac{-z}{\frac{a}{t}}\\ t_2 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2050000000000:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	68.2%
Cost	912

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{+88}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;x \leq -8200000000000:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	93.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \end{array} \]

Alternative 9
Accuracy	51.2%
Cost	320

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

Alternative 10
Accuracy	51.2%
Cost	320

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

Reproduce

herbie shell --seed 2023263 
(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))