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

Percentage Accurate: 91.2% → 97.5%

Time: 10.9s

Precision: binary64

Cost: 8072

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

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+244}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \end{array} \]

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))
     (- (* x (/ y a)) (/ z (/ a t)))
     (if (<= t_1 5e+244)
       (/ (fma (- z) t (* x y)) a)
       (fma x (/ y a) (/ (- 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 = (x * (y / a)) - (z / (a / t));
	} else if (t_1 <= 5e+244) {
		tmp = fma(-z, t, (x * y)) / a;
	} else {
		tmp = fma(x, (y / a), (-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 = Float64(Float64(x * Float64(y / a)) - Float64(z / Float64(a / t)));
	elseif (t_1 <= 5e+244)
		tmp = Float64(fma(Float64(-z), t, Float64(x * y)) / a);
	else
		tmp = fma(x, Float64(y / a), Float64(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[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+244], N[(N[((-z) * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision] + N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	91.2%
Target	91.8%
Herbie	97.5%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.8%

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

   2. Applied egg-rr 99.8%

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

      [Start] 67.8%	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-sub [=>] 67.8%	\[ \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]
      associate-/l* [=>] 80.4%	\[ \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{z \cdot t}{a} \]
      associate-/l* [=>] 99.8%	\[ \frac{x}{\frac{a}{y}} - \color{blue}{\frac{z}{\frac{a}{t}}} \]

   3. Applied egg-rr 99.8%

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

      [Start] 99.8%	\[ \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}} \]
      div-inv [=>] 99.8%	\[ \color{blue}{x \cdot \frac{1}{\frac{a}{y}}} - \frac{z}{\frac{a}{t}} \]
      clear-num [<=] 99.8%	\[ x \cdot \color{blue}{\frac{y}{a}} - \frac{z}{\frac{a}{t}} \]
      *-commutative [<=] 99.8%	\[ \color{blue}{\frac{y}{a} \cdot x} - \frac{z}{\frac{a}{t}} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.00000000000000022e244

   1. Initial program 98.2%

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

   2. Applied egg-rr 98.2%

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

      [Start] 98.2%	\[ \frac{x \cdot y - z \cdot t}{a} \]
      sub-neg [=>] 98.2%	\[ \frac{\color{blue}{x \cdot y + \left(-z \cdot t\right)}}{a} \]
      +-commutative [=>] 98.2%	\[ \frac{\color{blue}{\left(-z \cdot t\right) + x \cdot y}}{a} \]
      distribute-lft-neg-in [=>] 98.2%	\[ \frac{\color{blue}{\left(-z\right) \cdot t} + x \cdot y}{a} \]
      fma-def [=>] 98.2%	\[ \frac{\color{blue}{\mathsf{fma}\left(-z, t, x \cdot y\right)}}{a} \]

   if 5.00000000000000022e244 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 74.0%

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

   2. Applied egg-rr 91.8%

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

      [Start] 74.0%	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-sub [=>] 74.0%	\[ \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]
      *-un-lft-identity [=>] 74.0%	\[ \frac{x \cdot y}{\color{blue}{1 \cdot a}} - \frac{z \cdot t}{a} \]
      times-frac [=>] 84.1%	\[ \color{blue}{\frac{x}{1} \cdot \frac{y}{a}} - \frac{z \cdot t}{a} \]
      fma-neg [=>] 86.8%	\[ \color{blue}{\mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\frac{z \cdot t}{a}\right)} \]
      associate-/l* [=>] 91.8%	\[ \mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\color{blue}{\frac{z}{\frac{a}{t}}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+244}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.5%
Cost	8072

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+244}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \end{array} \]

Alternative 2
Accuracy	97.2%
Cost	7945

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+244}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, t, x \cdot y\right)}{a}\\ \end{array} \]

Alternative 3
Accuracy	97.2%
Cost	1737

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+244}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]

Alternative 4
Accuracy	74.3%
Cost	904

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+60}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-34}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 5
Accuracy	93.2%
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+280}:\\ \;\;\;\;\frac{\frac{x}{a}}{\frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 6
Accuracy	52.3%
Cost	320

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

Alternative 7
Accuracy	52.2%
Cost	320

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

Reproduce

herbie shell --seed 2023271 
(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))