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

?

Percentage Accurate: 91.5% → 96.7%
Time: 10.5s
Precision: binary64
Cost: 8137

?

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]
\[\frac{x \cdot y - z \cdot t}{a} \]
↓
\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+219} \lor \neg \left(t_1 \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]
(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 -1e+219) (not (<= t_1 5e+305)))
     (fma -1.0 (/ t (/ a z)) (/ y (/ a x)))
     (/ t_1 a))))
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 <= -1e+219) || !(t_1 <= 5e+305)) {
		tmp = fma(-1.0, (t / (a / z)), (y / (a / x)));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

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 <= -1e+219) || !(t_1 <= 5e+305))
		tmp = fma(-1.0, Float64(t / Float64(a / z)), Float64(y / Float64(a / x)));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

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, -1e+219], N[Not[LessEqual[t$95$1, 5e+305]], $MachinePrecision]], N[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

\frac{x \cdot y - z \cdot t}{a}
↓
\begin{array}{l}
t_1 := x \cdot y - z \cdot t\\
\mathbf{if}\;t_1 \leq -1 \cdot 10^{+219} \lor \neg \left(t_1 \leq 5 \cdot 10^{+305}\right):\\
\;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{t_1}{a}\\


\end{array}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 8 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Target

Original91.5%
Target91.5%
Herbie96.7%
\[\begin{array}{l} \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array} \]

Derivation?

  1. Split input into 2 regimes

  2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -9.99999999999999965e218 or 5.00000000000000009e305 < (-.f64 (*.f64 x y) (*.f64 z t))

    1. Initial program 76.4%

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

    2. Taylor expanded in x around 0 75.2%

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

    3. Simplified96.3%

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

      [Start]75.2%

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

      fma-def [=>]75.2%

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

      associate-/l* [=>]87.3%

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

      associate-/l* [=>]96.3%

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

    if -9.99999999999999965e218 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.00000000000000009e305

    1. Initial program 99.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification98.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{+219} \lor \neg \left(x \cdot y - z \cdot t \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy96.7%
Cost8137
\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+219} \lor \neg \left(t_1 \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]
Alternative 2
Accuracy94.4%
Cost969
\[\begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{+16} \lor \neg \left(a \leq 10^{-85}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]
Alternative 3
Accuracy93.5%
Cost836
\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \]
Alternative 4
Accuracy68.0%
Cost648
\[\begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-123}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]
Alternative 5
Accuracy68.1%
Cost648
\[\begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-122}:\\ \;\;\;\;\frac{z}{-\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]
Alternative 6
Accuracy67.7%
Cost648
\[\begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{t \cdot \left(-z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]
Alternative 7
Accuracy51.7%
Cost320
\[y \cdot \frac{x}{a} \]
Alternative 8
Accuracy51.4%
Cost320
\[x \cdot \frac{y}{a} \]

Reproduce?

herbie shell --seed 2023272 
(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))