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

Average Error: 7.6 → 1.3

Time: 12.6s

Precision: binary64

Cost: 8136

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+187}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 10^{+159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

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 -5e+187)
     (- (/ x (/ a y)) (/ z (/ a t)))
     (if (<= t_1 1e+159)
       (/ (fma x y (* z (- t))) a)
       (fma -1.0 (/ t (/ a z)) (/ y (/ a x)))))))

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 <= -5e+187) {
		tmp = (x / (a / y)) - (z / (a / t));
	} else if (t_1 <= 1e+159) {
		tmp = fma(x, y, (z * -t)) / a;
	} else {
		tmp = fma(-1.0, (t / (a / z)), (y / (a / x)));
	}
	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 <= -5e+187)
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	elseif (t_1 <= 1e+159)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a);
	else
		tmp = fma(-1.0, Float64(t / Float64(a / z)), Float64(y / Float64(a / x)));
	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, -5e+187], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+159], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.6
Target	5.9
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -5.0000000000000001e187

   1. Initial program 25.2

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

   2. Applied egg-rr 2.0

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

   if -5.0000000000000001e187 < (-.f64 (*.f64 x y) (*.f64 z t)) < 9.9999999999999993e158

   1. Initial program 0.9

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

   2. Simplified 0.9

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

      [Start] 0.9	\[ \frac{x \cdot y - z \cdot t}{a} \]
      fma-neg [=>] 0.9	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y, -z \cdot t\right)}}{a} \]
      distribute-rgt-neg-in [=>] 0.9	\[ \frac{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-t\right)}\right)}{a} \]

   if 9.9999999999999993e158 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 21.5

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

   2. Taylor expanded in x around 0 21.5

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

   3. Simplified 2.4

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

      [Start] 21.5	\[ -1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a} \]
      fma-def [=>] 21.5	\[ \color{blue}{\mathsf{fma}\left(-1, \frac{t \cdot z}{a}, \frac{y \cdot x}{a}\right)} \]
      associate-/l* [=>] 12.8	\[ \mathsf{fma}\left(-1, \color{blue}{\frac{t}{\frac{a}{z}}}, \frac{y \cdot x}{a}\right) \]
      associate-/l* [=>] 2.4	\[ \mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \color{blue}{\frac{y}{\frac{a}{x}}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+187}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.2
Cost	7945

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+187} \lor \neg \left(t_1 \leq 2 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \end{array} \]

Alternative 2
Error	1.3
Cost	1737

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+187} \lor \neg \left(t_1 \leq 2 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]

Alternative 3
Error	25.5
Cost	1705

\[\begin{array}{l} t_1 := x \cdot \frac{y}{a}\\ t_2 := \frac{t}{a} \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+225}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq -3.85 \cdot 10^{+186}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.3 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.95 \cdot 10^{+53}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-85} \lor \neg \left(z \leq -1.15 \cdot 10^{-113}\right) \land z \leq 2.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \]

Alternative 4
Error	4.2
Cost	1608

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{t}{a} \cdot \left(-z\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{a}}{\frac{1}{x}}\\ \end{array} \]

Alternative 5
Error	24.8
Cost	1441

\[\begin{array}{l} t_1 := \frac{z}{\frac{-a}{t}}\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{t}{a} \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -1.28 \cdot 10^{-85} \lor \neg \left(z \leq -1.1 \cdot 10^{-113}\right) \land z \leq 3.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \end{array} \]

Alternative 6
Error	25.1
Cost	1440

\[\begin{array}{l} t_1 := x \cdot \frac{y}{a}\\ t_2 := \frac{t}{a} \cdot \left(-z\right)\\ t_3 := \frac{-t}{\frac{a}{z}}\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+225}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -3.85 \cdot 10^{+186}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	18.8
Cost	1424

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+67}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;z \cdot t \leq 10^{+87}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \end{array} \]

Alternative 8
Error	24.3
Cost	780

\[\begin{array}{l} t_1 := \frac{-t}{\frac{a}{z}}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-141}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+84}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	31.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 10
Error	33.0
Cost	320

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

Alternative 11
Error	32.8
Cost	320

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

Alternative 12
Error	32.8
Cost	320

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

Reproduce

herbie shell --seed 2023055 
(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))