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

Average Error: 11.39% → 2.14%

Time: 14.4s

Precision: binary64

Cost: 8072

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, t \cdot \frac{-z}{a}\right)\\ \end{array} \]

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+245)
     (fma -1.0 (/ t (/ a z)) (/ y (/ a x)))
     (if (<= t_1 2e+127) (/ t_1 a) (fma (/ x a) y (* t (/ (- z) a)))))))

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+245) {
		tmp = fma(-1.0, (t / (a / z)), (y / (a / x)));
	} else if (t_1 <= 2e+127) {
		tmp = t_1 / a;
	} else {
		tmp = fma((x / a), y, (t * (-z / a)));
	}
	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+245)
		tmp = fma(-1.0, Float64(t / Float64(a / z)), Float64(y / Float64(a / x)));
	elseif (t_1 <= 2e+127)
		tmp = Float64(t_1 / a);
	else
		tmp = fma(Float64(x / a), y, Float64(t * Float64(Float64(-z) / a)));
	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+245], N[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+127], N[(t$95$1 / a), $MachinePrecision], N[(N[(x / a), $MachinePrecision] * y + N[(t * N[((-z) / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	11.39%
Target	8.5%
Herbie	2.14%

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -5.00000000000000034e245

   1. Initial program 58.17

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

   2. Taylor expanded in x around 0 58.17

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

   3. Simplified 0.84

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

      [Start] 58.17	\[ -1 \cdot \frac{t \cdot z}{a} + \frac{y \cdot x}{a} \]
      fma-def [=>] 58.17	\[ \color{blue}{\mathsf{fma}\left(-1, \frac{t \cdot z}{a}, \frac{y \cdot x}{a}\right)} \]
      associate-/l* [=>] 30.34	\[ \mathsf{fma}\left(-1, \color{blue}{\frac{t}{\frac{a}{z}}}, \frac{y \cdot x}{a}\right) \]
      associate-/l* [=>] 0.84	\[ \mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \color{blue}{\frac{y}{\frac{a}{x}}}\right) \]

   if -5.00000000000000034e245 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.99999999999999991e127

   1. Initial program 1.41

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

   if 1.99999999999999991e127 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 29.05

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

   2. Applied egg-rr 30.22

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x \cdot y - z \cdot t}{a}}\right)}^{3}} \]

   3. Applied egg-rr 4.96

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}} + \left(-\frac{z}{\frac{a}{t}}\right)} \]

   4. Simplified 5.47

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

      [Start] 4.96	\[ \frac{x}{\frac{a}{y}} + \left(-\frac{z}{\frac{a}{t}}\right) \]
      sub-neg [<=] 4.96	\[ \color{blue}{\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}} \]
      associate-/r/ [=>] 5.53	\[ \color{blue}{\frac{x}{a} \cdot y} - \frac{z}{\frac{a}{t}} \]
      associate-/r/ [=>] 5.47	\[ \frac{x}{a} \cdot y - \color{blue}{\frac{z}{a} \cdot t} \]

   5. Applied egg-rr 5.47

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{a}, y, \frac{z}{a} \cdot \left(-t\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 2.14

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, t \cdot \frac{-z}{a}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.15%
Cost	8072

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+245}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{y}}{x}} - t \cdot \frac{z}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, y, t \cdot \frac{-z}{a}\right)\\ \end{array} \]

Alternative 2
Error	7.1%
Cost	1864

\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 3
Error	2.13%
Cost	1737

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+245} \lor \neg \left(t_1 \leq 2 \cdot 10^{+127}\right):\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]

Alternative 4
Error	2.11%
Cost	1736

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t \cdot \frac{z}{a}\\ \end{array} \]

Alternative 5
Error	2.14%
Cost	1736

\[\begin{array}{l} t_1 := y \cdot \frac{x}{a}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+245}:\\ \;\;\;\;t_1 + \frac{-1}{\frac{\frac{a}{z}}{t}}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1 - t \cdot \frac{z}{a}\\ \end{array} \]

Alternative 6
Error	2.15%
Cost	1736

\[\begin{array}{l} t_1 := t \cdot \frac{z}{a}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+245}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{y}}{x}} - t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a} - t_1\\ \end{array} \]

Alternative 7
Error	39.06%
Cost	1440

\[\begin{array}{l} t_1 := \frac{x \cdot y}{a}\\ t_2 := \frac{y}{\frac{a}{x}}\\ t_3 := \frac{-t}{\frac{a}{z}}\\ \mathbf{if}\;x \leq -2.25 \cdot 10^{+172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+147}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq -370000000000:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-23}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-174}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -3.05 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{-103}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	39.74%
Cost	1440

\[\begin{array}{l} t_1 := \frac{x \cdot y}{a}\\ t_2 := \frac{-t}{\frac{a}{z}}\\ t_3 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+207}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.9 \cdot 10^{+34}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.28 \cdot 10^{-23}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-174}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.42 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-104}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	39.59%
Cost	1440

\[\begin{array}{l} t_1 := \frac{x \cdot y}{a}\\ t_2 := \frac{-t}{\frac{a}{z}}\\ t_3 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;x \leq -5.7 \cdot 10^{+205}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+34}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-174}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.05 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{-103}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Error	38.55%
Cost	1440

\[\begin{array}{l} t_1 := \frac{x \cdot y}{a}\\ t_2 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+205}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-24}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-174}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-102}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	49.73%
Cost	584

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+225}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 12
Error	51.05%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+172}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 13
Error	50.99%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+172}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 14
Error	51%
Cost	320

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

Alternative 15
Error	51.01%
Cost	320

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

Reproduce

herbie shell --seed 2023102 
(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))