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

Average Accuracy: 88.4% → 97.0%

Time: 12.7s

Precision: binary64

Cost: 15824

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+280}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{elif}\;t_1 \leq 1.6 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\frac{x \cdot y + \mathsf{fma}\left(z, -t, \mathsf{fma}\left(z, -t, z \cdot t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

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))) (t_2 (fma x (/ y a) (/ (- z) (/ a t)))))
   (if (<= t_1 -5e+280)
     t_2
     (if (<= t_1 -1e-285)
       (/ t_1 a)
       (if (<= t_1 1.6e-102)
         t_2
         (if (<= t_1 5e+55)
           (/ (+ (* x y) (fma z (- t) (fma z (- t) (* z t)))) 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 t_2 = fma(x, (y / a), (-z / (a / t)));
	double tmp;
	if (t_1 <= -5e+280) {
		tmp = t_2;
	} else if (t_1 <= -1e-285) {
		tmp = t_1 / a;
	} else if (t_1 <= 1.6e-102) {
		tmp = t_2;
	} else if (t_1 <= 5e+55) {
		tmp = ((x * y) + fma(z, -t, fma(z, -t, (z * t)))) / 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))
	t_2 = fma(x, Float64(y / a), Float64(Float64(-z) / Float64(a / t)))
	tmp = 0.0
	if (t_1 <= -5e+280)
		tmp = t_2;
	elseif (t_1 <= -1e-285)
		tmp = Float64(t_1 / a);
	elseif (t_1 <= 1.6e-102)
		tmp = t_2;
	elseif (t_1 <= 5e+55)
		tmp = Float64(Float64(Float64(x * y) + fma(z, Float64(-t), fma(z, Float64(-t), Float64(z * t)))) / 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]}, Block[{t$95$2 = N[(x * N[(y / a), $MachinePrecision] + N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+280], t$95$2, If[LessEqual[t$95$1, -1e-285], N[(t$95$1 / a), $MachinePrecision], If[LessEqual[t$95$1, 1.6e-102], t$95$2, If[LessEqual[t$95$1, 5e+55], N[(N[(N[(x * y), $MachinePrecision] + N[(z * (-t) + N[(z * (-t) + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	88.4%
Target	91.5%
Herbie	97.0%

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

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -5.0000000000000002e280 or -1.00000000000000007e-285 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.59999999999999993e-102

   1. Initial program 64.8%

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

   2. Applied egg-rr 97.3%

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

      [Start] 64.8	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-sub [=>] 64.8	\[ \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]
      *-un-lft-identity [=>] 64.8	\[ \frac{x \cdot y}{\color{blue}{1 \cdot a}} - \frac{z \cdot t}{a} \]
      times-frac [=>] 80.3	\[ \color{blue}{\frac{x}{1} \cdot \frac{y}{a}} - \frac{z \cdot t}{a} \]
      fma-neg [=>] 80.3	\[ \color{blue}{\mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\frac{z \cdot t}{a}\right)} \]
      associate-/l* [=>] 97.3	\[ \mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\color{blue}{\frac{z}{\frac{a}{t}}}\right) \]

   if -5.0000000000000002e280 < (-.f64 (*.f64 x y) (*.f64 z t)) < -1.00000000000000007e-285

   1. Initial program 99.6%

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

   if 1.59999999999999993e-102 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.00000000000000046e55

   1. Initial program 99.6%

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

   2. Applied egg-rr 99.5%

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

      [Start] 99.6	\[ \frac{x \cdot y - z \cdot t}{a} \]
      prod-diff [=>] 99.6	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y, -t \cdot z\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)}}{a} \]
      *-commutative [<=] 99.6	\[ \frac{\mathsf{fma}\left(x, y, -\color{blue}{z \cdot t}\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)}{a} \]
      fma-def [<=] 99.6	\[ \frac{\color{blue}{\left(x \cdot y + \left(-z \cdot t\right)\right)} + \mathsf{fma}\left(-t, z, t \cdot z\right)}{a} \]
      associate-+l+ [=>] 99.6	\[ \frac{\color{blue}{x \cdot y + \left(\left(-z \cdot t\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)\right)}}{a} \]
      distribute-rgt-neg-in [=>] 99.6	\[ \frac{x \cdot y + \left(\color{blue}{z \cdot \left(-t\right)} + \mathsf{fma}\left(-t, z, t \cdot z\right)\right)}{a} \]
      fma-def [=>] 99.5	\[ \frac{x \cdot y + \color{blue}{\mathsf{fma}\left(z, -t, \mathsf{fma}\left(-t, z, t \cdot z\right)\right)}}{a} \]
      *-commutative [<=] 99.5	\[ \frac{x \cdot y + \mathsf{fma}\left(z, -t, \mathsf{fma}\left(-t, z, \color{blue}{z \cdot t}\right)\right)}{a} \]
      fma-udef [=>] 99.6	\[ \frac{x \cdot y + \mathsf{fma}\left(z, -t, \color{blue}{\left(-t\right) \cdot z + z \cdot t}\right)}{a} \]
      distribute-lft-neg-in [<=] 99.6	\[ \frac{x \cdot y + \mathsf{fma}\left(z, -t, \color{blue}{\left(-t \cdot z\right)} + z \cdot t\right)}{a} \]
      *-commutative [<=] 99.6	\[ \frac{x \cdot y + \mathsf{fma}\left(z, -t, \left(-\color{blue}{z \cdot t}\right) + z \cdot t\right)}{a} \]
      distribute-rgt-neg-in [=>] 99.6	\[ \frac{x \cdot y + \mathsf{fma}\left(z, -t, \color{blue}{z \cdot \left(-t\right)} + z \cdot t\right)}{a} \]
      fma-def [=>] 99.5	\[ \frac{x \cdot y + \mathsf{fma}\left(z, -t, \color{blue}{\mathsf{fma}\left(z, -t, z \cdot t\right)}\right)}{a} \]

   if 5.00000000000000046e55 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 78.1%

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

   2. Applied egg-rr 91.3%

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

      [Start] 78.1	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-sub [=>] 78.1	\[ \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]
      associate-/l* [=>] 83.4	\[ \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{z \cdot t}{a} \]
      associate-/l* [=>] 91.3	\[ \frac{x}{\frac{a}{y}} - \color{blue}{\frac{z}{\frac{a}{t}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+280}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 1.6 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\frac{x \cdot y + \mathsf{fma}\left(z, -t, \mathsf{fma}\left(z, -t, z \cdot t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.3%
Cost	8588

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{t_1}{a}\\ t_3 := \mathsf{fma}\left(x, \frac{y}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+280}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{-139}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternative 2
Accuracy	96.8%
Cost	7620

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+169}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-285} \lor \neg \left(t_1 \leq 10^{-159}\right) \land t_1 \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternative 3
Accuracy	97.3%
Cost	2770

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+280} \lor \neg \left(t_1 \leq -1 \cdot 10^{-285} \lor \neg \left(t_1 \leq 10^{-159}\right) \land t_1 \leq 5 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]

Alternative 4
Accuracy	89.9%
Cost	1616

\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ t_2 := \frac{-z}{\frac{a}{t}}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+242}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-181}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+247}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	62.0%
Cost	912

\[\begin{array}{l} t_1 := t \cdot \frac{-z}{a}\\ \mathbf{if}\;t \leq -1.85 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.4 \cdot 10^{-268}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-247}:\\ \;\;\;\;-\frac{z \cdot t}{a}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+51}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	62.6%
Cost	649

\[\begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{-94} \lor \neg \left(t \leq 1.05 \cdot 10^{+25}\right):\\ \;\;\;\;-\frac{t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 7
Accuracy	62.6%
Cost	649

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-91} \lor \neg \left(t \leq 6.8 \cdot 10^{+51}\right):\\ \;\;\;\;-\frac{z}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 8
Accuracy	47.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 9
Accuracy	47.8%
Cost	320

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

Reproduce

herbie shell --seed 2023151 
(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))