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

Average Accuracy: 87.7% → 98.5%

Time: 10.7s

Precision: binary64

Cost: 7944

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{z}{\frac{a}{t}}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{a} - t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+175}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - t_1\\ \end{array} \]

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 (/ z (/ a t))) (t_2 (- (* x y) (* z t))))
   (if (<= t_2 (- INFINITY))
     (- (* y (/ x a)) t_1)
     (if (<= t_2 2e+175) (/ (fma x y (* z (- t))) a) (- (/ x (/ a y)) t_1)))))

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 = z / (a / t);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (y * (x / a)) - t_1;
	} else if (t_2 <= 2e+175) {
		tmp = fma(x, y, (z * -t)) / a;
	} else {
		tmp = (x / (a / y)) - t_1;
	}
	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(z / Float64(a / t))
	t_2 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(y * Float64(x / a)) - t_1);
	elseif (t_2 <= 2e+175)
		tmp = Float64(fma(x, y, Float64(z * Float64(-t))) / a);
	else
		tmp = Float64(Float64(x / Float64(a / y)) - t_1);
	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[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2e+175], N[(N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Target

Original	87.7%
Target	91.3%
Herbie	98.5%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 0.0%

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

   2. Applied egg-rr 99.6%

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

      [Start] 0.0	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-sub [=>] 0.0	\[ \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]
      associate-/l* [=>] 44.6	\[ \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{z \cdot t}{a} \]
      associate-/l* [=>] 99.6	\[ \frac{x}{\frac{a}{y}} - \color{blue}{\frac{z}{\frac{a}{t}}} \]

   3. Applied egg-rr 99.6%

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

      [Start] 99.6	\[ \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}} \]
      associate-/r/ [=>] 99.6	\[ \color{blue}{\frac{x}{a} \cdot y} - \frac{z}{\frac{a}{t}} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.9999999999999999e175

   1. Initial program 98.7%

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

   2. Simplified 98.7%

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

      [Start] 98.7	\[ \frac{x \cdot y - z \cdot t}{a} \]
      fma-neg [=>] 98.7	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y, -z \cdot t\right)}}{a} \]
      distribute-rgt-neg-in [=>] 98.7	\[ \frac{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-t\right)}\right)}{a} \]

   if 1.9999999999999999e175 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 60.0%

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

   2. Applied egg-rr 97.5%

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

      [Start] 60.0	\[ \frac{x \cdot y - z \cdot t}{a} \]
      div-sub [=>] 60.0	\[ \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]
      associate-/l* [=>] 76.7	\[ \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{z \cdot t}{a} \]
      associate-/l* [=>] 97.5	\[ \frac{x}{\frac{a}{y}} - \color{blue}{\frac{z}{\frac{a}{t}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{a} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 2 \cdot 10^{+175}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.2%
Cost	1865

\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{+270}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	98.8%
Cost	1737

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 10^{+251}\right):\\ \;\;\;\;y \cdot \frac{x}{a} - \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \]

Alternative 3
Accuracy	98.5%
Cost	1736

\[\begin{array}{l} t_1 := \frac{z}{\frac{a}{t}}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{a} - t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+175}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - t_1\\ \end{array} \]

Alternative 4
Accuracy	69.5%
Cost	1164

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{z}{\frac{-a}{t}}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-98}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq \infty:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \end{array} \]

Alternative 5
Accuracy	63.0%
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-16} \lor \neg \left(z \leq 3.8 \cdot 10^{-62}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]

Alternative 6
Accuracy	63.9%
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{-16}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;z \leq 1.88 \cdot 10^{-62}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \end{array} \]

Alternative 7
Accuracy	63.8%
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-15}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \]

Alternative 8
Accuracy	48.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-67}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 9
Accuracy	48.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-67}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 10
Accuracy	48.2%
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 11
Accuracy	48.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -5.7 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]

Alternative 12
Accuracy	48.2%
Cost	320

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

Reproduce

herbie shell --seed 2023135 
(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))