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

Percentage Accurate: 97.8% → 98.2%

Time: 23.0s

Precision: binary64

Cost: 7040

• Unsound rule application detected in e-graph. Results may not be sound.

• 24.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = 5.508523892515958e+271
  2. y = -8.24990851045654e+153
  3. z = -3.219310755323823e+247
  4. t = -3.3340147131301236e-124
  5. a = 1.8418394727125254e+257

Math

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

↓

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

FPCore

(FPCore (x y z t a) :precision binary64 (/ (- (* x y) (* z t)) a))

↓

(FPCore (x y z t a) :precision binary64 (* (fma x y (* t (- z))) (/ 1.0 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) {
	return fma(x, y, (t * -z)) * (1.0 / a);
}

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)
	return Float64(fma(x, y, Float64(t * Float64(-z))) * Float64(1.0 / a))
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_] := N[(N[(x * y + N[(t * (-z)), $MachinePrecision]), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]

Target

Original	97.8%
Target	92.4%
Herbie	98.2%

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

Derivation

1. Initial program 97.5%

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

2. Applied egg-rr 98.2%

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

   [Start] 97.5%	\[ \frac{x \cdot y - z \cdot t}{a} \]
   div-inv [=>] 97.4%	\[ \color{blue}{\left(x \cdot y - z \cdot t\right) \cdot \frac{1}{a}} \]
   fma-neg [=>] 98.2%	\[ \color{blue}{\mathsf{fma}\left(x, y, -z \cdot t\right)} \cdot \frac{1}{a} \]
   *-commutative [=>] 98.2%	\[ \mathsf{fma}\left(x, y, -\color{blue}{t \cdot z}\right) \cdot \frac{1}{a} \]
   distribute-rgt-neg-in [=>] 98.2%	\[ \mathsf{fma}\left(x, y, \color{blue}{t \cdot \left(-z\right)}\right) \cdot \frac{1}{a} \]

3. Final simplification 98.2%

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

Alternatives

Alternative 1
Accuracy	98.2%
Cost	7040

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

Alternative 2
Accuracy	64.6%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-92}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 3
Accuracy	66.4%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-92}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{t \cdot \left(-z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 4
Accuracy	97.8%
Cost	576

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

Alternative 5
Accuracy	50.9%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-304}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 6
Accuracy	52.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 7
Accuracy	51.0%
Cost	320

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

Alternative 8
Accuracy	52.8%
Cost	320

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

Reproduce

herbie shell --seed 2023277 
(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))