Average Error: 7.5 → 7.5
Time: 4.2s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\frac{\left(x \cdot y - z \cdot t\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)}{a}\]
\frac{x \cdot y - z \cdot t}{a}
↓
\frac{\left(x \cdot y - z \cdot t\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)}{a}
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 ((((x * y) - (z * t)) + fma(-t, z, (t * z))) / a);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.5
Target5.9
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;z \lt -2.46868496869954822 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.30983112197837121 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

  1. Initial program 7.5

    \[\frac{x \cdot y - z \cdot t}{a}\]
  2. Using strategy rm

  3. Applied prod-diff7.5

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

  4. Simplified7.5

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

  5. Final simplification7.5

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

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(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))