Average Error: 7.7 → 7.7
Time: 3.9s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\frac{\mathsf{fma}\left(-y, x, z \cdot t\right)}{-a}\]
\frac{x \cdot y - z \cdot t}{a}
↓
\frac{\mathsf{fma}\left(-y, x, z \cdot t\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 (fma(-y, x, (z * t)) / -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

Original7.7
Target6.2
Herbie7.7
\[\begin{array}{l} \mathbf{if}\;z \lt -2.46868496869954822 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.30983112197837121 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

  1. Initial program 7.7

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

  3. Applied frac-2neg7.7

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

  4. Simplified7.7

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

  5. Final simplification7.7

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

Reproduce

herbie shell --seed 2020078 +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))