Average Error: 45.1 → 13.7

Time: 50.6s

Precision: 64

Internal Precision: 2368

\[(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]

↓

\[\log \left(e^{\left(\left((x \cdot y + z)_* - y \cdot x\right) - z\right) - 1}\right)\]

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	45.1
Target	0
Herbie	13.7

\[-1\]

Derivation

Initial program 45.1
\[(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Using strategy rm
Applied add-log-exp46.8
\[\leadsto (x \cdot y + z)_* - \color{blue}{\log \left(e^{1 + \left(x \cdot y + z\right)}\right)}\]
Applied add-log-exp47.3
\[\leadsto \color{blue}{\log \left(e^{(x \cdot y + z)_*}\right)} - \log \left(e^{1 + \left(x \cdot y + z\right)}\right)\]
Applied diff-log47.3
\[\leadsto \color{blue}{\log \left(\frac{e^{(x \cdot y + z)_*}}{e^{1 + \left(x \cdot y + z\right)}}\right)}\]
Applied simplify34.6
\[\leadsto \log \color{blue}{\left(e^{\left((x \cdot y + z)_* - y \cdot x\right) - \left(z + 1\right)}\right)}\]
Using strategy rm
Applied associate--r+13.7
\[\leadsto \log \left(e^{\color{blue}{\left(\left((x \cdot y + z)_* - y \cdot x\right) - z\right) - 1}}\right)\]

Runtime

herbie shell --seed '#(1072743783 989954326 4239155542 3782239461 3602631542 1719177920)' 
(FPCore (x y z)
  :name "simple fma test"

  :herbie-target
  -1

  (- (fma x y z) (+ 1 (+ (* x y) z))))