fma_test1

Average Error: 61.8 → 0.3

Time: 8.0s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

\[0.900000000000000022 \le t \le 1.1000000000000001\]

Math

\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]

↓

\[\left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t\]

C

double code(double t) {
	return ((double) (((double) (((double) (1.0 + ((double) (t * 2e-16)))) * ((double) (1.0 + ((double) (t * 2e-16)))))) + ((double) (-1.0 - ((double) (2.0 * ((double) (t * 2e-16))))))));
}

↓

double code(double t) {
	return ((double) (((double) (3.9999999999999997e-32 * t)) * t));
}

Error

Bits error versus t

Target

Original	61.8
Target	50.6
Herbie	0.3

\[\mathsf{fma}\left(1 + t \cdot 2 \cdot 10^{-16}, 1 + t \cdot 2 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]

Derivation

1. Initial program 61.8

   \[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]

2. Taylor expanded around 0 0.3

   \[\leadsto \color{blue}{3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}}\]
3. Using strategy rm

4. Applied unpow2 0.3

   \[\leadsto 3.9999999999999997 \cdot 10^{-32} \cdot \color{blue}{\left(t \cdot t\right)}\]

5. Applied associate-*r* 0.3

   \[\leadsto \color{blue}{\left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t}\]

6. Final simplification 0.3

   \[\leadsto \left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t\]

Reproduce

herbie shell --seed 2020182 
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :pre (<= 0.9 t 1.1)

  :herbie-target
  (fma (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16)) (- -1.0 (* 2.0 (* t 2e-16))))

  (+ (* (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16))) (- -1.0 (* 2.0 (* t 2e-16)))))