Expression 3, p15

Average Error: 0.0 → 0.0

Time: 1.3s

Precision: binary64

\[0 \leq x \land x \leq 2\]

Math

\[x \cdot \left(x \cdot x\right) + x \cdot x \]

↓

\[\mathsf{fma}\left(x, x, {x}^{3}\right) \]

FPCore

(FPCore (x) :precision binary64 (+ (* x (* x x)) (* x x)))

↓

(FPCore (x) :precision binary64 (fma x x (pow x 3.0)))

C

double code(double x) {
	return (x * (x * x)) + (x * x);
}

↓

double code(double x) {
	return fma(x, x, pow(x, 3.0));
}

Error

Bits error versus x

Target

Original	0.0
Target	0.0
Herbie	0.0

\[\left(\left(1 + x\right) \cdot x\right) \cdot x \]

Derivation

1. Initial program 0.0

   \[x \cdot \left(x \cdot x\right) + x \cdot x \]

2. Taylor expanded in x around 0 0.0

   \[\leadsto \color{blue}{{x}^{2} + {x}^{3}} \]

3. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, {x}^{3}\right)} \]

4. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(x, x, {x}^{3}\right) \]

Reproduce

herbie shell --seed 2022067 
(FPCore (x)
  :name "Expression 3, p15"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 2.0))

  :herbie-target
  (* (* (+ 1.0 x) x) x)

  (+ (* x (* x x)) (* x x)))