(+ (+ x (* v dt)) (* (* a (* dt dt)) 0.5))

Average Error: 4.3 → 4.3

Time: 1.2s

Precision: binary64

Math

\[\left(x + v \cdot dt\right) + \left(a \cdot \left(dt \cdot dt\right)\right) \cdot 0.5\]

↓

\[\left(x + v \cdot dt\right) + \left(a \cdot \left(dt \cdot dt\right)\right) \cdot 0.5\]

C

double code(double x, double v, double dt, double a) {
	return ((double) (((double) (x + ((double) (v * dt)))) + ((double) (((double) (a * ((double) (dt * dt)))) * 0.5))));
}

↓

double code(double x, double v, double dt, double a) {
	return ((double) (((double) (x + ((double) (v * dt)))) + ((double) (((double) (a * ((double) (dt * dt)))) * 0.5))));
}

Error

Bits error versus x

Bits error versus v

Bits error versus dt

Bits error versus a

Derivation

1. Initial program 4.3

   \[\left(x + v \cdot dt\right) + \left(a \cdot \left(dt \cdot dt\right)\right) \cdot 0.5\]

2. Final simplification 4.3

   \[\leadsto \left(x + v \cdot dt\right) + \left(a \cdot \left(dt \cdot dt\right)\right) \cdot 0.5\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x v dt a)
  :name "(+ (+ x (* v dt)) (* (* a (* dt dt)) 0.5))"
  :precision binary64
  (+ (+ x (* v dt)) (* (* a (* dt dt)) 0.5)))