Average Error: 0.0 → 0.0
Time: 3.0s
Precision: binary64
\[x + \frac{y - x}{z} \]
↓
\[\mathsf{fma}\left(1, \frac{y}{z}, x\right) - \frac{x}{z} \]
x + \frac{y - x}{z}
↓
\mathsf{fma}\left(1, \frac{y}{z}, x\right) - \frac{x}{z}

(FPCore (x y z) :precision binary64 (+ x (/ (- y x) z)))
↓
(FPCore (x y z) :precision binary64 (- (fma 1.0 (/ y z) x) (/ x z)))
double code(double x, double y, double z) {
	return x + ((y - x) / z);
}
↓
double code(double x, double y, double z) {
	return fma(1.0, (y / z), x) - (x / z);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[x + \frac{y - x}{z} \]

  2. Taylor expanded in x around 0 0.0

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

  3. Applied *-un-lft-identity_binary640.0

    \[\leadsto \left(\color{blue}{1 \cdot \frac{y}{z}} + x\right) - \frac{x}{z} \]

  4. Applied fma-def_binary640.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{y}{z}, x\right)} - \frac{x}{z} \]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(1, \frac{y}{z}, x\right) - \frac{x}{z} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))