Average Error: 0.0 → 0.0
Time: 4.7s
Precision: binary64
\[x + \left(y - z\right) \cdot \left(t - x\right) \]
↓
\[\left(y \cdot t + \mathsf{fma}\left(z, x, x\right)\right) - \left(y \cdot x + t \cdot z\right) \]
x + \left(y - z\right) \cdot \left(t - x\right)
↓
\left(y \cdot t + \mathsf{fma}\left(z, x, x\right)\right) - \left(y \cdot x + t \cdot z\right)

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original0.0
Target0.0
Herbie0.0
\[x + \left(t \cdot \left(y - z\right) + \left(-x\right) \cdot \left(y - z\right)\right) \]

Derivation

  1. Initial program 0.0

    \[x + \left(y - z\right) \cdot \left(t - x\right) \]

  2. Simplified0.0

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

  3. Taylor expanded in y around 0 0.0

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

  4. Taylor expanded in x around 0 0.0

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

  5. Simplified0.0

    \[\leadsto \left(y \cdot t + \color{blue}{\mathsf{fma}\left(z, x, x\right)}\right) - \left(y \cdot x + t \cdot z\right) \]

  6. Final simplification0.0

    \[\leadsto \left(y \cdot t + \mathsf{fma}\left(z, x, x\right)\right) - \left(y \cdot x + t \cdot z\right) \]

Reproduce

herbie shell --seed 2021307 
(FPCore (x y z t)
  :name "Data.Metrics.Snapshot:quantile from metrics-0.3.0.2"
  :precision binary64

  :herbie-target
  (+ x (+ (* t (- y z)) (* (- x) (- y z))))

  (+ x (* (- y z) (- t x))))