Average Error: 0.0 → 0.0
Time: 1.5s
Precision: binary64
\[a \cdot a - b \cdot b \]
↓
\[\mathsf{fma}\left(a, a, -b \cdot b\right) + \mathsf{fma}\left(-b, b, b \cdot b\right) \]
a \cdot a - b \cdot b
↓
\mathsf{fma}\left(a, a, -b \cdot b\right) + \mathsf{fma}\left(-b, b, b \cdot b\right)

(FPCore (a b) :precision binary64 (- (* a a) (* b b)))
↓
(FPCore (a b)
 :precision binary64
 (+ (fma a a (- (* b b))) (fma (- b) b (* b b))))
double code(double a, double b) {
	return (a * a) - (b * b);
}
↓
double code(double a, double b) {
	return fma(a, a, -(b * b)) + fma(-b, b, (b * b));
}

Error

Bits error versus a

Bits error versus b

Target

Original0.0
Target0.0
Herbie0.0
\[\left(a + b\right) \cdot \left(a - b\right) \]

Derivation

  1. Initial program 0.0

    \[a \cdot a - b \cdot b \]

  2. Applied prod-diff_binary640.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, -b \cdot b\right) + \mathsf{fma}\left(-b, b, b \cdot b\right)} \]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(a, a, -b \cdot b\right) + \mathsf{fma}\left(-b, b, b \cdot b\right) \]

Reproduce

herbie shell --seed 2022076 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

  :herbie-target
  (* (+ a b) (- a b))

  (- (* a a) (* b b)))