Average Error: 61.8 → 0.3
Time: 11.8s
Precision: 64
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t\]
double f(double t) {
        double r3254933 = 1.0;
        double r3254934 = t;
        double r3254935 = 2e-16;
        double r3254936 = r3254934 * r3254935;
        double r3254937 = r3254933 + r3254936;
        double r3254938 = r3254937 * r3254937;
        double r3254939 = -1.0;
        double r3254940 = 2.0;
        double r3254941 = r3254940 * r3254936;
        double r3254942 = r3254939 - r3254941;
        double r3254943 = r3254938 + r3254942;
        return r3254943;
}
↓
double f(double t) {
        double r3254944 = 3.9999999999999997e-32;
        double r3254945 = t;
        double r3254946 = r3254944 * r3254945;
        double r3254947 = r3254946 * r3254945;
        return r3254947;
}


\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)
↓
\left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t

Error

Bits error versus t

Target

Original61.8
Target50.6
Herbie0.3
\[(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right))_*\]

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\left(2 \cdot 10^{-16} \cdot t\right) \cdot \left(2 \cdot 10^{-16} \cdot t\right)}\]
  3. Using strategy rm

  4. Applied associate-*r*0.3

    \[\leadsto \color{blue}{\left(\left(2 \cdot 10^{-16} \cdot t\right) \cdot 2 \cdot 10^{-16}\right) \cdot t}\]

  5. Taylor expanded around -inf 0.3

    \[\leadsto \color{blue}{\left(3.9999999999999997 \cdot 10^{-32} \cdot t\right)} \cdot t\]

  6. Final simplification0.3

    \[\leadsto \left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t\]

Reproduce

herbie shell --seed 2019101 
(FPCore (t)
  :name "fma_test1"
  :pre (<= 0.9 t 1.1)

  :herbie-target
  (fma (+ 1 (* t 2e-16)) (+ 1 (* t 2e-16)) (- -1 (* 2 (* t 2e-16))))

  (+ (* (+ 1 (* t 2e-16)) (+ 1 (* t 2e-16))) (- -1 (* 2 (* t 2e-16)))))