Average Error: 61.8 → 0.3
Time: 2.5s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right) \cdot t\]
\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(t \cdot 3.9999999999999997 \cdot 10^{-32}\right) \cdot t
double f(double t) {
        double r74158 = 1.0;
        double r74159 = t;
        double r74160 = 2e-16;
        double r74161 = r74159 * r74160;
        double r74162 = r74158 + r74161;
        double r74163 = r74162 * r74162;
        double r74164 = -1.0;
        double r74165 = 2.0;
        double r74166 = r74165 * r74161;
        double r74167 = r74164 - r74166;
        double r74168 = r74163 + r74167;
        return r74168;
}
↓
double f(double t) {
        double r74169 = t;
        double r74170 = 3.9999999999999997e-32;
        double r74171 = r74169 * r74170;
        double r74172 = r74171 * r74169;
        return r74172;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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. Simplified57.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot t, 2 \cdot 10^{-16}, \mathsf{fma}\left(\mathsf{fma}\left(2 \cdot 10^{-16}, t, 1\right), \mathsf{fma}\left(2 \cdot 10^{-16}, t, 1\right), -1\right)\right)}\]

  3. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}}\]
  4. Using strategy rm

  5. Applied unpow20.4

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

  6. Applied associate-*r*0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :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)))))