Average Error: 61.8 → 0.3
Time: 10.8s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)\]
\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)
↓
t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)
double f(double t) {
        double r1235729 = 1.0;
        double r1235730 = t;
        double r1235731 = 2e-16;
        double r1235732 = r1235730 * r1235731;
        double r1235733 = r1235729 + r1235732;
        double r1235734 = r1235733 * r1235733;
        double r1235735 = -1.0;
        double r1235736 = 2.0;
        double r1235737 = r1235736 * r1235732;
        double r1235738 = r1235735 - r1235737;
        double r1235739 = r1235734 + r1235738;
        return r1235739;
}
↓
double f(double t) {
        double r1235740 = t;
        double r1235741 = 3.9999999999999997e-32;
        double r1235742 = r1235740 * r1235741;
        double r1235743 = r1235740 * r1235742;
        return r1235743;
}

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. Simplified61.8

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

  3. Taylor expanded around -inf 0.4

    \[\leadsto \color{blue}{3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}}\]

  4. Simplified0.4

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

  5. Taylor expanded around -inf 0.4

    \[\leadsto \color{blue}{3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}}\]

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(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)))))