Average Error: 61.8 → 0.4
Time: 10.4s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(t \cdot t\right)\right)\]
\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)
↓
\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(t \cdot t\right)\right)
double f(double t) {
        double r18153955 = 1.0;
        double r18153956 = t;
        double r18153957 = 2e-16;
        double r18153958 = r18153956 * r18153957;
        double r18153959 = r18153955 + r18153958;
        double r18153960 = r18153959 * r18153959;
        double r18153961 = -1.0;
        double r18153962 = 2.0;
        double r18153963 = r18153962 * r18153958;
        double r18153964 = r18153961 - r18153963;
        double r18153965 = r18153960 + r18153964;
        return r18153965;
}
↓
double f(double t) {
        double r18153966 = 3.9999999999999997e-32;
        double r18153967 = sqrt(r18153966);
        double r18153968 = t;
        double r18153969 = r18153968 * r18153968;
        double r18153970 = r18153967 * r18153969;
        double r18153971 = r18153967 * r18153970;
        return r18153971;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.6
Herbie0.4
\[(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right))_*\]

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. Simplified50.3

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

    \[\leadsto \color{blue}{\left(t \cdot t\right) \cdot 3.9999999999999997 \cdot 10^{-32}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.3

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

  7. Applied associate-*r*0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019112 +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)))))