Average Error: 61.8 → 0.3
Time: 2.2s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(3.9999999999999997 \cdot 10^{-32} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\]
\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 \left|t\right|\right) \cdot \sqrt{{t}^{2}}
double f(double t) {
        double r70962 = 1.0;
        double r70963 = t;
        double r70964 = 2e-16;
        double r70965 = r70963 * r70964;
        double r70966 = r70962 + r70965;
        double r70967 = r70966 * r70966;
        double r70968 = -1.0;
        double r70969 = 2.0;
        double r70970 = r70969 * r70965;
        double r70971 = r70968 - r70970;
        double r70972 = r70967 + r70971;
        return r70972;
}
↓
double f(double t) {
        double r70973 = 3.9999999999999997e-32;
        double r70974 = t;
        double r70975 = fabs(r70974);
        double r70976 = r70973 * r70975;
        double r70977 = 2.0;
        double r70978 = pow(r70974, r70977);
        double r70979 = sqrt(r70978);
        double r70980 = r70976 * r70979;
        return r70980;
}

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 add-sqr-sqrt0.4

    \[\leadsto 3.9999999999999997 \cdot 10^{-32} \cdot \color{blue}{\left(\sqrt{{t}^{2}} \cdot \sqrt{{t}^{2}}\right)}\]

  6. Applied associate-*r*0.3

    \[\leadsto \color{blue}{\left(3.9999999999999997 \cdot 10^{-32} \cdot \sqrt{{t}^{2}}\right) \cdot \sqrt{{t}^{2}}}\]

  7. Simplified0.3

    \[\leadsto \color{blue}{\left(3.9999999999999997 \cdot 10^{-32} \cdot \left|t\right|\right)} \cdot \sqrt{{t}^{2}}\]

  8. Final simplification0.3

    \[\leadsto \left(3.9999999999999997 \cdot 10^{-32} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\]

Reproduce

herbie shell --seed 2020018 +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)))))