Average Error: 61.8 → 0.3
Time: 20.7s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\left(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot {t}^{2}\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\]
\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)
↓
\left(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot {t}^{2}\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}
double f(double t) {
        double r95140 = 1.0;
        double r95141 = t;
        double r95142 = 2e-16;
        double r95143 = r95141 * r95142;
        double r95144 = r95140 + r95143;
        double r95145 = r95144 * r95144;
        double r95146 = -1.0;
        double r95147 = 2.0;
        double r95148 = r95147 * r95143;
        double r95149 = r95146 - r95148;
        double r95150 = r95145 + r95149;
        return r95150;
}
↓
double f(double t) {
        double r95151 = 3.9999999999999997e-32;
        double r95152 = sqrt(r95151);
        double r95153 = t;
        double r95154 = 2.0;
        double r95155 = pow(r95153, r95154);
        double r95156 = r95152 * r95155;
        double r95157 = r95156 * r95152;
        return r95157;
}

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
\[\mathsf{fma}\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

  2. Simplified50.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1\right), \mathsf{fma}\left(t, 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1\right), -1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)}\]

  3. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.3

    \[\leadsto \color{blue}{\left(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right)} \cdot {t}^{2}\]

  6. Applied associate-*l*0.3

    \[\leadsto \color{blue}{\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot {t}^{2}\right)}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity0.3

    \[\leadsto \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \color{blue}{\left(1 \cdot \left(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot {t}^{2}\right)\right)}\]

  9. Final simplification0.3

    \[\leadsto \left(\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot {t}^{2}\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :pre (<= 0.900000000000000022 t 1.1000000000000001)

  :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)))))