Average Error: 61.8 → 0.3
Time: 11.9s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left(\left(\left|t\right| \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right) \cdot \sqrt{{t}^{2}}\right)\]
\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)
↓
\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left(\left(\left|t\right| \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right) \cdot \sqrt{{t}^{2}}\right)
double f(double t) {
        double r44126 = 1.0;
        double r44127 = t;
        double r44128 = 2e-16;
        double r44129 = r44127 * r44128;
        double r44130 = r44126 + r44129;
        double r44131 = r44130 * r44130;
        double r44132 = -1.0;
        double r44133 = 2.0;
        double r44134 = r44133 * r44129;
        double r44135 = r44132 - r44134;
        double r44136 = r44131 + r44135;
        return r44136;
}
↓
double f(double t) {
        double r44137 = 3.9999999999999997e-32;
        double r44138 = sqrt(r44137);
        double r44139 = t;
        double r44140 = fabs(r44139);
        double r44141 = r44140 * r44138;
        double r44142 = 2.0;
        double r44143 = pow(r44139, r44142);
        double r44144 = sqrt(r44143);
        double r44145 = r44141 * r44144;
        double r44146 = r44138 * r44145;
        return r44146;
}

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*0.4

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

  8. Applied add-sqr-sqrt0.4

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

  9. Applied associate-*r*0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019235 +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)))))