Average Error: 61.8 → 0.3
Time: 9.5s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\left(\left(t \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right) \cdot t\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\]
\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(\left(t \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right) \cdot t\right) \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}
double f(double t) {
        double r79135 = 1.0;
        double r79136 = t;
        double r79137 = 2e-16;
        double r79138 = r79136 * r79137;
        double r79139 = r79135 + r79138;
        double r79140 = r79139 * r79139;
        double r79141 = -1.0;
        double r79142 = 2.0;
        double r79143 = r79142 * r79138;
        double r79144 = r79141 - r79143;
        double r79145 = r79140 + r79144;
        return r79145;
}
↓
double f(double t) {
        double r79146 = t;
        double r79147 = 3.9999999999999997e-32;
        double r79148 = sqrt(r79147);
        double r79149 = r79146 * r79148;
        double r79150 = r79149 * r79146;
        double r79151 = r79150 * r79148;
        return r79151;
}

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. Simplified61.8

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

  3. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}}\]

  4. Simplified0.3

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

  5. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}}\]

  6. Simplified0.3

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

  8. Applied add-sqr-sqrt0.3

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

  9. Applied associate-*r*0.4

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 
(FPCore (t)
  :name "fma_test1"
  :pre (<= 0.9 t 1.1)

  :herbie-target
  (fma (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16)) (- -1.0 (* 2.0 (* t 2e-16))))

  (+ (* (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16))) (- -1.0 (* 2.0 (* t 2e-16)))))