Average Error: 61.8 → 0.3
Time: 10.8s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}} \cdot \left(t \cdot \left(t \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right)\right)\]
\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(t \cdot \left(t \cdot \sqrt{3.999999999999999676487027278085939408227 \cdot 10^{-32}}\right)\right)
double f(double t) {
        double r3345378 = 1.0;
        double r3345379 = t;
        double r3345380 = 2e-16;
        double r3345381 = r3345379 * r3345380;
        double r3345382 = r3345378 + r3345381;
        double r3345383 = r3345382 * r3345382;
        double r3345384 = -1.0;
        double r3345385 = 2.0;
        double r3345386 = r3345385 * r3345381;
        double r3345387 = r3345384 - r3345386;
        double r3345388 = r3345383 + r3345387;
        return r3345388;
}
↓
double f(double t) {
        double r3345389 = 3.9999999999999997e-32;
        double r3345390 = sqrt(r3345389);
        double r3345391 = t;
        double r3345392 = r3345391 * r3345390;
        double r3345393 = r3345391 * r3345392;
        double r3345394 = r3345390 * r3345393;
        return r3345394;
}

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

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  6. Applied add-sqr-sqrt0.4

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

  7. Applied associate-*l*0.4

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

  9. Applied associate-*r*0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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)))))