Average Error: 61.8 → 0.3
Time: 14.0s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\left(-1 + 1 \cdot 1\right) + \frac{\sqrt{3653754093327257}}{\sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}} \cdot \sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}} \cdot \left(\frac{\sqrt{3653754093327257}}{\sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}} \cdot {t}^{2}\right)\]
\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)
↓
\left(-1 + 1 \cdot 1\right) + \frac{\sqrt{3653754093327257}}{\sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}} \cdot \sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}} \cdot \left(\frac{\sqrt{3653754093327257}}{\sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}} \cdot {t}^{2}\right)
double f(double t) {
        double r61818 = 1.0;
        double r61819 = t;
        double r61820 = 2e-16;
        double r61821 = r61819 * r61820;
        double r61822 = r61818 + r61821;
        double r61823 = r61822 * r61822;
        double r61824 = -1.0;
        double r61825 = 2.0;
        double r61826 = r61825 * r61821;
        double r61827 = r61824 - r61826;
        double r61828 = r61823 + r61827;
        return r61828;
}
↓
double f(double t) {
        double r61829 = -1.0;
        double r61830 = 1.0;
        double r61831 = r61830 * r61830;
        double r61832 = r61829 + r61831;
        double r61833 = 3653754093327257.0;
        double r61834 = sqrt(r61833);
        double r61835 = 9.134385233318143e+46;
        double r61836 = cbrt(r61835);
        double r61837 = r61836 * r61836;
        double r61838 = r61834 / r61837;
        double r61839 = r61834 / r61836;
        double r61840 = t;
        double r61841 = 2.0;
        double r61842 = pow(r61840, r61841);
        double r61843 = r61839 * r61842;
        double r61844 = r61838 * r61843;
        double r61845 = r61832 + r61844;
        return r61845;
}

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

    \[\leadsto \color{blue}{\left(-1 + 1 \cdot 1\right) + \left(t \cdot \frac{2028240960365167}{10141204801825835211973625643008}\right) \cdot \left(\left(\left(1 + t \cdot \frac{2028240960365167}{10141204801825835211973625643008}\right) + 1\right) - 2\right)}\]

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

    \[\leadsto \left(-1 + 1 \cdot 1\right) + \color{blue}{\frac{3653754093327257}{9.13438523331814323877303020447676887285 \cdot 10^{46}} \cdot {t}^{2}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.3

    \[\leadsto \left(-1 + 1 \cdot 1\right) + \frac{3653754093327257}{\color{blue}{\left(\sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}} \cdot \sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}\right) \cdot \sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}}} \cdot {t}^{2}\]

  7. Applied add-sqr-sqrt0.3

    \[\leadsto \left(-1 + 1 \cdot 1\right) + \frac{\color{blue}{\sqrt{3653754093327257} \cdot \sqrt{3653754093327257}}}{\left(\sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}} \cdot \sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}\right) \cdot \sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}} \cdot {t}^{2}\]

  8. Applied times-frac0.3

    \[\leadsto \left(-1 + 1 \cdot 1\right) + \color{blue}{\left(\frac{\sqrt{3653754093327257}}{\sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}} \cdot \sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}} \cdot \frac{\sqrt{3653754093327257}}{\sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}}\right)} \cdot {t}^{2}\]

  9. Applied associate-*l*0.3

    \[\leadsto \left(-1 + 1 \cdot 1\right) + \color{blue}{\frac{\sqrt{3653754093327257}}{\sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}} \cdot \sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}} \cdot \left(\frac{\sqrt{3653754093327257}}{\sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}} \cdot {t}^{2}\right)}\]

  10. Final simplification0.3

    \[\leadsto \left(-1 + 1 \cdot 1\right) + \frac{\sqrt{3653754093327257}}{\sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}} \cdot \sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}} \cdot \left(\frac{\sqrt{3653754093327257}}{\sqrt[3]{9.13438523331814323877303020447676887285 \cdot 10^{46}}} \cdot {t}^{2}\right)\]

Reproduce

herbie shell --seed 2019303 
(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)))))