Average Error: 61.8 → 0.3
Time: 13.6s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\right)\]
\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)
↓
\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\right)
double f(double t) {
        double r76764 = 1.0;
        double r76765 = t;
        double r76766 = 2e-16;
        double r76767 = r76765 * r76766;
        double r76768 = r76764 + r76767;
        double r76769 = r76768 * r76768;
        double r76770 = -1.0;
        double r76771 = 2.0;
        double r76772 = r76771 * r76767;
        double r76773 = r76770 - r76772;
        double r76774 = r76769 + r76773;
        return r76774;
}
↓
double f(double t) {
        double r76775 = 3.9999999999999997e-32;
        double r76776 = sqrt(r76775);
        double r76777 = t;
        double r76778 = fabs(r76777);
        double r76779 = r76776 * r76778;
        double r76780 = 2.0;
        double r76781 = pow(r76777, r76780);
        double r76782 = sqrt(r76781);
        double r76783 = r76779 * r76782;
        double r76784 = r76776 * r76783;
        return r76784;
}

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

Derivation

  1. Initial program 61.8

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.3

    \[\leadsto \color{blue}{\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \sqrt{3.9999999999999997 \cdot 10^{-32}}\right)} \cdot {t}^{2}\]

  5. Applied associate-*l*0.4

    \[\leadsto \color{blue}{\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot {t}^{2}\right)}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.4

    \[\leadsto \sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \color{blue}{\left(\sqrt{{t}^{2}} \cdot \sqrt{{t}^{2}}\right)}\right)\]

  8. Applied associate-*r*0.3

    \[\leadsto \sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \color{blue}{\left(\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \sqrt{{t}^{2}}\right) \cdot \sqrt{{t}^{2}}\right)}\]

  9. Simplified0.3

    \[\leadsto \sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\color{blue}{\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left|t\right|\right)} \cdot \sqrt{{t}^{2}}\right)\]

  10. Final simplification0.3

    \[\leadsto \sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left(\left(\sqrt{3.9999999999999997 \cdot 10^{-32}} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\right)\]

Reproduce

herbie shell --seed 2020047 
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :pre (<= 0.9 t 1.1)

  :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)))))