(- (/ x0 (- 1 x1)) x0)

Average Error: 7.9 → 6.1

Time: 10.5s

Precision: 64

• could not determine a ground truth for program pre

  1. x0 = 1.855
  2. x1 = 0.0186

\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]

Math

\[\frac{x0}{1 - x1} - x0\]

↓

\[\begin{array}{l} \mathbf{if}\;x1 \le 0.01820459765624999823319107861152588156983:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\\ \end{array}\]

C

double f(double x0, double x1) {
        double r5742643 = x0;
        double r5742644 = 1.0;
        double r5742645 = x1;
        double r5742646 = r5742644 - r5742645;
        double r5742647 = r5742643 / r5742646;
        double r5742648 = r5742647 - r5742643;
        return r5742648;
}

↓

double f(double x0, double x1) {
        double r5742649 = x1;
        double r5742650 = 0.018204597656249998;
        bool r5742651 = r5742649 <= r5742650;
        double r5742652 = x0;
        double r5742653 = cbrt(r5742652);
        double r5742654 = r5742653 * r5742653;
        double r5742655 = 1.0;
        double r5742656 = r5742655 - r5742649;
        double r5742657 = r5742653 / r5742656;
        double r5742658 = -r5742652;
        double r5742659 = fma(r5742654, r5742657, r5742658);
        double r5742660 = sqrt(r5742652);
        double r5742661 = sqrt(r5742655);
        double r5742662 = sqrt(r5742649);
        double r5742663 = r5742661 + r5742662;
        double r5742664 = r5742660 / r5742663;
        double r5742665 = r5742661 - r5742662;
        double r5742666 = r5742660 / r5742665;
        double r5742667 = fma(r5742664, r5742666, r5742658);
        double r5742668 = r5742651 ? r5742659 : r5742667;
        return r5742668;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.3
Herbie	6.1

\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

1. Split input into 2 regimes

2. if x1 < 0.018204597656249998

   1. Initial program 11.2

      \[\frac{x0}{1 - x1} - x0\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 11.2

      \[\leadsto \frac{x0}{\color{blue}{1 \cdot \left(1 - x1\right)}} - x0\]

   4. Applied add-cube-cbrt 11.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}\right) \cdot \sqrt[3]{x0}}}{1 \cdot \left(1 - x1\right)} - x0\]

   5. Applied times-frac 10.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{1} \cdot \frac{\sqrt[3]{x0}}{1 - x1}} - x0\]

   6. Applied fma-neg 8.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{1}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)}\]

   if 0.018204597656249998 < x1

   1. Initial program 4.6

      \[\frac{x0}{1 - x1} - x0\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 4.6

      \[\leadsto \frac{x0}{1 - \color{blue}{\sqrt{x1} \cdot \sqrt{x1}}} - x0\]

   4. Applied add-sqr-sqrt 4.6

      \[\leadsto \frac{x0}{\color{blue}{\sqrt{1} \cdot \sqrt{1}} - \sqrt{x1} \cdot \sqrt{x1}} - x0\]

   5. Applied difference-of-squares 4.6

      \[\leadsto \frac{x0}{\color{blue}{\left(\sqrt{1} + \sqrt{x1}\right) \cdot \left(\sqrt{1} - \sqrt{x1}\right)}} - x0\]

   6. Applied add-sqr-sqrt 4.6

      \[\leadsto \frac{\color{blue}{\sqrt{x0} \cdot \sqrt{x0}}}{\left(\sqrt{1} + \sqrt{x1}\right) \cdot \left(\sqrt{1} - \sqrt{x1}\right)} - x0\]

   7. Applied times-frac 5.2

      \[\leadsto \color{blue}{\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}} \cdot \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}} - x0\]

   8. Applied fma-neg 3.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 0.01820459765624999823319107861152588156983:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :pre (or (and (== x0 1.855) (== x1 0.000209)) (and (== x0 2.985) (== x1 0.0186)))

  :herbie-target
  (/ (* x0 x1) (- 1.0 x1))

  (- (/ x0 (- 1.0 x1)) x0))