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

Average Error: 7.8 → 6.9

Time: 7.8s

Precision: 64

• could not determine a ground truth for program pre

  1. x0 = 1.855
  2. x1 = 0.0186

\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]

Math

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

↓

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

C

double f(double x0, double x1) {
        double r6244948 = x0;
        double r6244949 = 1.0;
        double r6244950 = x1;
        double r6244951 = r6244949 - r6244950;
        double r6244952 = r6244948 / r6244951;
        double r6244953 = r6244952 - r6244948;
        return r6244953;
}

↓

double f(double x0, double x1) {
        double r6244954 = x0;
        double r6244955 = cbrt(r6244954);
        double r6244956 = r6244955 * r6244955;
        double r6244957 = 1.0;
        double r6244958 = x1;
        double r6244959 = r6244957 - r6244958;
        double r6244960 = r6244955 / r6244959;
        double r6244961 = -r6244954;
        double r6244962 = fma(r6244956, r6244960, r6244961);
        return r6244962;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.8
Target	0.3
Herbie	6.9

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

Derivation

1. Initial program 7.8

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

3. Applied *-un-lft-identity 7.8

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

4. Applied *-un-lft-identity 7.8

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

5. Applied distribute-lft-out-- 7.8

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

6. Applied add-cube-cbrt 7.8

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

7. Applied times-frac 8.2

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

8. Applied fma-neg 6.9

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

9. Final simplification 6.9

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

Reproduce

herbie shell --seed 2019133 +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 x1))

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