Average Error: 0.5 → 0.5
Time: 11.5s
Precision: 64
\[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]
↓
\[d1 \cdot d2 + \left(d4 \cdot d1 + \left(-\left(d1 + d3\right)\right) \cdot d1\right)\]
\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)
↓
d1 \cdot d2 + \left(d4 \cdot d1 + \left(-\left(d1 + d3\right)\right) \cdot d1\right)
double f(double d1, double d2, double d3, double d4) {
        double r1949109 = d1;
        double r1949110 = d2;
        double r1949111 = r1949109 * r1949110;
        double r1949112 = d3;
        double r1949113 = r1949109 * r1949112;
        double r1949114 = r1949111 - r1949113;
        double r1949115 = d4;
        double r1949116 = r1949115 * r1949109;
        double r1949117 = r1949114 + r1949116;
        double r1949118 = r1949109 * r1949109;
        double r1949119 = r1949117 - r1949118;
        return r1949119;
}
↓
double f(double d1, double d2, double d3, double d4) {
        double r1949120 = d1;
        double r1949121 = d2;
        double r1949122 = r1949120 * r1949121;
        double r1949123 = d4;
        double r1949124 = r1949123 * r1949120;
        double r1949125 = d3;
        double r1949126 = r1949120 + r1949125;
        double r1949127 = -r1949126;
        double r1949128 = r1949127 * r1949120;
        double r1949129 = r1949124 + r1949128;
        double r1949130 = r1949122 + r1949129;
        return r1949130;
}

Error

Bits error versus d1

Bits error versus d2

Bits error versus d3

Bits error versus d4

Derivation

  1. Initial program 0.5

    \[\left(\frac{\left(\left(d1 \cdot d2\right) - \left(d1 \cdot d3\right)\right)}{\left(d4 \cdot d1\right)}\right) - \left(d1 \cdot d1\right)\]

  2. Simplified0.4

    \[\leadsto \color{blue}{d1 \cdot \left(\frac{\left(d2 - d3\right)}{\left(d4 - d1\right)}\right)}\]
  3. Using strategy rm

  4. Applied sub-neg0.4

    \[\leadsto d1 \cdot \left(\frac{\color{blue}{\left(\frac{d2}{\left(-d3\right)}\right)}}{\left(d4 - d1\right)}\right)\]

  5. Applied associate-+l+0.4

    \[\leadsto d1 \cdot \color{blue}{\left(\frac{d2}{\left(\frac{\left(-d3\right)}{\left(d4 - d1\right)}\right)}\right)}\]
  6. Using strategy rm

  7. Applied distribute-lft-in0.4

    \[\leadsto \color{blue}{\frac{\left(d1 \cdot d2\right)}{\left(d1 \cdot \left(\frac{\left(-d3\right)}{\left(d4 - d1\right)}\right)\right)}}\]

  8. Simplified0.4

    \[\leadsto \frac{\left(d1 \cdot d2\right)}{\color{blue}{\left(d1 \cdot \left(d4 - \left(\frac{d1}{d3}\right)\right)\right)}}\]
  9. Using strategy rm

  10. Applied sub-neg0.4

    \[\leadsto \frac{\left(d1 \cdot d2\right)}{\left(d1 \cdot \color{blue}{\left(\frac{d4}{\left(-\left(\frac{d1}{d3}\right)\right)}\right)}\right)}\]

  11. Applied distribute-rgt-in0.5

    \[\leadsto \frac{\left(d1 \cdot d2\right)}{\color{blue}{\left(\frac{\left(d4 \cdot d1\right)}{\left(\left(-\left(\frac{d1}{d3}\right)\right) \cdot d1\right)}\right)}}\]

  12. Final simplification0.5

    \[\leadsto d1 \cdot d2 + \left(d4 \cdot d1 + \left(-\left(d1 + d3\right)\right) \cdot d1\right)\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  (-.p16 (+.p16 (-.p16 (*.p16 d1 d2) (*.p16 d1 d3)) (*.p16 d4 d1)) (*.p16 d1 d1)))