Average Error: 0.0 → 0.0
Time: 17.3s
Precision: 64
\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
↓
\[\left(\sqrt[3]{{\left(\frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right)}\right)}^{3}} - x\right) \cdot 0.707110000000000016\]
0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)
↓
\left(\sqrt[3]{{\left(\frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right)}\right)}^{3}} - x\right) \cdot 0.707110000000000016
double f(double x) {
        double r117560 = 0.70711;
        double r117561 = 2.30753;
        double r117562 = x;
        double r117563 = 0.27061;
        double r117564 = r117562 * r117563;
        double r117565 = r117561 + r117564;
        double r117566 = 1.0;
        double r117567 = 0.99229;
        double r117568 = 0.04481;
        double r117569 = r117562 * r117568;
        double r117570 = r117567 + r117569;
        double r117571 = r117562 * r117570;
        double r117572 = r117566 + r117571;
        double r117573 = r117565 / r117572;
        double r117574 = r117573 - r117562;
        double r117575 = r117560 * r117574;
        return r117575;
}
↓
double f(double x) {
        double r117576 = 0.27061;
        double r117577 = x;
        double r117578 = 2.30753;
        double r117579 = fma(r117576, r117577, r117578);
        double r117580 = 0.04481;
        double r117581 = 0.99229;
        double r117582 = fma(r117580, r117577, r117581);
        double r117583 = 1.0;
        double r117584 = fma(r117582, r117577, r117583);
        double r117585 = r117579 / r117584;
        double r117586 = 3.0;
        double r117587 = pow(r117585, r117586);
        double r117588 = cbrt(r117587);
        double r117589 = r117588 - r117577;
        double r117590 = 0.70711;
        double r117591 = r117589 * r117590;
        return r117591;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right)} - x\right) \cdot 0.707110000000000016}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube0.0

    \[\leadsto \left(\frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right)}}} - x\right) \cdot 0.707110000000000016\]

  5. Applied add-cbrt-cube21.7

    \[\leadsto \left(\frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right) \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)\right) \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}{\sqrt[3]{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right)}} - x\right) \cdot 0.707110000000000016\]

  6. Applied cbrt-undiv21.7

    \[\leadsto \left(\color{blue}{\sqrt[3]{\frac{\left(\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right) \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)\right) \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right)}}} - x\right) \cdot 0.707110000000000016\]

  7. Simplified0.0

    \[\leadsto \left(\sqrt[3]{\color{blue}{{\left(\frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right)}\right)}^{3}}} - x\right) \cdot 0.707110000000000016\]

  8. Final simplification0.0

    \[\leadsto \left(\sqrt[3]{{\left(\frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), x, 1\right)}\right)}^{3}} - x\right) \cdot 0.707110000000000016\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* x (+ 0.99229 (* x 0.04481))))) x)))