Average Error: 0.0 → 0.0
Time: 5.1s
Precision: 64
\[\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\]
↓
\[\sqrt[3]{{\left(\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)}^{3}} - x\]
\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x
↓
\sqrt[3]{{\left(\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)}^{3}} - x
double f(double x) {
        double r58087 = 2.30753;
        double r58088 = x;
        double r58089 = 0.27061;
        double r58090 = r58088 * r58089;
        double r58091 = r58087 + r58090;
        double r58092 = 1.0;
        double r58093 = 0.99229;
        double r58094 = 0.04481;
        double r58095 = r58088 * r58094;
        double r58096 = r58093 + r58095;
        double r58097 = r58088 * r58096;
        double r58098 = r58092 + r58097;
        double r58099 = r58091 / r58098;
        double r58100 = r58099 - r58088;
        return r58100;
}
↓
double f(double x) {
        double r58101 = 0.27061;
        double r58102 = x;
        double r58103 = 2.30753;
        double r58104 = fma(r58101, r58102, r58103);
        double r58105 = 1.0;
        double r58106 = 0.04481;
        double r58107 = 0.99229;
        double r58108 = fma(r58106, r58102, r58107);
        double r58109 = 1.0;
        double r58110 = fma(r58102, r58108, r58109);
        double r58111 = r58105 / r58110;
        double r58112 = r58104 * r58111;
        double r58113 = 3.0;
        double r58114 = pow(r58112, r58113);
        double r58115 = cbrt(r58114);
        double r58116 = r58115 - r58102;
        return r58116;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\]
  2. Using strategy rm

  3. Applied add-cbrt-cube0.0

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

  4. Applied add-cbrt-cube21.7

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

  5. Applied cbrt-undiv21.7

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

  6. Simplified0.0

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

  8. Applied div-inv0.0

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

  9. Final simplification0.0

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

Reproduce

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