Average Error: 0.0 → 0.0
Time: 16.2s
Precision: 64
\[\frac{2.30753 + x \cdot 0.27061}{1.0 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\]
↓
\[\mathsf{fma}\left(x, -1, x\right) + \left(\frac{\frac{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}} - x\right)\]
\frac{2.30753 + x \cdot 0.27061}{1.0 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x
↓
\mathsf{fma}\left(x, -1, x\right) + \left(\frac{\frac{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}} - x\right)
double f(double x) {
        double r3264300 = 2.30753;
        double r3264301 = x;
        double r3264302 = 0.27061;
        double r3264303 = r3264301 * r3264302;
        double r3264304 = r3264300 + r3264303;
        double r3264305 = 1.0;
        double r3264306 = 0.99229;
        double r3264307 = 0.04481;
        double r3264308 = r3264301 * r3264307;
        double r3264309 = r3264306 + r3264308;
        double r3264310 = r3264301 * r3264309;
        double r3264311 = r3264305 + r3264310;
        double r3264312 = r3264304 / r3264311;
        double r3264313 = r3264312 - r3264301;
        return r3264313;
}
↓
double f(double x) {
        double r3264314 = x;
        double r3264315 = -1.0;
        double r3264316 = fma(r3264314, r3264315, r3264314);
        double r3264317 = 0.27061;
        double r3264318 = 2.30753;
        double r3264319 = fma(r3264314, r3264317, r3264318);
        double r3264320 = 0.04481;
        double r3264321 = 0.99229;
        double r3264322 = fma(r3264320, r3264314, r3264321);
        double r3264323 = 1.0;
        double r3264324 = fma(r3264322, r3264314, r3264323);
        double r3264325 = cbrt(r3264324);
        double r3264326 = r3264319 / r3264325;
        double r3264327 = r3264326 / r3264325;
        double r3264328 = r3264327 / r3264325;
        double r3264329 = r3264328 - r3264314;
        double r3264330 = r3264316 + r3264329;
        return r3264330;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{2.30753 + x \cdot 0.27061}{1.0 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)} - x}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.7

    \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)} - \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\]

  5. Applied add-cube-cbrt0.7

    \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)}}} - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\]

  6. Applied add-sqr-sqrt16.2

    \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(x, 0.27061, 2.30753\right)} \cdot \sqrt{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}}{\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)}} - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\]

  7. Applied times-frac16.2

    \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)}}} - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\]

  8. Applied prod-diff16.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)}}, \frac{\sqrt{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1.0\right)}}, -\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}\]

  9. Simplified0.0

    \[\leadsto \color{blue}{\left(\frac{\frac{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}} - x\right)} + \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\]

  10. Simplified0.0

    \[\leadsto \left(\frac{\frac{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}} - x\right) + \color{blue}{\mathsf{fma}\left(x, -1, x\right)}\]

  11. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, -1, x\right) + \left(\frac{\frac{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}}}{\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1.0\right)}} - x\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))