\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]

↓

\[\begin{array}{l} \mathbf{if}\;wj \le 7.79759099112930591 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj - x \cdot 2, x\right) + \left(wj \cdot x\right) \cdot \left(\left(-2\right) + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, \sqrt{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, -\frac{wj}{wj + 1}\right)\\ \end{array}\]

wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}

↓

\begin{array}{l}
\mathbf{if}\;wj \le 7.79759099112930591 \cdot 10^{-8}:\\
\;\;\;\;\mathsf{fma}\left(wj, wj - x \cdot 2, x\right) + \left(wj \cdot x\right) \cdot \left(\left(-2\right) + 2\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, \sqrt{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, -\frac{wj}{wj + 1}\right)\\

\end{array}

double f(double wj, double x) {
        double r294087 = wj;
        double r294088 = exp(r294087);
        double r294089 = r294087 * r294088;
        double r294090 = x;
        double r294091 = r294089 - r294090;
        double r294092 = r294088 + r294089;
        double r294093 = r294091 / r294092;
        double r294094 = r294087 - r294093;
        return r294094;
}

↓

double f(double wj, double x) {
        double r294095 = wj;
        double r294096 = 7.797590991129306e-08;
        bool r294097 = r294095 <= r294096;
        double r294098 = x;
        double r294099 = 2.0;
        double r294100 = r294098 * r294099;
        double r294101 = r294095 - r294100;
        double r294102 = fma(r294095, r294101, r294098);
        double r294103 = r294095 * r294098;
        double r294104 = -r294099;
        double r294105 = r294104 + r294099;
        double r294106 = r294103 * r294105;
        double r294107 = r294102 + r294106;
        double r294108 = 1.0;
        double r294109 = r294095 + r294108;
        double r294110 = r294098 / r294109;
        double r294111 = exp(r294095);
        double r294112 = r294110 / r294111;
        double r294113 = r294112 + r294095;
        double r294114 = sqrt(r294113);
        double r294115 = r294095 / r294109;
        double r294116 = -r294115;
        double r294117 = fma(r294114, r294114, r294116);
        double r294118 = r294097 ? r294107 : r294117;
        return r294118;
}

Original	13.4
Target	12.7
Herbie	1.2

Derivation

Split input into 2 regimes
if wj < 7.797590991129306e-08
1. Initial program 13.0
  \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
2. Simplified13.0
  \[\leadsto \color{blue}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1}}\]
3. Taylor expanded around 0 0.8
  \[\leadsto \color{blue}{\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)}\]
4. Using strategy rm
5. Applied add-sqr-sqrt28.9
  \[\leadsto \color{blue}{\sqrt{x + {wj}^{2}} \cdot \sqrt{x + {wj}^{2}}} - 2 \cdot \left(wj \cdot x\right)\]
6. Applied prod-diff28.9
  \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x + {wj}^{2}}, \sqrt{x + {wj}^{2}}, -\left(wj \cdot x\right) \cdot 2\right) + \mathsf{fma}\left(-wj \cdot x, 2, \left(wj \cdot x\right) \cdot 2\right)}\]
7. Simplified0.9
  \[\leadsto \color{blue}{\mathsf{fma}\left(wj, wj - x \cdot 2, x\right)} + \mathsf{fma}\left(-wj \cdot x, 2, \left(wj \cdot x\right) \cdot 2\right)\]
8. Simplified0.9
  \[\leadsto \mathsf{fma}\left(wj, wj - x \cdot 2, x\right) + \color{blue}{\left(wj \cdot x\right) \cdot \left(\left(-2\right) + 2\right)}\]
if 7.797590991129306e-08 < wj
1. Initial program 26.1
  \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
2. Simplified2.7
  \[\leadsto \color{blue}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt11.0
  \[\leadsto \color{blue}{\sqrt{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj} \cdot \sqrt{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}} - \frac{wj}{wj + 1}\]
5. Applied fma-neg10.9
  \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, \sqrt{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, -\frac{wj}{wj + 1}\right)}\]
Recombined 2 regimes into one program.
Final simplification1.2
\[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 7.79759099112930591 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj - x \cdot 2, x\right) + \left(wj \cdot x\right) \cdot \left(\left(-2\right) + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, \sqrt{\frac{\frac{x}{wj + 1}}{e^{wj}} + wj}, -\frac{wj}{wj + 1}\right)\\ \end{array}\]

could not determine a ground truth for program body (more)

Error

Target

Derivation

`if wj < 7.797590991129306e-08`

`if 7.797590991129306e-08 < wj`

Reproduce