Average Error: 9.9 → 0.2
Time: 8.7s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -20890976261.996433258056640625 \lor \neg \left(x \le 4.003006293405846964686451842961573369429 \cdot 10^{-55}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, x \cdot \frac{y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -20890976261.996433258056640625 \lor \neg \left(x \le 4.003006293405846964686451842961573369429 \cdot 10^{-55}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, x \cdot \frac{y}{z}\right) - x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{x \cdot y}{z}\right) - x\\

\end{array}
double f(double x, double y, double z) {
        double r484000 = x;
        double r484001 = y;
        double r484002 = z;
        double r484003 = r484001 - r484002;
        double r484004 = 1.0;
        double r484005 = r484003 + r484004;
        double r484006 = r484000 * r484005;
        double r484007 = r484006 / r484002;
        return r484007;
}
↓
double f(double x, double y, double z) {
        double r484008 = x;
        double r484009 = -20890976261.996433;
        bool r484010 = r484008 <= r484009;
        double r484011 = 4.003006293405847e-55;
        bool r484012 = r484008 <= r484011;
        double r484013 = !r484012;
        bool r484014 = r484010 || r484013;
        double r484015 = z;
        double r484016 = r484008 / r484015;
        double r484017 = 1.0;
        double r484018 = y;
        double r484019 = r484018 / r484015;
        double r484020 = r484008 * r484019;
        double r484021 = fma(r484016, r484017, r484020);
        double r484022 = r484021 - r484008;
        double r484023 = r484008 * r484018;
        double r484024 = r484023 / r484015;
        double r484025 = fma(r484016, r484017, r484024);
        double r484026 = r484025 - r484008;
        double r484027 = r484014 ? r484022 : r484026;
        return r484027;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original9.9
Target0.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -20890976261.996433 or 4.003006293405847e-55 < x

    1. Initial program 22.4

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*0.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]

    4. Taylor expanded around 0 7.3

      \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x}\]

    5. Simplified7.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, 1, \frac{x \cdot y}{z}\right) - x}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity7.3

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 1, \frac{x \cdot y}{\color{blue}{1 \cdot z}}\right) - x\]

    8. Applied times-frac0.3

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 1, \color{blue}{\frac{x}{1} \cdot \frac{y}{z}}\right) - x\]

    9. Simplified0.3

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 1, \color{blue}{x} \cdot \frac{y}{z}\right) - x\]

    if -20890976261.996433 < x < 4.003006293405847e-55

    1. Initial program 0.1

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*5.6

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]

    4. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x}\]

    5. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, 1, \frac{x \cdot y}{z}\right) - x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -20890976261.996433258056640625 \lor \neg \left(x \le 4.003006293405846964686451842961573369429 \cdot 10^{-55}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, x \cdot \frac{y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.7148310671343599e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.87410881643954616e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1)) z))