Average Error: 10.9 → 0.4
Time: 3.4s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -6.816496712142369533985882229910807371226 \cdot 10^{110} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 4.167561421650247042871141348207441053161 \cdot 10^{-52}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1 + y, -x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{z}{\left(y - z\right) + 1}\right)\right)}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -6.816496712142369533985882229910807371226 \cdot 10^{110} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 4.167561421650247042871141348207441053161 \cdot 10^{-52}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1 + y, -x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{z}{\left(y - z\right) + 1}\right)\right)}\\

\end{array}
double f(double x, double y, double z) {
        double r694379 = x;
        double r694380 = y;
        double r694381 = z;
        double r694382 = r694380 - r694381;
        double r694383 = 1.0;
        double r694384 = r694382 + r694383;
        double r694385 = r694379 * r694384;
        double r694386 = r694385 / r694381;
        return r694386;
}
↓
double f(double x, double y, double z) {
        double r694387 = x;
        double r694388 = y;
        double r694389 = z;
        double r694390 = r694388 - r694389;
        double r694391 = 1.0;
        double r694392 = r694390 + r694391;
        double r694393 = r694387 * r694392;
        double r694394 = r694393 / r694389;
        double r694395 = -6.81649671214237e+110;
        bool r694396 = r694394 <= r694395;
        double r694397 = 4.167561421650247e-52;
        bool r694398 = r694394 <= r694397;
        double r694399 = !r694398;
        bool r694400 = r694396 || r694399;
        double r694401 = r694387 / r694389;
        double r694402 = r694391 + r694388;
        double r694403 = -r694387;
        double r694404 = fma(r694401, r694402, r694403);
        double r694405 = r694389 / r694392;
        double r694406 = expm1(r694405);
        double r694407 = log1p(r694406);
        double r694408 = r694387 / r694407;
        double r694409 = r694400 ? r694404 : r694408;
        return r694409;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.9
Target0.5
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* x (+ (- y z) 1.0)) z) < -6.81649671214237e+110 or 4.167561421650247e-52 < (/ (* x (+ (- y z) 1.0)) z)

    1. Initial program 19.3

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

    3. Applied associate-/l*4.9

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]
    4. Using strategy rm

    5. Applied log1p-expm1-u4.9

      \[\leadsto \frac{x}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{z}{\left(y - z\right) + 1}\right)\right)}}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt5.6

      \[\leadsto \frac{x}{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\sqrt[3]{\frac{z}{\left(y - z\right) + 1}} \cdot \sqrt[3]{\frac{z}{\left(y - z\right) + 1}}\right) \cdot \sqrt[3]{\frac{z}{\left(y - z\right) + 1}}}\right)\right)}\]

    8. Taylor expanded around 0 6.3

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

    9. Simplified0.2

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

    if -6.81649671214237e+110 < (/ (* x (+ (- y z) 1.0)) z) < 4.167561421650247e-52

    1. Initial program 0.2

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

    3. Applied associate-/l*0.6

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]
    4. Using strategy rm

    5. Applied log1p-expm1-u0.6

      \[\leadsto \frac{x}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{z}{\left(y - z\right) + 1}\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -6.816496712142369533985882229910807371226 \cdot 10^{110} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 4.167561421650247042871141348207441053161 \cdot 10^{-52}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1 + y, -x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{z}{\left(y - z\right) + 1}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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 -2.71483106713436e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

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