\[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(y - z\right) + 1.0\right) \cdot x}{z} \le -1.7501339300690804 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, \frac{x}{z} \cdot y\right) - x\\ \mathbf{elif}\;\frac{\left(\left(y - z\right) + 1.0\right) \cdot x}{z} \le 2.4940001452283625 \cdot 10^{-171}:\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1.0}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, \frac{x}{z} \cdot y\right) - x\\ \end{array}\]

\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}

↓

\begin{array}{l}
\mathbf{if}\;\frac{\left(\left(y - z\right) + 1.0\right) \cdot x}{z} \le -1.7501339300690804 \cdot 10^{-115}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, \frac{x}{z} \cdot y\right) - x\\

\mathbf{elif}\;\frac{\left(\left(y - z\right) + 1.0\right) \cdot x}{z} \le 2.4940001452283625 \cdot 10^{-171}:\\
\;\;\;\;x \cdot \frac{\left(y - z\right) + 1.0}{z}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, \frac{x}{z} \cdot y\right) - x\\

\end{array}

double f(double x, double y, double z) {
        double r31637488 = x;
        double r31637489 = y;
        double r31637490 = z;
        double r31637491 = r31637489 - r31637490;
        double r31637492 = 1.0;
        double r31637493 = r31637491 + r31637492;
        double r31637494 = r31637488 * r31637493;
        double r31637495 = r31637494 / r31637490;
        return r31637495;
}

↓

double f(double x, double y, double z) {
        double r31637496 = y;
        double r31637497 = z;
        double r31637498 = r31637496 - r31637497;
        double r31637499 = 1.0;
        double r31637500 = r31637498 + r31637499;
        double r31637501 = x;
        double r31637502 = r31637500 * r31637501;
        double r31637503 = r31637502 / r31637497;
        double r31637504 = -1.7501339300690804e-115;
        bool r31637505 = r31637503 <= r31637504;
        double r31637506 = r31637501 / r31637497;
        double r31637507 = r31637506 * r31637496;
        double r31637508 = fma(r31637506, r31637499, r31637507);
        double r31637509 = r31637508 - r31637501;
        double r31637510 = 2.4940001452283625e-171;
        bool r31637511 = r31637503 <= r31637510;
        double r31637512 = r31637500 / r31637497;
        double r31637513 = r31637501 * r31637512;
        double r31637514 = r31637511 ? r31637513 : r31637509;
        double r31637515 = r31637505 ? r31637509 : r31637514;
        return r31637515;
}

Original	9.2
Target	0.4
Herbie	0.7

Derivation

Split input into 2 regimes
if (/ (* x (+ (- y z) 1.0)) z) < -1.7501339300690804e-115 or 2.4940001452283625e-171 < (/ (* x (+ (- y z) 1.0)) z)
1. Initial program 11.1
  \[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]
2. Taylor expanded around 0 3.6
  \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z} + 1.0 \cdot \frac{x}{z}\right) - x}\]
3. Simplified0.8
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, 1.0, y \cdot \frac{x}{z}\right) - x}\]
if -1.7501339300690804e-115 < (/ (* x (+ (- y z) 1.0)) z) < 2.4940001452283625e-171
1. Initial program 0.1
  \[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]
2. Using strategy rm
3. Applied *-un-lft-identity0.1
  \[\leadsto \frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{\color{blue}{1 \cdot z}}\]
4. Applied times-frac0.1
  \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{\left(y - z\right) + 1.0}{z}}\]
5. Simplified0.1
  \[\leadsto \color{blue}{x} \cdot \frac{\left(y - z\right) + 1.0}{z}\]
Recombined 2 regimes into one program.
Final simplification0.7
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(y - z\right) + 1.0\right) \cdot x}{z} \le -1.7501339300690804 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, \frac{x}{z} \cdot y\right) - x\\ \mathbf{elif}\;\frac{\left(\left(y - z\right) + 1.0\right) \cdot x}{z} \le 2.4940001452283625 \cdot 10^{-171}:\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1.0}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, \frac{x}{z} \cdot y\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))

Error

Target

Derivation

`if (/ (* x (+ (- y z) 1.0)) z) < -1.7501339300690804e-115 or 2.4940001452283625e-171 < (/ (* x (+ (- y z) 1.0)) z)`

`if -1.7501339300690804e-115 < (/ (* x (+ (- y z) 1.0)) z) < 2.4940001452283625e-171`

Reproduce