Average Error: 3.4 → 3.4
Time: 24.2s
Precision: 64
\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\]
↓
\[\mathsf{fma}\left(z, y - 1, 1\right) \cdot x\]
x \cdot \left(1 - \left(1 - y\right) \cdot z\right)
↓
\mathsf{fma}\left(z, y - 1, 1\right) \cdot x
double f(double x, double y, double z) {
        double r496570 = x;
        double r496571 = 1.0;
        double r496572 = y;
        double r496573 = r496571 - r496572;
        double r496574 = z;
        double r496575 = r496573 * r496574;
        double r496576 = r496571 - r496575;
        double r496577 = r496570 * r496576;
        return r496577;
}
↓
double f(double x, double y, double z) {
        double r496578 = z;
        double r496579 = y;
        double r496580 = 1.0;
        double r496581 = r496579 - r496580;
        double r496582 = fma(r496578, r496581, r496580);
        double r496583 = x;
        double r496584 = r496582 * r496583;
        return r496584;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.4
Target0.2
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt -1.618195973607048970493874632750554853795 \cdot 10^{50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt 3.892237649663902900973248011051357504727 \cdot 10^{134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

  1. Initial program 3.4

    \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\]

  2. Simplified3.4

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

  3. Final simplification3.4

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :herbie-target
  (if (< (* x (- 1 (* (- 1 y) z))) -1.618195973607049e+50) (+ x (* (- 1 y) (* (- z) x))) (if (< (* x (- 1 (* (- 1 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1 y) (* (- z) x)))))

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