Average Error: 3.5 → 3.5
Time: 37.3s
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 r39725026 = x;
        double r39725027 = 1.0;
        double r39725028 = y;
        double r39725029 = r39725027 - r39725028;
        double r39725030 = z;
        double r39725031 = r39725029 * r39725030;
        double r39725032 = r39725027 - r39725031;
        double r39725033 = r39725026 * r39725032;
        return r39725033;
}
↓
double f(double x, double y, double z) {
        double r39725034 = z;
        double r39725035 = y;
        double r39725036 = 1.0;
        double r39725037 = r39725035 - r39725036;
        double r39725038 = fma(r39725034, r39725037, r39725036);
        double r39725039 = x;
        double r39725040 = r39725038 * r39725039;
        return r39725040;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.5
Target0.2
Herbie3.5
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt -1.618195973607048970493874632750554853795 \cdot 10^{50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt 3.892237649663902900973248011051357504727 \cdot 10^{134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

  1. Initial program 3.5

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

  2. Simplified3.5

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

  3. Final simplification3.5

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"

  :herbie-target
  (if (< (* x (- 1.0 (* (- 1.0 y) z))) -1.618195973607049e+50) (+ x (* (- 1.0 y) (* (- z) x))) (if (< (* x (- 1.0 (* (- 1.0 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1.0 y) (* (- z) x)))))

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