Average Error: 10.3 → 1.3
Time: 20.7s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r25180807 = x;
        double r25180808 = y;
        double r25180809 = z;
        double r25180810 = t;
        double r25180811 = r25180809 - r25180810;
        double r25180812 = r25180808 * r25180811;
        double r25180813 = a;
        double r25180814 = r25180809 - r25180813;
        double r25180815 = r25180812 / r25180814;
        double r25180816 = r25180807 + r25180815;
        return r25180816;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r25180817 = z;
        double r25180818 = t;
        double r25180819 = r25180817 - r25180818;
        double r25180820 = a;
        double r25180821 = r25180817 - r25180820;
        double r25180822 = r25180819 / r25180821;
        double r25180823 = y;
        double r25180824 = x;
        double r25180825 = fma(r25180822, r25180823, r25180824);
        return r25180825;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.3
Target1.2
Herbie1.3
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.3

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

  2. Simplified3.1

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

  4. Applied clear-num3.3

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

  6. Applied fma-udef3.3

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

  7. Simplified3.1

    \[\leadsto \color{blue}{\frac{z - t}{\frac{z - a}{y}}} + x\]
  8. Using strategy rm

  9. Applied associate-/r/1.3

    \[\leadsto \color{blue}{\frac{z - t}{z - a} \cdot y} + x\]

  10. Applied fma-def1.3

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

  11. Final simplification1.3

    \[\leadsto \mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

  (+ x (/ (* y (- z t)) (- z a))))