Average Error: 2.0 → 0.2
Time: 21.2s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1.0}{a}}\]
↓
\[\mathsf{fma}\left(\frac{z - y}{\left(1.0 + t\right) - z}, a, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1.0}{a}}
↓
\mathsf{fma}\left(\frac{z - y}{\left(1.0 + t\right) - z}, a, x\right)
double f(double x, double y, double z, double t, double a) {
        double r26015019 = x;
        double r26015020 = y;
        double r26015021 = z;
        double r26015022 = r26015020 - r26015021;
        double r26015023 = t;
        double r26015024 = r26015023 - r26015021;
        double r26015025 = 1.0;
        double r26015026 = r26015024 + r26015025;
        double r26015027 = a;
        double r26015028 = r26015026 / r26015027;
        double r26015029 = r26015022 / r26015028;
        double r26015030 = r26015019 - r26015029;
        return r26015030;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r26015031 = z;
        double r26015032 = y;
        double r26015033 = r26015031 - r26015032;
        double r26015034 = 1.0;
        double r26015035 = t;
        double r26015036 = r26015034 + r26015035;
        double r26015037 = r26015036 - r26015031;
        double r26015038 = r26015033 / r26015037;
        double r26015039 = a;
        double r26015040 = x;
        double r26015041 = fma(r26015038, r26015039, r26015040);
        return r26015041;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original2.0
Target0.2
Herbie0.2
\[x - \frac{y - z}{\left(t - z\right) + 1.0} \cdot a\]

Derivation

  1. Initial program 2.0

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{\left(1.0 + t\right) - z}, a, x\right)}\]

  3. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\frac{z - y}{\left(1.0 + t\right) - z}, a, x\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"

  :herbie-target
  (- x (* (/ (- y z) (+ (- t z) 1.0)) a))

  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))