Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Average Error: 2.2 → 0.2

Time: 21.6s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(a, \frac{z - y}{\left(t - z\right) + 1}, x\right)\]

C

double f(double x, double y, double z, double t, double a) {
        double r25771943 = x;
        double r25771944 = y;
        double r25771945 = z;
        double r25771946 = r25771944 - r25771945;
        double r25771947 = t;
        double r25771948 = r25771947 - r25771945;
        double r25771949 = 1.0;
        double r25771950 = r25771948 + r25771949;
        double r25771951 = a;
        double r25771952 = r25771950 / r25771951;
        double r25771953 = r25771946 / r25771952;
        double r25771954 = r25771943 - r25771953;
        return r25771954;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r25771955 = a;
        double r25771956 = z;
        double r25771957 = y;
        double r25771958 = r25771956 - r25771957;
        double r25771959 = t;
        double r25771960 = r25771959 - r25771956;
        double r25771961 = 1.0;
        double r25771962 = r25771960 + r25771961;
        double r25771963 = r25771958 / r25771962;
        double r25771964 = x;
        double r25771965 = fma(r25771955, r25771963, r25771964);
        return r25771965;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	2.2
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 2.2

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

2. Simplified 0.2

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

3. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(a, \frac{z - y}{\left(t - z\right) + 1}, x\right)\]

Reproduce

herbie shell --seed 2019171 +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))))