Average Error: 2.0 → 0.3
Time: 15.0s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[\mathsf{fma}\left(a, \frac{1}{\frac{\left(t - z\right) + 1}{z - y}}, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
\mathsf{fma}\left(a, \frac{1}{\frac{\left(t - z\right) + 1}{z - y}}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r352119 = x;
        double r352120 = y;
        double r352121 = z;
        double r352122 = r352120 - r352121;
        double r352123 = t;
        double r352124 = r352123 - r352121;
        double r352125 = 1.0;
        double r352126 = r352124 + r352125;
        double r352127 = a;
        double r352128 = r352126 / r352127;
        double r352129 = r352122 / r352128;
        double r352130 = r352119 - r352129;
        return r352130;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r352131 = a;
        double r352132 = 1.0;
        double r352133 = t;
        double r352134 = z;
        double r352135 = r352133 - r352134;
        double r352136 = 1.0;
        double r352137 = r352135 + r352136;
        double r352138 = y;
        double r352139 = r352134 - r352138;
        double r352140 = r352137 / r352139;
        double r352141 = r352132 / r352140;
        double r352142 = x;
        double r352143 = fma(r352131, r352141, r352142);
        return r352143;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original2.0
Target0.3
Herbie0.3
\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a\]

Derivation

  1. Initial program 2.0

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

  2. Simplified0.3

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

  4. Applied clear-num0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- x (* (/ (- y z) (+ (- t z) 1)) a))

  (- x (/ (- y z) (/ (+ (- t z) 1) a))))