Average Error: 28.6 → 0.2
Time: 26.2s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\left(\left(z - x\right) \cdot \left(\frac{1}{y} \cdot \left(z + x\right)\right)\right) \cdot \frac{x - z}{z - x} + y}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\left(\left(z - x\right) \cdot \left(\frac{1}{y} \cdot \left(z + x\right)\right)\right) \cdot \frac{x - z}{z - x} + y}{2}
double f(double x, double y, double z) {
        double r34481720 = x;
        double r34481721 = r34481720 * r34481720;
        double r34481722 = y;
        double r34481723 = r34481722 * r34481722;
        double r34481724 = r34481721 + r34481723;
        double r34481725 = z;
        double r34481726 = r34481725 * r34481725;
        double r34481727 = r34481724 - r34481726;
        double r34481728 = 2.0;
        double r34481729 = r34481722 * r34481728;
        double r34481730 = r34481727 / r34481729;
        return r34481730;
}
↓
double f(double x, double y, double z) {
        double r34481731 = z;
        double r34481732 = x;
        double r34481733 = r34481731 - r34481732;
        double r34481734 = 1.0;
        double r34481735 = y;
        double r34481736 = r34481734 / r34481735;
        double r34481737 = r34481731 + r34481732;
        double r34481738 = r34481736 * r34481737;
        double r34481739 = r34481733 * r34481738;
        double r34481740 = r34481732 - r34481731;
        double r34481741 = r34481740 / r34481733;
        double r34481742 = r34481739 * r34481741;
        double r34481743 = r34481742 + r34481735;
        double r34481744 = 2.0;
        double r34481745 = r34481743 / r34481744;
        return r34481745;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.6
Target0.2
Herbie0.2
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 28.6

    \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{\frac{x - z}{\frac{y}{z + x}} + y}{2}}\]
  3. Using strategy rm

  4. Applied flip-+12.5

    \[\leadsto \frac{\frac{x - z}{\frac{y}{\color{blue}{\frac{z \cdot z - x \cdot x}{z - x}}}} + y}{2}\]

  5. Applied associate-/r/12.5

    \[\leadsto \frac{\frac{x - z}{\color{blue}{\frac{y}{z \cdot z - x \cdot x} \cdot \left(z - x\right)}} + y}{2}\]

  6. Applied *-un-lft-identity12.5

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(x - z\right)}}{\frac{y}{z \cdot z - x \cdot x} \cdot \left(z - x\right)} + y}{2}\]

  7. Applied times-frac12.5

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{y}{z \cdot z - x \cdot x}} \cdot \frac{x - z}{z - x}} + y}{2}\]

  8. Simplified0.2

    \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\right)} \cdot \frac{x - z}{z - x} + y}{2}\]

  9. Final simplification0.2

    \[\leadsto \frac{\left(\left(z - x\right) \cdot \left(\frac{1}{y} \cdot \left(z + x\right)\right)\right) \cdot \frac{x - z}{z - x} + y}{2}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))