Average Error: 6.7 → 2.2
Time: 45.5s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\mathsf{fma}\left(1, \mathsf{fma}\left(\frac{z}{t}, y, x\right), \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{-1}{\frac{t}{z}}\right) \cdot \sqrt[3]{x}\right) + \mathsf{fma}\left(\frac{-1}{\frac{t}{z}}, x, \frac{1}{\frac{t}{z}} \cdot x\right)\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\mathsf{fma}\left(1, \mathsf{fma}\left(\frac{z}{t}, y, x\right), \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{-1}{\frac{t}{z}}\right) \cdot \sqrt[3]{x}\right) + \mathsf{fma}\left(\frac{-1}{\frac{t}{z}}, x, \frac{1}{\frac{t}{z}} \cdot x\right)
double f(double x, double y, double z, double t) {
        double r22527739 = x;
        double r22527740 = y;
        double r22527741 = r22527740 - r22527739;
        double r22527742 = z;
        double r22527743 = r22527741 * r22527742;
        double r22527744 = t;
        double r22527745 = r22527743 / r22527744;
        double r22527746 = r22527739 + r22527745;
        return r22527746;
}
↓
double f(double x, double y, double z, double t) {
        double r22527747 = 1.0;
        double r22527748 = z;
        double r22527749 = t;
        double r22527750 = r22527748 / r22527749;
        double r22527751 = y;
        double r22527752 = x;
        double r22527753 = fma(r22527750, r22527751, r22527752);
        double r22527754 = cbrt(r22527752);
        double r22527755 = r22527754 * r22527754;
        double r22527756 = -1.0;
        double r22527757 = r22527749 / r22527748;
        double r22527758 = r22527756 / r22527757;
        double r22527759 = r22527755 * r22527758;
        double r22527760 = r22527759 * r22527754;
        double r22527761 = fma(r22527747, r22527753, r22527760);
        double r22527762 = r22527747 / r22527757;
        double r22527763 = r22527762 * r22527752;
        double r22527764 = fma(r22527758, r22527752, r22527763);
        double r22527765 = r22527761 + r22527764;
        return r22527765;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.7
Target2.0
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Initial program 6.7

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

  2. Simplified6.4

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

  3. Taylor expanded around 0 6.7

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

  4. Simplified2.0

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

  6. Applied div-inv2.0

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

  7. Applied *-un-lft-identity2.0

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

  8. Applied prod-diff2.0

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

  10. Applied add-cube-cbrt2.2

    \[\leadsto \mathsf{fma}\left(1, \mathsf{fma}\left(\frac{z}{t}, y, x\right), -\frac{1}{\frac{t}{z}} \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}\right) + \mathsf{fma}\left(-\frac{1}{\frac{t}{z}}, x, \frac{1}{\frac{t}{z}} \cdot x\right)\]

  11. Applied associate-*r*2.2

    \[\leadsto \mathsf{fma}\left(1, \mathsf{fma}\left(\frac{z}{t}, y, x\right), -\color{blue}{\left(\frac{1}{\frac{t}{z}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}}\right) + \mathsf{fma}\left(-\frac{1}{\frac{t}{z}}, x, \frac{1}{\frac{t}{z}} \cdot x\right)\]

  12. Final simplification2.2

    \[\leadsto \mathsf{fma}\left(1, \mathsf{fma}\left(\frac{z}{t}, y, x\right), \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{-1}{\frac{t}{z}}\right) \cdot \sqrt[3]{x}\right) + \mathsf{fma}\left(\frac{-1}{\frac{t}{z}}, x, \frac{1}{\frac{t}{z}} \cdot x\right)\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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