Average Error: 7.3 → 2.0
Time: 14.5s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\ \;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, t, -x\right)}, z, x\right)}} - \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{t \cdot z - x}\right)\right)}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 2.600138962860817447856891281313435992358 \cdot 10^{205}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\
\;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, t, -x\right)}, z, x\right)}} - \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{t \cdot z - x}\right)\right)}{x + 1}\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 2.600138962860817447856891281313435992358 \cdot 10^{205}:\\
\;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r594061 = x;
        double r594062 = y;
        double r594063 = z;
        double r594064 = r594062 * r594063;
        double r594065 = r594064 - r594061;
        double r594066 = t;
        double r594067 = r594066 * r594063;
        double r594068 = r594067 - r594061;
        double r594069 = r594065 / r594068;
        double r594070 = r594061 + r594069;
        double r594071 = 1.0;
        double r594072 = r594061 + r594071;
        double r594073 = r594070 / r594072;
        return r594073;
}
↓
double f(double x, double y, double z, double t) {
        double r594074 = x;
        double r594075 = y;
        double r594076 = z;
        double r594077 = r594075 * r594076;
        double r594078 = r594077 - r594074;
        double r594079 = t;
        double r594080 = r594079 * r594076;
        double r594081 = r594080 - r594074;
        double r594082 = r594078 / r594081;
        double r594083 = r594074 + r594082;
        double r594084 = 1.0;
        double r594085 = r594074 + r594084;
        double r594086 = r594083 / r594085;
        double r594087 = -inf.0;
        bool r594088 = r594086 <= r594087;
        double r594089 = 1.0;
        double r594090 = -r594074;
        double r594091 = fma(r594076, r594079, r594090);
        double r594092 = r594075 / r594091;
        double r594093 = fma(r594092, r594076, r594074);
        double r594094 = r594085 / r594093;
        double r594095 = r594089 / r594094;
        double r594096 = r594074 / r594081;
        double r594097 = expm1(r594096);
        double r594098 = log1p(r594097);
        double r594099 = r594098 / r594085;
        double r594100 = r594095 - r594099;
        double r594101 = 2.6001389628608174e+205;
        bool r594102 = r594086 <= r594101;
        double r594103 = r594075 / r594079;
        double r594104 = r594074 + r594103;
        double r594105 = r594104 / r594085;
        double r594106 = r594102 ? r594086 : r594105;
        double r594107 = r594088 ? r594100 : r594106;
        return r594107;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.3
Target0.3
Herbie2.0
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < -inf.0

    1. Initial program 64.0

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
    2. Using strategy rm

    3. Applied div-sub64.0

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

    4. Applied associate-+r-64.0

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

    5. Applied div-sub64.0

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

    6. Simplified4.7

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

    8. Applied clear-num4.7

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

    10. Applied log1p-expm1-u4.7

      \[\leadsto \frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, t, -x\right)}, z, x\right)}} - \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{t \cdot z - x}\right)\right)}}{x + 1}\]

    if -inf.0 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 2.6001389628608174e+205

    1. Initial program 0.7

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

    if 2.6001389628608174e+205 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

    1. Initial program 52.6

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

    2. Taylor expanded around inf 13.7

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\ \;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, t, -x\right)}, z, x\right)}} - \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{t \cdot z - x}\right)\right)}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 2.600138962860817447856891281313435992358 \cdot 10^{205}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

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

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