Average Error: 24.7 → 9.7
Time: 41.2s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -2.735236830308558818018448627428718325221 \cdot 10^{-193} \lor \neg \left(a \le 1.352585029372079948136707219245541043528 \cdot 10^{-160}\right):\\ \;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;a \le -2.735236830308558818018448627428718325221 \cdot 10^{-193} \lor \neg \left(a \le 1.352585029372079948136707219245541043528 \cdot 10^{-160}\right):\\
\;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\

\mathbf{else}:\\
\;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r602113 = x;
        double r602114 = y;
        double r602115 = z;
        double r602116 = r602114 - r602115;
        double r602117 = t;
        double r602118 = r602117 - r602113;
        double r602119 = r602116 * r602118;
        double r602120 = a;
        double r602121 = r602120 - r602115;
        double r602122 = r602119 / r602121;
        double r602123 = r602113 + r602122;
        return r602123;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r602124 = a;
        double r602125 = -2.735236830308559e-193;
        bool r602126 = r602124 <= r602125;
        double r602127 = 1.35258502937208e-160;
        bool r602128 = r602124 <= r602127;
        double r602129 = !r602128;
        bool r602130 = r602126 || r602129;
        double r602131 = x;
        double r602132 = t;
        double r602133 = r602132 - r602131;
        double r602134 = z;
        double r602135 = r602124 - r602134;
        double r602136 = y;
        double r602137 = r602136 - r602134;
        double r602138 = r602135 / r602137;
        double r602139 = r602133 / r602138;
        double r602140 = r602131 + r602139;
        double r602141 = r602131 / r602134;
        double r602142 = r602132 / r602134;
        double r602143 = r602141 - r602142;
        double r602144 = r602136 * r602143;
        double r602145 = r602144 + r602132;
        double r602146 = r602130 ? r602140 : r602145;
        return r602146;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original24.7
Target11.8
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;z \lt -1.253613105609503593846459977496550767343 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -2.735236830308559e-193 or 1.35258502937208e-160 < a

    1. Initial program 23.4

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

    2. Simplified9.5

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

    4. Applied fma-udef9.5

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

    5. Simplified9.6

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

    if -2.735236830308559e-193 < a < 1.35258502937208e-160

    1. Initial program 30.5

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

    2. Simplified20.0

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

    3. Taylor expanded around inf 12.6

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

    4. Simplified10.4

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.735236830308558818018448627428718325221 \cdot 10^{-193} \lor \neg \left(a \le 1.352585029372079948136707219245541043528 \cdot 10^{-160}\right):\\ \;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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