Average Error: 24.4 → 10.3
Time: 33.6s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -9.56944376288991126 \cdot 10^{-141} \lor \neg \left(a \le 8.7838325036490706 \cdot 10^{-87}\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{y - z}{a - z} \cdot \left(t - x\right)} \cdot \sqrt[3]{\frac{y - z}{a - z} \cdot \left(t - x\right)}\right) \cdot \left(\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{t - x}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;a \le -9.56944376288991126 \cdot 10^{-141} \lor \neg \left(a \le 8.7838325036490706 \cdot 10^{-87}\right):\\
\;\;\;\;\left(\sqrt[3]{\frac{y - z}{a - z} \cdot \left(t - x\right)} \cdot \sqrt[3]{\frac{y - z}{a - z} \cdot \left(t - x\right)}\right) \cdot \left(\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{t - x}\right) + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\

\end{array}
double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z))))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((a <= -9.569443762889911e-141) || !(a <= 8.78383250364907e-87))) {
		VAR = ((double) (((double) (((double) (((double) cbrt(((double) (((double) (((double) (y - z)) / ((double) (a - z)))) * ((double) (t - x)))))) * ((double) cbrt(((double) (((double) (((double) (y - z)) / ((double) (a - z)))) * ((double) (t - x)))))))) * ((double) (((double) cbrt(((double) (((double) (y - z)) / ((double) (a - z)))))) * ((double) cbrt(((double) (t - x)))))))) + x));
	} else {
		VAR = ((double) fma(y, ((double) (((double) (x / z)) - ((double) (t / z)))), t));
	}
	return VAR;
}

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

Original24.4
Target12.3
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;z \lt -1.25361310560950359 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -9.569443762889911e-141 or 8.78383250364907e-87 < a

    1. Initial program 22.5

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

    2. Simplified8.5

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

    4. Applied fma-udef8.5

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

    6. Applied add-cube-cbrt9.0

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{y - z}{a - z} \cdot \left(t - x\right)} \cdot \sqrt[3]{\frac{y - z}{a - z} \cdot \left(t - x\right)}\right) \cdot \sqrt[3]{\frac{y - z}{a - z} \cdot \left(t - x\right)}} + x\]
    7. Using strategy rm

    8. Applied cbrt-prod9.0

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

    if -9.569443762889911e-141 < a < 8.78383250364907e-87

    1. Initial program 29.4

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

    2. Simplified19.8

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

    3. Taylor expanded around inf 16.2

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

    4. Simplified13.8

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -9.56944376288991126 \cdot 10^{-141} \lor \neg \left(a \le 8.7838325036490706 \cdot 10^{-87}\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{y - z}{a - z} \cdot \left(t - x\right)} \cdot \sqrt[3]{\frac{y - z}{a - z} \cdot \left(t - x\right)}\right) \cdot \left(\sqrt[3]{\frac{y - z}{a - z}} \cdot \sqrt[3]{t - x}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :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))))