Average Error: 24.3 → 10.2
Time: 9.7s
Precision: binary64
\[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.0460208328950513 \cdot 10^{+194} \lor \neg \left(z \leq 6.37987987931022 \cdot 10^{+72} \lor \neg \left(z \leq 1.0088239616805826 \cdot 10^{+149}\right) \land z \leq 3.558503535078779 \cdot 10^{+212}\right):\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1.0460208328950513 \cdot 10^{+194} \lor \neg \left(z \leq 6.37987987931022 \cdot 10^{+72} \lor \neg \left(z \leq 1.0088239616805826 \cdot 10^{+149}\right) \land z \leq 3.558503535078779 \cdot 10^{+212}\right):\\
\;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\

\mathbf{else}:\\
\;\;\;\;x + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\

\end{array}
(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) (- t x)) (- a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.0460208328950513e+194)
         (not
          (or (<= z 6.37987987931022e+72)
              (and (not (<= z 1.0088239616805826e+149))
                   (<= z 3.558503535078779e+212)))))
   (+ t (* y (- (/ x z) (/ t z))))
   (+
    x
    (*
     (*
      (- y z)
      (/ (* (cbrt (- t x)) (cbrt (- t x))) (* (cbrt (- a z)) (cbrt (- a z)))))
     (/ (cbrt (- t x)) (cbrt (- a z)))))))
double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}
↓
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.0460208328950513e+194) || !((z <= 6.37987987931022e+72) || (!(z <= 1.0088239616805826e+149) && (z <= 3.558503535078779e+212)))) {
		tmp = t + (y * ((x / z) - (t / z)));
	} else {
		tmp = x + (((y - z) * ((cbrt(t - x) * cbrt(t - x)) / (cbrt(a - z) * cbrt(a - z)))) * (cbrt(t - x) / cbrt(a - z)));
	}
	return tmp;
}

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

Original24.3
Target11.7
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.0460208328950513e194 or 6.3798798793102201e72 < z < 1.00882396168058256e149 or 3.55850353507877901e212 < z

    1. Initial program 45.9

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

    2. Taylor expanded around inf 24.5

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

    3. Simplified15.6

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

    if -1.0460208328950513e194 < z < 6.3798798793102201e72 or 1.00882396168058256e149 < z < 3.55850353507877901e212

    1. Initial program 16.9

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

    3. Applied *-un-lft-identity_binary64_1928616.9

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

    4. Applied times-frac_binary64_1929210.9

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

    5. Simplified10.9

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

    7. Applied add-cube-cbrt_binary64_1931811.4

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

    8. Applied add-cube-cbrt_binary64_1931811.6

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

    9. Applied times-frac_binary64_1929211.6

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

    10. Applied associate-*r*_binary64_192288.4

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.0460208328950513 \cdot 10^{+194} \lor \neg \left(z \leq 6.37987987931022 \cdot 10^{+72} \lor \neg \left(z \leq 1.0088239616805826 \cdot 10^{+149}\right) \land z \leq 3.558503535078779 \cdot 10^{+212}\right):\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(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))))