Average Error: 25.3 → 11.0
Time: 10.1s
Precision: binary64
\[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \leq -5.284160521582155 \cdot 10^{-116}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot \left(t - x\right)\\ \mathbf{elif}\;a \leq 1.5807917828594455 \cdot 10^{-115}:\\ \;\;\;\;\left(t + \frac{x \cdot y}{z}\right) - \frac{y \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;a \leq -5.284160521582155 \cdot 10^{-116}:\\
\;\;\;\;x + \frac{y - z}{a - z} \cdot \left(t - x\right)\\

\mathbf{elif}\;a \leq 1.5807917828594455 \cdot 10^{-115}:\\
\;\;\;\;\left(t + \frac{x \cdot y}{z}\right) - \frac{y \cdot t}{z}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{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 (<= a -5.284160521582155e-116)
   (+ x (* (/ (- y z) (- a z)) (- t x)))
   (if (<= a 1.5807917828594455e-115)
     (- (+ t (/ (* x y) z)) (/ (* y t) z))
     (+
      x
      (*
       (/ (- y z) (* (cbrt (- a z)) (cbrt (- a z))))
       (/ (- 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 (a <= -5.284160521582155e-116) {
		tmp = x + (((y - z) / (a - z)) * (t - x));
	} else if (a <= 1.5807917828594455e-115) {
		tmp = (t + ((x * y) / z)) - ((y * t) / z);
	} else {
		tmp = x + (((y - z) / (cbrt(a - z) * cbrt(a - z))) * ((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

Original25.3
Target12.1
Herbie11.0
\[\begin{array}{l} \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -5.28416052158215461e-116

    1. Initial program 23.0

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

    3. Applied associate-/l*_binary6411.4

      \[\leadsto x + \color{blue}{\frac{y - z}{\frac{a - z}{t - x}}}\]
    4. Using strategy rm

    5. Applied associate-/r/_binary648.9

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

    if -5.28416052158215461e-116 < a < 1.5807917828594455e-115

    1. Initial program 29.8

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

    3. Applied associate-/l*_binary6424.9

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

    4. Taylor expanded around inf 15.6

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

    5. Simplified15.6

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

    if 1.5807917828594455e-115 < a

    1. Initial program 24.1

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

    3. Applied add-cube-cbrt_binary6424.5

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

    4. Applied times-frac_binary649.6

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.284160521582155 \cdot 10^{-116}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot \left(t - x\right)\\ \mathbf{elif}\;a \leq 1.5807917828594455 \cdot 10^{-115}:\\ \;\;\;\;\left(t + \frac{x \cdot y}{z}\right) - \frac{y \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020273 
(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))))