Average Error: 2.0 → 1.5
Time: 6.8s
Precision: binary64
\[\frac{x}{y} \cdot \left(z - t\right) + t \]
↓
\[\begin{array}{l} t_1 := \sqrt[3]{z - t}\\ t + \left(x \cdot \frac{t_1 \cdot t_1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{t_1}{\sqrt[3]{y}} \end{array} \]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
t_1 := \sqrt[3]{z - t}\\
t + \left(x \cdot \frac{t_1 \cdot t_1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{t_1}{\sqrt[3]{y}}
\end{array}

(FPCore (x y z t) :precision binary64 (+ (* (/ x y) (- z t)) t))
↓
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (cbrt (- z t))))
   (+ t (* (* x (/ (* t_1 t_1) (* (cbrt y) (cbrt y)))) (/ t_1 (cbrt y))))))
double code(double x, double y, double z, double t) {
	return ((x / y) * (z - t)) + t;
}
↓
double code(double x, double y, double z, double t) {
	double t_1 = cbrt((z - t));
	return t + ((x * ((t_1 * t_1) / (cbrt(y) * cbrt(y)))) * (t_1 / cbrt(y)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.0
Target2.3
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z < 2.759456554562692 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z < 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array} \]

Derivation

  1. Initial program 2.0

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

  2. Simplified2.0

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

  3. Applied fma-udef_binary642.0

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

  4. Simplified6.6

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

  5. Applied add-cube-cbrt_binary647.0

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

  6. Applied add-cube-cbrt_binary647.1

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

  7. Applied times-frac_binary647.1

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

  8. Applied associate-*r*_binary641.5

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

  9. Final simplification1.5

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

Reproduce

herbie shell --seed 2022125 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

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