Average Error: 1.4 → 0.6

Time: 9.2s

Precision: binary64

Cost: 40128

\[x + y \cdot \frac{z - t}{a - t}\]

↓

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

x + y \cdot \frac{z - t}{a - t}

↓

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

(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- a t)))))

↓

(FPCore (x y z t a)
 :precision binary64
 (+
  x
  (*
   (*
    y
    (/ (* (cbrt (- z t)) (cbrt (- z t))) (* (cbrt (- a t)) (cbrt (- a t)))))
   (/ (cbrt (- z t)) (cbrt (- a t))))))

double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (a - t)));
}

↓

double code(double x, double y, double z, double t, double a) {
	return x + ((y * ((cbrt(z - t) * cbrt(z - t)) / (cbrt(a - t) * cbrt(a - t)))) * (cbrt(z - t) / cbrt(a - t)));
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	1.4
Target	0.5
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Alternatives

Alternative 1
Error	1.9
Cost	21184

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

Alternative 2
Error	2.4
Cost	20416

\[x + \frac{y}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{z - t}{\sqrt[3]{a - t}}\]

Alternative 3
Error	29.9
Cost	14144

\[x + \sqrt{y \cdot \frac{z - t}{a - t}} \cdot \sqrt{y \cdot \frac{z - t}{a - t}}\]

Alternative 4
Error	32.1
Cost	13632

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

Alternative 5
Error	11.6
Cost	13568

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

Alternative 6
Error	22.3
Cost	13568

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

Alternative 7
Error	32.1
Cost	8960

\[x + \left(\left(\left(1 + \frac{t}{a}\right) \cdot \frac{y \cdot z}{a} + \left(\frac{t}{a} \cdot \frac{t}{a}\right) \cdot \left(\frac{y \cdot z}{a} - y\right)\right) - y \cdot \left(\frac{t}{a} + {\left(\frac{t}{a}\right)}^{3}\right)\right)\]

Alternative 8
Error	30.5
Cost	8704

\[x + y \cdot \left(\frac{z}{a} \cdot \left(\frac{t}{a} \cdot \left(1 + \frac{t}{a}\right)\right) + \left(\frac{z}{a} + \left(\frac{t}{a} \cdot \left(-1 - \frac{t}{a}\right) - {\left(\frac{t}{a}\right)}^{3}\right)\right)\right)\]

Alternative 9
Error	19.6
Cost	1216

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

Alternative 10
Error	10.5
Cost	832

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

Alternative 11
Error	1.5
Cost	832

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

Alternative 12
Error	10.5
Cost	832

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

Alternative 13
Error	1.4
Cost	704

\[x + y \cdot \frac{z - t}{a - t}\]

Alternative 14
Error	1.3
Cost	704

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

Alternative 15
Error	10.5
Cost	704

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

Alternative 16
Error	18.4
Cost	576

\[x + \frac{y \cdot z}{a - t}\]

Alternative 17
Error	26.4
Cost	576

\[x + y \cdot \frac{z - t}{a}\]

Alternative 18
Error	28.3
Cost	576

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

Alternative 19
Error	28.3
Cost	576

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

Alternative 20
Error	16.0
Cost	576

\[x + y \cdot \frac{z}{a - t}\]

Alternative 21
Error	14.8
Cost	576

\[x - y \cdot \frac{t}{a - t}\]

Alternative 22
Error	25.3
Cost	448

\[x + y \cdot \frac{z}{a}\]

Alternative 23
Error	26.5
Cost	448

\[x + \frac{y \cdot z}{a}\]

Alternative 24
Error	22.7
Cost	192

\[x + y\]

Alternative 25
Error	61.8
Cost	64

\[1\]

Alternative 26
Error	62.2
Cost	64

\[0\]

Alternative 27
Error	61.9
Cost	64

\[-1\]

Derivation

Initial program 1.4
\[x + y \cdot \frac{z - t}{a - t}\]
Using strategy rm
Applied add-cube-cbrt_binary64_117612.0
\[\leadsto x + y \cdot \frac{z - t}{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}}\]
Applied add-cube-cbrt_binary64_117611.8
\[\leadsto x + y \cdot \frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}\]
Applied times-frac_binary64_117321.8
\[\leadsto x + y \cdot \color{blue}{\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}\right)}\]
Applied associate-*r*_binary64_116660.6
\[\leadsto x + \color{blue}{\left(y \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\]
Simplified0.6
\[\leadsto \color{blue}{x + \left(y \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\]
Final simplification0.6
\[\leadsto x + \left(y \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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