Average Error: 1.3 → 0.5
Time: 6.1s
Precision: binary64
\[x + y \cdot \frac{z - t}{z - a} \]
↓
\[\begin{array}{l} t_1 := \sqrt[3]{z - a}\\ t_2 := \sqrt[3]{z - t}\\ x + \left(y \cdot \frac{t_2 \cdot t_2}{{t_1}^{2}}\right) \cdot \frac{t_2}{t_1} \end{array} \]
(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- z a)))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (cbrt (- z a))) (t_2 (cbrt (- z t))))
   (+ x (* (* y (/ (* t_2 t_2) (pow t_1 2.0))) (/ t_2 t_1)))))
double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (z - a)));
}
↓
double code(double x, double y, double z, double t, double a) {
	double t_1 = cbrt((z - a));
	double t_2 = cbrt((z - t));
	return x + ((y * ((t_2 * t_2) / pow(t_1, 2.0))) * (t_2 / t_1));
}

public static double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (z - a)));
}
↓
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.cbrt((z - a));
	double t_2 = Math.cbrt((z - t));
	return x + ((y * ((t_2 * t_2) / Math.pow(t_1, 2.0))) * (t_2 / t_1));
}

function code(x, y, z, t, a)
	return Float64(x + Float64(y * Float64(Float64(z - t) / Float64(z - a))))
end
↓
function code(x, y, z, t, a)
	t_1 = cbrt(Float64(z - a))
	t_2 = cbrt(Float64(z - t))
	return Float64(x + Float64(Float64(y * Float64(Float64(t_2 * t_2) / (t_1 ^ 2.0))) * Float64(t_2 / t_1)))
end

code[x_, y_, z_, t_, a_] := N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Power[N[(z - a), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(z - t), $MachinePrecision], 1/3], $MachinePrecision]}, N[(x + N[(N[(y * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

x + y \cdot \frac{z - t}{z - a}
↓
\begin{array}{l}
t_1 := \sqrt[3]{z - a}\\
t_2 := \sqrt[3]{z - t}\\
x + \left(y \cdot \frac{t_2 \cdot t_2}{{t_1}^{2}}\right) \cdot \frac{t_2}{t_1}
\end{array}

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

Original1.3
Target1.1
Herbie0.5
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation

  1. Initial program 1.3

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

  2. Applied add-cube-cbrt_binary641.8

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

  3. Applied add-cube-cbrt_binary641.6

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

  4. Applied times-frac_binary641.6

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

  5. Applied associate-*r*_binary640.5

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

  6. Applied pow1_binary640.5

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

  7. Applied pow1_binary640.5

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

  8. Applied pow-sqr_binary640.5

    \[\leadsto x + \left(y \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\color{blue}{{\left(\sqrt[3]{z - a}\right)}^{\left(2 \cdot 1\right)}}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{z - a}} \]

  9. Final simplification0.5

    \[\leadsto x + \left(y \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{{\left(\sqrt[3]{z - a}\right)}^{2}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{z - a}} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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