Average Error: 11.3 → 1.1
Time: 8.0s
Precision: binary64
\[\frac{x \cdot \left(y - z\right)}{t - z} \]
↓
\[\begin{array}{l} t_1 := \sqrt[3]{y - z}\\ t_2 := \sqrt[3]{t - z}\\ \left(x \cdot \frac{t_1 \cdot t_1}{t_2 \cdot t_2}\right) \cdot \frac{t_1}{t_2} \end{array} \]
(FPCore (x y z t) :precision binary64 (/ (* x (- y z)) (- t z)))
↓
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (cbrt (- y z))) (t_2 (cbrt (- t z))))
   (* (* x (/ (* t_1 t_1) (* t_2 t_2))) (/ t_1 t_2))))
double code(double x, double y, double z, double t) {
	return (x * (y - z)) / (t - z);
}
↓
double code(double x, double y, double z, double t) {
	double t_1 = cbrt((y - z));
	double t_2 = cbrt((t - z));
	return (x * ((t_1 * t_1) / (t_2 * t_2))) * (t_1 / t_2);
}

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

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

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

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

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

Original11.3
Target2.1
Herbie1.1
\[\frac{x}{\frac{t - z}{y - z}} \]

Derivation

  1. Initial program 11.3

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

  2. Applied *-un-lft-identity_binary6411.3

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

  3. Applied times-frac_binary642.1

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

  4. Simplified2.1

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

  5. Applied add-cube-cbrt_binary643.1

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

  6. Applied add-cube-cbrt_binary642.8

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

  7. Applied times-frac_binary642.8

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

  8. Applied associate-*r*_binary641.1

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

  9. Final simplification1.1

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

Reproduce

herbie shell --seed 2022130 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

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

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