2cbrt (problem 3.3.4)

Percentage Accurate: 53.1% → 99.1%
Time: 15.6s
Alternatives: 13
Speedup: 0.5×

Specification

?
\[\begin{array}{l} \\ \sqrt[3]{x + 1} - \sqrt[3]{x} \end{array} \]
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
double code(double x) {
	return cbrt((x + 1.0)) - cbrt(x);
}

public static double code(double x) {
	return Math.cbrt((x + 1.0)) - Math.cbrt(x);
}

function code(x)
	return Float64(cbrt(Float64(x + 1.0)) - cbrt(x))
end

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt[3]{x + 1} - \sqrt[3]{x}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 13 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 53.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sqrt[3]{x + 1} - \sqrt[3]{x} \end{array} \]
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
double code(double x) {
	return cbrt((x + 1.0)) - cbrt(x);
}

public static double code(double x) {
	return Math.cbrt((x + 1.0)) - Math.cbrt(x);
}

function code(x)
	return Float64(cbrt(Float64(x + 1.0)) - cbrt(x))
end

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt[3]{x + 1} - \sqrt[3]{x}
\end{array}

Alternative 1: 99.1% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \frac{1}{{t_0}^{2} + \frac{1}{\frac{1}{\sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}}} \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (/ 1.0 (+ (pow t_0 2.0) (/ 1.0 (/ 1.0 (* (cbrt x) (+ t_0 (cbrt x)))))))))
double code(double x) {
	double t_0 = cbrt((1.0 + x));
	return 1.0 / (pow(t_0, 2.0) + (1.0 / (1.0 / (cbrt(x) * (t_0 + cbrt(x))))));
}

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	return 1.0 / (Math.pow(t_0, 2.0) + (1.0 / (1.0 / (Math.cbrt(x) * (t_0 + Math.cbrt(x))))));
}

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	return Float64(1.0 / Float64((t_0 ^ 2.0) + Float64(1.0 / Float64(1.0 / Float64(cbrt(x) * Float64(t_0 + cbrt(x)))))))
end

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(1.0 / N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt[3]{1 + x}\\
\frac{1}{{t_0}^{2} + \frac{1}{\frac{1}{\sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}}}
\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 99.2% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}}\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (/ 1.0 (+ (pow t_0 2.0) (* (cbrt x) (+ (cbrt x) (cbrt x)))))
     (/ 1.0 (+ (* (cbrt x) (+ t_0 (cbrt x))) (cbrt (pow (+ 1.0 x) 2.0)))))))
double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 1.0 / (pow(t_0, 2.0) + (cbrt(x) * (cbrt(x) + cbrt(x))));
	} else {
		tmp = 1.0 / ((cbrt(x) * (t_0 + cbrt(x))) + cbrt(pow((1.0 + x), 2.0)));
	}
	return tmp;
}

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double tmp;
	if ((t_0 - Math.cbrt(x)) <= 0.0) {
		tmp = 1.0 / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (Math.cbrt(x) + Math.cbrt(x))));
	} else {
		tmp = 1.0 / ((Math.cbrt(x) * (t_0 + Math.cbrt(x))) + Math.cbrt(Math.pow((1.0 + x), 2.0)));
	}
	return tmp;
}

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(1.0 / Float64((t_0 ^ 2.0) + Float64(cbrt(x) * Float64(cbrt(x) + cbrt(x)))));
	else
		tmp = Float64(1.0 / Float64(Float64(cbrt(x) * Float64(t_0 + cbrt(x))) + cbrt((Float64(1.0 + x) ^ 2.0))));
	end
	return tmp
end

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt[3]{1 + x}\\
\mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\
\;\;\;\;\frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 98.0% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{1}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (/ 1.0 (+ (pow t_0 2.0) (* (cbrt x) (+ (cbrt x) (cbrt x)))))
     (- (* (cbrt (fma x x -1.0)) (/ 1.0 (cbrt (+ x -1.0)))) (cbrt x)))))
double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 1.0 / (pow(t_0, 2.0) + (cbrt(x) * (cbrt(x) + cbrt(x))));
	} else {
		tmp = (cbrt(fma(x, x, -1.0)) * (1.0 / cbrt((x + -1.0)))) - cbrt(x);
	}
	return tmp;
}

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(1.0 / Float64((t_0 ^ 2.0) + Float64(cbrt(x) * Float64(cbrt(x) + cbrt(x)))));
	else
		tmp = Float64(Float64(cbrt(fma(x, x, -1.0)) * Float64(1.0 / cbrt(Float64(x + -1.0)))) - cbrt(x));
	end
	return tmp
end

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(x * x + -1.0), $MachinePrecision], 1/3], $MachinePrecision] * N[(1.0 / N[Power[N[(x + -1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt[3]{1 + x}\\
\mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\
\;\;\;\;\frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x}\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{1}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 4: 60.6% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{1}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (/ 1.0 (+ 1.0 (* (cbrt x) (+ t_0 (cbrt x)))))
     (- (* (cbrt (fma x x -1.0)) (/ 1.0 (cbrt (+ x -1.0)))) (cbrt x)))))
double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 1.0 / (1.0 + (cbrt(x) * (t_0 + cbrt(x))));
	} else {
		tmp = (cbrt(fma(x, x, -1.0)) * (1.0 / cbrt((x + -1.0)))) - cbrt(x);
	}
	return tmp;
}

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))));
	else
		tmp = Float64(Float64(cbrt(fma(x, x, -1.0)) * Float64(1.0 / cbrt(Float64(x + -1.0)))) - cbrt(x));
	end
	return tmp
end

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(x * x + -1.0), $MachinePrecision], 1/3], $MachinePrecision] * N[(1.0 / N[Power[N[(x + -1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt[3]{1 + x}\\
\mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\
\;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{1}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 5: 60.6% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (/ 1.0 (+ 1.0 (* (cbrt x) (+ t_0 (cbrt x)))))
     (- (/ (cbrt (fma x x -1.0)) (cbrt (+ x -1.0))) (cbrt x)))))
double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 1.0 / (1.0 + (cbrt(x) * (t_0 + cbrt(x))));
	} else {
		tmp = (cbrt(fma(x, x, -1.0)) / cbrt((x + -1.0))) - cbrt(x);
	}
	return tmp;
}

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))));
	else
		tmp = Float64(Float64(cbrt(fma(x, x, -1.0)) / cbrt(Float64(x + -1.0))) - cbrt(x));
	end
	return tmp
end

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(x * x + -1.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(x + -1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt[3]{1 + x}\\
\mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\
\;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 6: 87.8% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x} + \sqrt[3]{x}\\ t_1 := \sqrt[3]{x} \cdot t_0\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + t_1}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{1}{t_1 + \sqrt[3]{{x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(1 + x\right)}^{0.6666666666666666} + \frac{\sqrt[3]{x}}{\frac{1}{t_0}}}\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (cbrt (+ 1.0 x)) (cbrt x))) (t_1 (* (cbrt x) t_0)))
   (if (<= x -1.35e+154)
     (/ 1.0 (+ 1.0 t_1))
     (if (<= x -1.0)
       (/ 1.0 (+ t_1 (cbrt (pow x 2.0))))
       (/
        1.0
        (+ (pow (+ 1.0 x) 0.6666666666666666) (/ (cbrt x) (/ 1.0 t_0))))))))
double code(double x) {
	double t_0 = cbrt((1.0 + x)) + cbrt(x);
	double t_1 = cbrt(x) * t_0;
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / (1.0 + t_1);
	} else if (x <= -1.0) {
		tmp = 1.0 / (t_1 + cbrt(pow(x, 2.0)));
	} else {
		tmp = 1.0 / (pow((1.0 + x), 0.6666666666666666) + (cbrt(x) / (1.0 / t_0)));
	}
	return tmp;
}

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x)) + Math.cbrt(x);
	double t_1 = Math.cbrt(x) * t_0;
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / (1.0 + t_1);
	} else if (x <= -1.0) {
		tmp = 1.0 / (t_1 + Math.cbrt(Math.pow(x, 2.0)));
	} else {
		tmp = 1.0 / (Math.pow((1.0 + x), 0.6666666666666666) + (Math.cbrt(x) / (1.0 / t_0)));
	}
	return tmp;
}

function code(x)
	t_0 = Float64(cbrt(Float64(1.0 + x)) + cbrt(x))
	t_1 = Float64(cbrt(x) * t_0)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(1.0 / Float64(1.0 + t_1));
	elseif (x <= -1.0)
		tmp = Float64(1.0 / Float64(t_1 + cbrt((x ^ 2.0))));
	else
		tmp = Float64(1.0 / Float64((Float64(1.0 + x) ^ 0.6666666666666666) + Float64(cbrt(x) / Float64(1.0 / t_0))));
	end
	return tmp
end

code[x_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], N[(1.0 / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.0], N[(1.0 / N[(t$95$1 + N[Power[N[Power[x, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt[3]{1 + x} + \sqrt[3]{x}\\
t_1 := \sqrt[3]{x} \cdot t_0\\
\mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\
\;\;\;\;\frac{1}{1 + t_1}\\

\mathbf{elif}\;x \leq -1:\\
\;\;\;\;\frac{1}{t_1 + \sqrt[3]{{x}^{2}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{\left(1 + x\right)}^{0.6666666666666666} + \frac{\sqrt[3]{x}}{\frac{1}{t_0}}}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 7: 60.6% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (- t_0 (cbrt x))))
   (if (<= t_1 0.0) (/ 1.0 (+ 1.0 (* (cbrt x) (+ t_0 (cbrt x))))) t_1)))
double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = t_0 - cbrt(x);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 1.0 / (1.0 + (cbrt(x) * (t_0 + cbrt(x))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double t_1 = t_0 - Math.cbrt(x);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (t_0 + Math.cbrt(x))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = Float64(t_0 - cbrt(x))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))));
	else
		tmp = t_1;
	end
	return tmp
end

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt[3]{1 + x}\\
t_1 := t_0 - \sqrt[3]{x}\\
\mathbf{if}\;t_1 \leq 0:\\
\;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 8: 99.1% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)} \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (/ 1.0 (+ (pow t_0 2.0) (* (cbrt x) (+ t_0 (cbrt x)))))))
double code(double x) {
	double t_0 = cbrt((1.0 + x));
	return 1.0 / (pow(t_0, 2.0) + (cbrt(x) * (t_0 + cbrt(x))));
}

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	return 1.0 / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (t_0 + Math.cbrt(x))));
}

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	return Float64(1.0 / Float64((t_0 ^ 2.0) + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))))
end

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt[3]{1 + x}\\
\frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}
\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 9: 78.2% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x} + \sqrt[3]{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(1 + x\right)}^{0.6666666666666666} + \frac{\sqrt[3]{x}}{\frac{1}{t_0}}}\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (cbrt (+ 1.0 x)) (cbrt x))))
   (if (<= x -1.0)
     (/ 1.0 (+ 1.0 (* (cbrt x) t_0)))
     (/ 1.0 (+ (pow (+ 1.0 x) 0.6666666666666666) (/ (cbrt x) (/ 1.0 t_0)))))))
double code(double x) {
	double t_0 = cbrt((1.0 + x)) + cbrt(x);
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (1.0 + (cbrt(x) * t_0));
	} else {
		tmp = 1.0 / (pow((1.0 + x), 0.6666666666666666) + (cbrt(x) / (1.0 / t_0)));
	}
	return tmp;
}

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x)) + Math.cbrt(x);
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * t_0));
	} else {
		tmp = 1.0 / (Math.pow((1.0 + x), 0.6666666666666666) + (Math.cbrt(x) / (1.0 / t_0)));
	}
	return tmp;
}

function code(x)
	t_0 = Float64(cbrt(Float64(1.0 + x)) + cbrt(x))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * t_0)));
	else
		tmp = Float64(1.0 / Float64((Float64(1.0 + x) ^ 0.6666666666666666) + Float64(cbrt(x) / Float64(1.0 / t_0))));
	end
	return tmp
end

code[x_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt[3]{1 + x} + \sqrt[3]{x}\\
\mathbf{if}\;x \leq -1:\\
\;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot t_0}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{\left(1 + x\right)}^{0.6666666666666666} + \frac{\sqrt[3]{x}}{\frac{1}{t_0}}}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 10: 78.2% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 + {\left(1 + x\right)}^{0.6666666666666666}}\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (cbrt x) (+ (cbrt (+ 1.0 x)) (cbrt x)))))
   (if (<= x -1.0)
     (/ 1.0 (+ 1.0 t_0))
     (/ 1.0 (+ t_0 (pow (+ 1.0 x) 0.6666666666666666))))))
double code(double x) {
	double t_0 = cbrt(x) * (cbrt((1.0 + x)) + cbrt(x));
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (1.0 + t_0);
	} else {
		tmp = 1.0 / (t_0 + pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

public static double code(double x) {
	double t_0 = Math.cbrt(x) * (Math.cbrt((1.0 + x)) + Math.cbrt(x));
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (1.0 + t_0);
	} else {
		tmp = 1.0 / (t_0 + Math.pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

function code(x)
	t_0 = Float64(cbrt(x) * Float64(cbrt(Float64(1.0 + x)) + cbrt(x)))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 / Float64(1.0 + t_0));
	else
		tmp = Float64(1.0 / Float64(t_0 + (Float64(1.0 + x) ^ 0.6666666666666666)));
	end
	return tmp
end

code[x_] := Block[{t$95$0 = N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\\
\mathbf{if}\;x \leq -1:\\
\;\;\;\;\frac{1}{1 + t_0}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{t_0 + {\left(1 + x\right)}^{0.6666666666666666}}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 11: 56.3% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (cbrt (+ 1.0 x)) (cbrt x))))
   (if (<= t_0 0.0) (cbrt (/ 1.0 (pow x 2.0))) t_0)))
double code(double x) {
	double t_0 = cbrt((1.0 + x)) - cbrt(x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = cbrt((1.0 / pow(x, 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.cbrt((1.0 / Math.pow(x, 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(x)
	t_0 = Float64(cbrt(Float64(1.0 + x)) - cbrt(x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = cbrt(Float64(1.0 / (x ^ 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

code[x_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$0]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt[3]{1 + x} - \sqrt[3]{x}\\
\mathbf{if}\;t_0 \leq 0:\\
\;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 12: 3.6% accurate, 205.0× speedup?

\[\begin{array}{l} \\ 0 \end{array} \]
(FPCore (x) :precision binary64 0.0)
double code(double x) {
	return 0.0;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

public static double code(double x) {
	return 0.0;
}

def code(x):
	return 0.0

function code(x)
	return 0.0
end

function tmp = code(x)
	tmp = 0.0;
end

code[x_] := 0.0

\begin{array}{l}

\\
0
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 13: 49.5% accurate, 205.0× speedup?

\[\begin{array}{l} \\ 1 \end{array} \]
(FPCore (x) :precision binary64 1.0)
double code(double x) {
	return 1.0;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

public static double code(double x) {
	return 1.0;
}

def code(x):
	return 1.0

function code(x)
	return 1.0
end

function tmp = code(x)
	tmp = 1.0;
end

code[x_] := 1.0

\begin{array}{l}

\\
1
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2024006 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))