2cbrt (problem 3.3.4)

Average Error: 29.7 → 29.7

Time: 45.5s

Precision: binary64

Cost: 33088

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \frac{t_0 - \sqrt[3]{x}}{2} - \left(\left(\sqrt[3]{x} + t_0\right) \cdot -0.5 + \sqrt[3]{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ x 1.0))))
   (- (/ (- t_0 (cbrt x)) 2.0) (+ (* (+ (cbrt x) t_0) -0.5) (cbrt x)))))

C

double code(double x) {
	return cbrt((x + 1.0)) - cbrt(x);
}

↓

double code(double x) {
	double t_0 = cbrt((x + 1.0));
	return ((t_0 - cbrt(x)) / 2.0) - (((cbrt(x) + t_0) * -0.5) + cbrt(x));
}

Java

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

↓

public static double code(double x) {
	double t_0 = Math.cbrt((x + 1.0));
	return ((t_0 - Math.cbrt(x)) / 2.0) - (((Math.cbrt(x) + t_0) * -0.5) + Math.cbrt(x));
}

Julia

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

↓

function code(x)
	t_0 = cbrt(Float64(x + 1.0))
	return Float64(Float64(Float64(t_0 - cbrt(x)) / 2.0) - Float64(Float64(Float64(cbrt(x) + t_0) * -0.5) + cbrt(x)))
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] * -0.5), $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.7

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

2. Applied egg-rr 29.7

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

3. Applied egg-rr 29.7

   \[\leadsto \color{blue}{\frac{\sqrt[3]{x + 1} - \sqrt[3]{x}}{2} - \left(\sqrt[3]{x} - \left(\sqrt[3]{x + 1} \cdot 0.5 + \frac{\sqrt[3]{x}}{2}\right)\right)} \]

4. Applied egg-rr 29.7

   \[\leadsto \frac{\sqrt[3]{x + 1} - \sqrt[3]{x}}{2} - \color{blue}{\left(\left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right) \cdot -0.5 + \sqrt[3]{x}\right)} \]

5. Final simplification 29.7

   \[\leadsto \frac{\sqrt[3]{x + 1} - \sqrt[3]{x}}{2} - \left(\left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right) \cdot -0.5 + \sqrt[3]{x}\right) \]

Alternatives

Alternative 1
Error	29.7
Cost	13120

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

Alternative 2
Error	61.7
Cost	64

\[0 \]

Alternative 3
Error	31.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023099 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))