2cbrt (problem 3.3.4)

Average Error: 30.1 → 29.8

Time: 13.7s

Precision: binary64

Cost: 13376

Math

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

↓

\[1 + \left(\sqrt[3]{1 + x} - \left(\sqrt[3]{x} - -1\right)\right) \]

FPCore

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

↓

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

C

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

↓

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

Java

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 30.1

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

2. Applied egg-rr 30.1

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

3. Simplified 29.8

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

   [Start] 30.1	\[ 1 + \left(-1 - \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)\right) \]
   rational.json-simplify-44 [=>] 29.8	\[ 1 + \color{blue}{\left(\sqrt[3]{x + 1} - \left(\sqrt[3]{x} - -1\right)\right)} \]
   rational.json-simplify-1 [=>] 29.8	\[ 1 + \left(\sqrt[3]{\color{blue}{1 + x}} - \left(\sqrt[3]{x} - -1\right)\right) \]

4. Final simplification 29.8

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

Alternatives

Alternative 1
Error	30.1
Cost	13120

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

Alternative 2
Error	61.7
Cost	64

\[0 \]

Alternative 3
Error	32.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))