2cbrt (problem 3.3.4)

Average Error: 29.7 → 29.4

Time: 3.6s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

function code(x)
	return Float64(Float64(cbrt(Float64(x + 1.0)) + Float64(-1.0 - 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[(N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] + N[(-1.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 29.7

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

2. Applied egg-rr 29.4

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

3. Final simplification 29.4

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

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	31.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023074 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))