2cbrt (problem 3.3.4)

Average Error: 29.5 → 29.5

Time: 10.1s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (- (cbrt (* (+ x (+ x 2.0)) 0.5)) (cbrt x)))

C

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

↓

double code(double x) {
	return cbrt(((x + (x + 2.0)) * 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) {
	return Math.cbrt(((x + (x + 2.0)) * 0.5)) - Math.cbrt(x);
}

Julia

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

↓

function code(x)
	return Float64(cbrt(Float64(Float64(x + Float64(x + 2.0)) * 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_] := N[(N[Power[N[(N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.5

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

2. Applied egg-rr 30.1

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

3. Simplified 29.5

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

   [Start] 30.1	\[ \sqrt[3]{x + \left(x + 2\right)} \cdot \sqrt[3]{0.5} - \sqrt[3]{x} \]
   exponential-simplify-18 [=>] 29.5	\[ \color{blue}{\sqrt[3]{\left(x + \left(x + 2\right)\right) \cdot 0.5}} - \sqrt[3]{x} \]

4. Final simplification 29.5

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

Alternatives

Alternative 1
Error	29.5
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 2023100 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))