2cbrt (problem 3.3.4)

Average Error: 46.1% → 0.84%

Time: 9.2s

Precision: binary64

Cost: 33024

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	return 1.0 / (pow(cbrt(x), 2.0) + (t_0 / (1.0 / (cbrt(x) + t_0))));
}

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((1.0 + x));
	return 1.0 / (Math.pow(Math.cbrt(x), 2.0) + (t_0 / (1.0 / (Math.cbrt(x) + t_0))));
}

Julia

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

↓

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	return Float64(1.0 / Float64((cbrt(x) ^ 2.0) + Float64(t_0 / Float64(1.0 / Float64(cbrt(x) + t_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_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$0 / N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 46.1

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

2. Applied egg-rr 45.08

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

3. Simplified 0.82

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}} \]
   Proof

   [Start] 45.08	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate-*r/ [=>] 45.08	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   *-rgt-identity [=>] 45.08	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 45.08	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate--l+ [=>] 0.82	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-inverses [=>] 0.82	\[ \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   metadata-eval [=>] 0.82	\[ \frac{\color{blue}{1}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 0.82	\[ \frac{1}{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {\left(\sqrt[3]{x + 1}\right)}^{2}}} \]
   fma-def [=>] 0.82	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)}} \]
   +-commutative [=>] 0.82	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)} \]
   +-commutative [=>] 0.82	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\color{blue}{1 + x}}\right)}^{2}\right)} \]

4. Applied egg-rr 0.83

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

5. Applied egg-rr 0.84

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

6. Final simplification 0.84

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

Alternatives

Alternative 1
Error	39.75%
Cost	39236

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{1}{1 + {\left(\sqrt[3]{x}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left({t_0}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternative 2
Error	10.87%
Cost	33096

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_1, 1\right)}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_1 + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot t_1 + e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}}\\ \end{array} \]

Alternative 3
Error	10.91%
Cost	33032

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_1, 1\right)}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_1 + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_1, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]

Alternative 4
Error	0.83%
Cost	32896

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

Alternative 5
Error	21.55%
Cost	26692

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_1 + \sqrt[3]{x \cdot x}}\\ \end{array} \]

Alternative 6
Error	38.7%
Cost	26440

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t_0, 1\right)}\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;{\left({\left(t_0 - \sqrt[3]{x}\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	39.75%
Cost	26372

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

Alternative 8
Error	46.1%
Cost	13120

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

Alternative 9
Error	48.7%
Cost	6592

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

Alternative 10
Error	96.37%
Cost	64

\[0 \]

Alternative 11
Error	49.47%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023115 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))