2cbrt (problem 3.3.4)

Average Error: 53.7% → 99.2%

Time: 10.6s

Precision: binary64

Cost: 39296.00

Math

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

↓

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

FPCore

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

↓

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

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 / fma(cbrt(x), (cbrt(x) + t_0), (1.0 / pow(t_0, -2.0)));
}

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 / fma(cbrt(x), Float64(cbrt(x) + t_0), Float64(1.0 / (t_0 ^ -2.0))))
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[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] + N[(1.0 / N[Power[t$95$0, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.7

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

2. Applied egg-rr 54.5

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

3. Simplified 99.2

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

   [Start] 54.5	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate-*r/ [=>] 54.5	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   *-rgt-identity [=>] 54.5	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 54.5	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate--l+ [=>] 99.2	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-inverses [=>] 99.2	\[ \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   metadata-eval [=>] 99.2	\[ \frac{\color{blue}{1}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 99.2	\[ \frac{1}{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {\left(\sqrt[3]{x + 1}\right)}^{2}}} \]
   fma-def [=>] 99.2	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)}} \]
   +-commutative [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)} \]
   +-commutative [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\color{blue}{1 + x}}\right)}^{2}\right)} \]

4. Applied egg-rr 76.6

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

5. Applied egg-rr 99.2

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

6. Applied egg-rr 99.2

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

7. Final simplification 99.2

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

Alternatives

Alternative 1
Error	99.2%
Cost	39168.00

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

Alternative 2
Error	89.1%
Cost	33224.00

\[\begin{array}{l} t_0 := \sqrt[3]{x} + \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, 1\right)}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, \sqrt[3]{\left(1 + x\right) \cdot \left(1 + x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]

Alternative 3
Error	88.3%
Cost	33032.00

\[\begin{array}{l} t_0 := \sqrt[3]{x} + \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+161}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, 1\right)}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]

Alternative 4
Error	80.3%
Cost	26704.00

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, 1\right)}\\ t_1 := 0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -62000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 27000000:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+161}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	76.8%
Cost	26441.00

\[\begin{array}{l} \mathbf{if}\;x \leq -62000000 \lor \neg \left(x \leq 27000000\right):\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \end{array} \]

Alternative 6
Error	76.8%
Cost	13769.00

\[\begin{array}{l} \mathbf{if}\;x \leq -25500000 \lor \neg \left(x \leq 27000000\right):\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1 - x \cdot x}{1 - x}} - \sqrt[3]{x}\\ \end{array} \]

Alternative 7
Error	76.8%
Cost	13385.00

\[\begin{array}{l} \mathbf{if}\;x \leq -24000000 \lor \neg \left(x \leq 27000000\right):\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array} \]

Alternative 8
Error	74.8%
Cost	7113.00

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.48\right):\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt[3]{x}\\ \end{array} \]

Alternative 9
Error	75.4%
Cost	7113.00

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.48\right):\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt[3]{x}\\ \end{array} \]

Alternative 10
Error	75.9%
Cost	7113.00

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot 0.3333333333333333 - \sqrt[3]{x}\right)\\ \end{array} \]

Alternative 11
Error	51.2%
Cost	6592.00

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

Alternative 12
Error	3.6%
Cost	64.00

\[0 \]

Alternative 13
Error	50.4%
Cost	64.00

\[1 \]

Reproduce

herbie shell --seed 2023098 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))