2cbrt (problem 3.3.4)

Average Error: 30.0 → 0.3

Time: 9.2s

Precision: binary64

Cost: 65472

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\sqrt[3]{x}\right)}^{2}\\ t_1 := \sqrt[3]{1 + x}\\ \frac{1}{t_0 + \frac{t_1}{\frac{\mathsf{fma}\left(t_1 - \sqrt[3]{x}, t_1, t_0\right)}{\mathsf{fma}\left(x, 2, 1\right)}}} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (cbrt x) 2.0)) (t_1 (cbrt (+ 1.0 x))))
   (/ 1.0 (+ t_0 (/ t_1 (/ (fma (- t_1 (cbrt x)) t_1 t_0) (fma x 2.0 1.0)))))))

C

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

↓

double code(double x) {
	double t_0 = pow(cbrt(x), 2.0);
	double t_1 = cbrt((1.0 + x));
	return 1.0 / (t_0 + (t_1 / (fma((t_1 - cbrt(x)), t_1, t_0) / fma(x, 2.0, 1.0))));
}

Julia

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

↓

function code(x)
	t_0 = cbrt(x) ^ 2.0
	t_1 = cbrt(Float64(1.0 + x))
	return Float64(1.0 / Float64(t_0 + Float64(t_1 / Float64(fma(Float64(t_1 - cbrt(x)), t_1, t_0) / fma(x, 2.0, 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_] := Block[{t$95$0 = N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[(t$95$0 + N[(t$95$1 / N[(N[(N[(t$95$1 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] / N[(x * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 30.0

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

2. Applied egg-rr 29.3

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

3. Simplified 0.5

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

   [Start] 29.3	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate-*r/ [=>] 29.3	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   *-rgt-identity [=>] 29.3	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 29.3	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate--l+ [=>] 0.5	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-inverses [=>] 0.5	\[ \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   metadata-eval [=>] 0.5	\[ \frac{\color{blue}{1}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 0.5	\[ \frac{1}{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {\left(\sqrt[3]{x + 1}\right)}^{2}}} \]
   fma-def [=>] 0.5	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)}} \]
   +-commutative [=>] 0.5	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)} \]
   +-commutative [=>] 0.5	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\color{blue}{1 + x}}\right)}^{2}\right)} \]

4. Applied egg-rr 0.5

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

5. Applied egg-rr 0.5

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

6. Applied egg-rr 0.3

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

7. Simplified 0.3

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

   [Start] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{x + 1}}{\frac{1}{1 + \left(x + x\right)} \cdot {\left(\sqrt[3]{x}\right)}^{2} + \frac{1}{1 + \left(x + x\right)} \cdot \left(\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)\right)}} \]
   distribute-lft-out [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{x + 1}}{\color{blue}{\frac{1}{1 + \left(x + x\right)} \cdot \left({\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)\right)}}} \]
   +-commutative [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{x + 1}}{\frac{1}{1 + \left(x + x\right)} \cdot \color{blue}{\left(\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right) + {\left(\sqrt[3]{x}\right)}^{2}\right)}}} \]
   associate-*l/ [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{x + 1}}{\color{blue}{\frac{1 \cdot \left(\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right) + {\left(\sqrt[3]{x}\right)}^{2}\right)}{1 + \left(x + x\right)}}}} \]
   *-lft-identity [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{x + 1}}{\frac{\color{blue}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right) + {\left(\sqrt[3]{x}\right)}^{2}}}{1 + \left(x + x\right)}}} \]
   *-commutative [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{x + 1}}{\frac{\color{blue}{\left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right) \cdot \sqrt[3]{1 + x}} + {\left(\sqrt[3]{x}\right)}^{2}}{1 + \left(x + x\right)}}} \]
   fma-def [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{x + 1}}{\frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{1 + x} - \sqrt[3]{x}, \sqrt[3]{1 + x}, {\left(\sqrt[3]{x}\right)}^{2}\right)}}{1 + \left(x + x\right)}}} \]
   +-commutative [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{x + 1}}{\frac{\mathsf{fma}\left(\sqrt[3]{1 + x} - \sqrt[3]{x}, \sqrt[3]{1 + x}, {\left(\sqrt[3]{x}\right)}^{2}\right)}{\color{blue}{\left(x + x\right) + 1}}}} \]
   count-2 [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{x + 1}}{\frac{\mathsf{fma}\left(\sqrt[3]{1 + x} - \sqrt[3]{x}, \sqrt[3]{1 + x}, {\left(\sqrt[3]{x}\right)}^{2}\right)}{\color{blue}{2 \cdot x} + 1}}} \]
   *-commutative [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{x + 1}}{\frac{\mathsf{fma}\left(\sqrt[3]{1 + x} - \sqrt[3]{x}, \sqrt[3]{1 + x}, {\left(\sqrt[3]{x}\right)}^{2}\right)}{\color{blue}{x \cdot 2} + 1}}} \]
   fma-def [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{x + 1}}{\frac{\mathsf{fma}\left(\sqrt[3]{1 + x} - \sqrt[3]{x}, \sqrt[3]{1 + x}, {\left(\sqrt[3]{x}\right)}^{2}\right)}{\color{blue}{\mathsf{fma}\left(x, 2, 1\right)}}}} \]

8. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	53056

\[\begin{array}{l} t_0 := {\left(\sqrt[3]{x}\right)}^{2}\\ t_1 := \sqrt[3]{1 + x}\\ \frac{1}{t_0 + \frac{t_1}{\frac{1}{1 + \left(x + x\right)} \cdot \left(t_0 + t_1 \cdot \left(t_1 - \sqrt[3]{x}\right)\right)}} \end{array} \]

Alternative 2
Error	25.1
Cost	39108

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log t_1}\\ \end{array} \]

Alternative 3
Error	7.5
Cost	33096

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ t_2 := \sqrt[3]{x} \cdot t_1\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + t_2}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_1 + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_2 + e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}}\\ \end{array} \]

Alternative 4
Error	0.5
Cost	33024

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

Alternative 5
Error	0.5
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 6
Error	13.3
Cost	26825

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

Alternative 7
Error	25.1
Cost	26632

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;e^{\log \left(t_0 - \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + t_0 \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \]

Alternative 8
Error	25.1
Cost	20169

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+15} \lor \neg \left(x \leq 1.95 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \sqrt[3]{x}\\ \end{array} \]

Alternative 9
Error	25.8
Cost	13449

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+15} \lor \neg \left(x \leq 2.2 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{1}{1 + {\left(\sqrt[3]{x}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array} \]

Alternative 10
Error	30.0
Cost	13120

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

Alternative 11
Error	61.7
Cost	64

\[0 \]

Alternative 12
Error	32.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023017 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))