2cbrt (problem 3.3.4)

Average Error: 29.6 → 0.3

Time: 10.2s

Precision: binary64

Cost: 59200

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Derivation

1. Initial program 29.6

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

2. Applied egg-rr 29.0

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

3. Taylor expanded in x around 0 0.5

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

4. Applied egg-rr 0.3

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

5. Simplified 0.3

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

   [Start] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \frac{\sqrt[3]{x}}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)}{x + \left(x + 1\right)}}} \]
   associate-/l* [<=] 7.7	\[ \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \color{blue}{\frac{\sqrt[3]{x} \cdot \left(x + \left(x + 1\right)\right)}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)}}} \]
   *-commutative [<=] 7.7	\[ \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \frac{\color{blue}{\left(x + \left(x + 1\right)\right) \cdot \sqrt[3]{x}}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)}} \]
   associate-/l* [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \color{blue}{\frac{x + \left(x + 1\right)}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)}{\sqrt[3]{x}}}}} \]
   associate-/r/ [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \color{blue}{\frac{x + \left(x + 1\right)}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)} \cdot \sqrt[3]{x}}} \]
   +-commutative [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \frac{x + \color{blue}{\left(1 + x\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)} \cdot \sqrt[3]{x}} \]
   +-commutative [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \frac{x + \left(1 + x\right)}{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right) + {\left(\sqrt[3]{x + 1}\right)}^{2}}} \cdot \sqrt[3]{x}} \]
   fma-def [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \frac{x + \left(1 + x\right)}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} - \sqrt[3]{x + 1}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)}} \cdot \sqrt[3]{x}} \]
   +-commutative [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \frac{x + \left(1 + x\right)}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} - \sqrt[3]{\color{blue}{1 + x}}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)} \cdot \sqrt[3]{x}} \]
   +-commutative [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \frac{x + \left(1 + x\right)}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} - \sqrt[3]{1 + x}, {\left(\sqrt[3]{\color{blue}{1 + x}}\right)}^{2}\right)} \cdot \sqrt[3]{x}} \]

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	52928

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

Alternative 2
Error	24.8
Cost	39364

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0 + \sqrt[3]{x}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1 + {x}^{3}}{1 + \left(x \cdot x - x\right)}} - \sqrt[3]{x}\\ \end{array} \]

Alternative 3
Error	0.5
Cost	39168

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

Alternative 4
Error	7.4
Cost	33096

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 + \sqrt[3]{x}\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_1, 1\right)}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot \left(1 + x\right)} + \sqrt[3]{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot t_1 + e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}}\\ \end{array} \]

Alternative 5
Error	0.5
Cost	32896

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

Alternative 6
Error	0.5
Cost	32896

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

Alternative 7
Error	13.0
Cost	26889

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+154} \lor \neg \left(x \leq 1.32 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0 + \sqrt[3]{x}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot \left(1 + x\right)} + \sqrt[3]{x \cdot x}\right)}\\ \end{array} \]

Alternative 8
Error	29.6
Cost	13120

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

Alternative 9
Error	61.7
Cost	64

\[0 \]

Alternative 10
Error	31.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023083 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))