2cbrt (problem 3.3.4)

Average Error: 46.23% → 0.86%

Time: 9.2s

Precision: binary64

Cost: 52032

Math

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

↓

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

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) (pow (cbrt (pow t_0 3.0)) 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), pow(cbrt(pow(t_0, 3.0)), 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), (cbrt((t_0 ^ 3.0)) ^ 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[Power[N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 46.23

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

2. Applied egg-rr 45.18

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

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] 45.18	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate-*r/ [=>] 45.18	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   *-rgt-identity [=>] 45.18	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 45.18	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate--l+ [=>] 0.83	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-inverses [=>] 0.83	\[ \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   metadata-eval [=>] 0.83	\[ \frac{\color{blue}{1}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 0.83	\[ \frac{1}{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {\left(\sqrt[3]{x + 1}\right)}^{2}}} \]
   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 26.72

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(e^{\mathsf{log1p}\left(x\right) \cdot 0.3333333333333333}\right)}}^{2}\right)} \]

5. Simplified 26.49

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(\sqrt[3]{e^{\mathsf{log1p}\left(x\right)}}\right)}}^{2}\right)} \]
   Proof

   [Start] 26.72	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\mathsf{log1p}\left(x\right) \cdot 0.3333333333333333}\right)}^{2}\right)} \]
   exp-prod [=>] 26.66	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left({\left(e^{\mathsf{log1p}\left(x\right)}\right)}^{0.3333333333333333}\right)}}^{2}\right)} \]
   unpow1/3 [=>] 26.49	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(\sqrt[3]{e^{\mathsf{log1p}\left(x\right)}}\right)}}^{2}\right)} \]

6. Applied egg-rr 0.86

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

7. Final simplification 0.86

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

Alternatives

Alternative 1
Error	0.83%
Cost	45956

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + t_0 \cdot \left(\sqrt[3]{x} + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x \cdot x} + \frac{t_0}{\frac{1}{\sqrt[3]{x} + t_0}}}\\ \end{array} \]

Alternative 2
Error	38.8%
Cost	39556

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + t_0 \cdot \left(1 + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.82%
Cost	39168

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

Alternative 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	11.53%
Cost	26952

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + t_0 \cdot \left(1 + \sqrt[3]{x}\right)}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x \cdot x} + \frac{t_0}{\frac{1}{t_1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_1 + {x}^{0.6666666666666666}}\\ \end{array} \]

Alternative 6
Error	11.52%
Cost	26824

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 \cdot \left(\sqrt[3]{x} + t_0\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + t_0 \cdot \left(1 + \sqrt[3]{x}\right)}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_1 + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + {x}^{0.6666666666666666}}\\ \end{array} \]

Alternative 7
Error	21.32%
Cost	26760

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + t_0 \cdot \left(1 + \sqrt[3]{x}\right)}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;{\left({\left(t_0 - \sqrt[3]{x}\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot \left(\sqrt[3]{x} + t_0\right) + {x}^{0.6666666666666666}}\\ \end{array} \]

Alternative 8
Error	39.86%
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 9
Error	43.14%
Cost	26308

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	39.87%
Cost	19912

\[\begin{array}{l} \mathbf{if}\;x \leq -1.82 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{1 + {\left(\sqrt[3]{x}\right)}^{2}}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, 1 + \sqrt[3]{x}, 1\right)}\\ \end{array} \]

Alternative 11
Error	47.2%
Cost	6848

\[\frac{1}{1 + \sqrt[3]{x \cdot x}} \]

Alternative 12
Error	96.36%
Cost	64

\[0 \]

Alternative 13
Error	49.63%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))