2cbrt (problem 3.3.4)

Average Accuracy: 53.2% → 99.2%

Time: 11.9s

Precision: binary64

Cost: 39296

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 53.2%

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

2. Applied egg-rr 54.2%

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

   [Start] 53.2	\[ \sqrt[3]{x + 1} - \sqrt[3]{x} \]
   flip3-- [=>] 53.2	\[ \color{blue}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}} \]
   div-inv [=>] 53.2	\[ \color{blue}{\left({\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}} \]
   rem-cube-cbrt [=>] 53.1	\[ \left(\color{blue}{\left(x + 1\right)} - {\left(\sqrt[3]{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
   rem-cube-cbrt [=>] 54.2	\[ \left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
   pow2 [=>] 54.2	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{2}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
   distribute-rgt-out [=>] 54.2	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}} \]
   +-commutative [<=] 54.2	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \color{blue}{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

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] 54.2	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate-*r/ [=>] 54.2	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   *-rgt-identity [=>] 54.2	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 54.2	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   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 76.0%

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

   [Start] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   flip-+ [=>] 75.9	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}}}\right)}^{2}\right)} \]
   cbrt-div [=>] 76.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(\frac{\sqrt[3]{1 \cdot 1 - x \cdot x}}{\sqrt[3]{1 - x}}\right)}}^{2}\right)} \]
   metadata-eval [=>] 76.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\frac{\sqrt[3]{\color{blue}{1} - x \cdot x}}{\sqrt[3]{1 - x}}\right)}^{2}\right)} \]

5. Applied egg-rr 99.2%

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

   [Start] 76.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\frac{\sqrt[3]{1 - x \cdot x}}{\sqrt[3]{1 - x}}\right)}^{2}\right)} \]
   clear-num [=>] 76.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(\frac{1}{\frac{\sqrt[3]{1 - x}}{\sqrt[3]{1 - x \cdot x}}}\right)}}^{2}\right)} \]
   inv-pow [=>] 76.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left({\left(\frac{\sqrt[3]{1 - x}}{\sqrt[3]{1 - x \cdot x}}\right)}^{-1}\right)}}^{2}\right)} \]
   clear-num [=>] 76.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left({\color{blue}{\left(\frac{1}{\frac{\sqrt[3]{1 - x \cdot x}}{\sqrt[3]{1 - x}}}\right)}}^{-1}\right)}^{2}\right)} \]
   cbrt-undiv [=>] 75.9	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left({\left(\frac{1}{\color{blue}{\sqrt[3]{\frac{1 - x \cdot x}{1 - x}}}}\right)}^{-1}\right)}^{2}\right)} \]
   metadata-eval [<=] 75.9	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left({\left(\frac{1}{\sqrt[3]{\frac{\color{blue}{1 \cdot 1} - x \cdot x}{1 - x}}}\right)}^{-1}\right)}^{2}\right)} \]
   flip-+ [<=] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left({\left(\frac{1}{\sqrt[3]{\color{blue}{1 + x}}}\right)}^{-1}\right)}^{2}\right)} \]

6. Applied egg-rr 99.2%

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

   [Start] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left({\left(\frac{1}{\sqrt[3]{1 + x}}\right)}^{-1}\right)}^{2}\right)} \]
   unpow2 [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\frac{1}{\sqrt[3]{1 + x}}\right)}^{-1} \cdot {\left(\frac{1}{\sqrt[3]{1 + x}}\right)}^{-1}}\right)} \]
   unpow-1 [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\frac{1}{\sqrt[3]{1 + x}}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{1}{\sqrt[3]{1 + x}}}}\right)} \]
   unpow-1 [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{\frac{1}{\frac{1}{\sqrt[3]{1 + x}}}} \cdot \frac{1}{\frac{1}{\sqrt[3]{1 + x}}}\right)} \]
   frac-times [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{\frac{1 \cdot 1}{\frac{1}{\sqrt[3]{1 + x}} \cdot \frac{1}{\sqrt[3]{1 + x}}}}\right)} \]
   metadata-eval [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \frac{\color{blue}{1}}{\frac{1}{\sqrt[3]{1 + x}} \cdot \frac{1}{\sqrt[3]{1 + x}}}\right)} \]
   *-un-lft-identity [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \frac{1}{\color{blue}{1 \cdot \left(\frac{1}{\sqrt[3]{1 + x}} \cdot \frac{1}{\sqrt[3]{1 + x}}\right)}}\right)} \]
   metadata-eval [<=] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \frac{1}{\color{blue}{\frac{1}{1}} \cdot \left(\frac{1}{\sqrt[3]{1 + x}} \cdot \frac{1}{\sqrt[3]{1 + x}}\right)}\right)} \]
   associate-/r/ [<=] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \frac{1}{\color{blue}{\frac{1}{\frac{1}{\frac{1}{\sqrt[3]{1 + x}} \cdot \frac{1}{\sqrt[3]{1 + x}}}}}}\right)} \]
   metadata-eval [<=] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \frac{1}{\frac{\color{blue}{1 \cdot 1}}{\frac{1}{\frac{1}{\sqrt[3]{1 + x}} \cdot \frac{1}{\sqrt[3]{1 + x}}}}}\right)} \]
   metadata-eval [<=] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \frac{1}{\frac{1 \cdot 1}{\frac{\color{blue}{1 \cdot 1}}{\frac{1}{\sqrt[3]{1 + x}} \cdot \frac{1}{\sqrt[3]{1 + x}}}}}\right)} \]
   frac-times [<=] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \frac{1}{\frac{1 \cdot 1}{\color{blue}{\frac{1}{\frac{1}{\sqrt[3]{1 + x}}} \cdot \frac{1}{\frac{1}{\sqrt[3]{1 + x}}}}}}\right)} \]
   unpow-1 [<=] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \frac{1}{\frac{1 \cdot 1}{\color{blue}{{\left(\frac{1}{\sqrt[3]{1 + x}}\right)}^{-1}} \cdot \frac{1}{\frac{1}{\sqrt[3]{1 + x}}}}}\right)} \]
   unpow-1 [<=] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \frac{1}{\frac{1 \cdot 1}{{\left(\frac{1}{\sqrt[3]{1 + x}}\right)}^{-1} \cdot \color{blue}{{\left(\frac{1}{\sqrt[3]{1 + x}}\right)}^{-1}}}}\right)} \]
   frac-times [<=] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \frac{1}{\color{blue}{\frac{1}{{\left(\frac{1}{\sqrt[3]{1 + x}}\right)}^{-1}} \cdot \frac{1}{{\left(\frac{1}{\sqrt[3]{1 + x}}\right)}^{-1}}}}\right)} \]

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
Accuracy	99.2%
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 2
Accuracy	60.7%
Cost	39108

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	87.7%
Cost	33096

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

Alternative 4
Accuracy	60.7%
Cost	33092

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

Alternative 5
Accuracy	99.2%
Cost	32896

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

Alternative 6
Accuracy	77.4%
Cost	26829

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + t_1}\\ \mathbf{elif}\;x \leq -32000000 \lor \neg \left(x \leq 32000000\right):\\ \;\;\;\;\frac{1}{t_1 + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_0 - \sqrt[3]{x}\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	87.6%
Cost	26760

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

Alternative 8
Accuracy	53.2%
Cost	13120

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

Alternative 9
Accuracy	3.6%
Cost	64

\[0 \]

Alternative 10
Accuracy	49.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))