2cbrt (problem 3.3.4)

Average Accuracy: 53.8% → 99.2%

Time: 9.5s

Precision: binary64

Cost: 39424.00

Math

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

↓

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

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

Derivation

1. Initial program 53.8%

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

2. Applied egg-rr 54.9%

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

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] 54.9	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate-*r/ [=>] 54.9	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   *-rgt-identity [=>] 54.9	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 54.9	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   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 48.5%

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

5. Simplified 49.5%

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

   [Start] 48.5	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + {\left(\sqrt{x}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{x}\right)}^{0.3333333333333333}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   unpow1/3 [=>] 48.9	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \color{blue}{\sqrt[3]{\sqrt{x}}} \cdot {\left(\sqrt{x}\right)}^{0.3333333333333333}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   unpow1/3 [=>] 49.5	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{\sqrt{x}} \cdot \color{blue}{\sqrt[3]{\sqrt{x}}}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]

6. Applied egg-rr 99.2%

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

7. Simplified 99.2%

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

   [Start] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \left(\left(1 + \sqrt[3]{x}\right) - 1\right), {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   associate--l+ [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \color{blue}{\left(1 + \left(\sqrt[3]{x} - 1\right)\right)}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]

8. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	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
Accuracy	88.7%
Cost	33096.00

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

Alternative 3
Accuracy	88.7%
Cost	33032.00

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

Alternative 4
Accuracy	99.1%
Cost	32896.00

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

Alternative 5
Accuracy	80.3%
Cost	26825.00

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

Alternative 6
Accuracy	59.1%
Cost	26368.00

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

Alternative 7
Accuracy	59.1%
Cost	26176.00

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

Alternative 8
Accuracy	58.0%
Cost	19648.00

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

Alternative 9
Accuracy	60.1%
Cost	13449.00

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+15} \lor \neg \left(x \leq 2.3 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{1}{1 + {\left(\sqrt[3]{x}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array} \]

Alternative 10
Accuracy	53.8%
Cost	13120.00

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

Alternative 11
Accuracy	51.0%
Cost	6592.00

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

Alternative 12
Accuracy	3.6%
Cost	64.00

\[0 \]

Alternative 13
Accuracy	50.2%
Cost	64.00

\[1 \]

Reproduce

herbie shell --seed 2023096 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))