2cbrt (problem 3.3.4)

Average Accuracy: 53.3% → 99.2%

Time: 10.7s

Precision: binary64

Cost: 58944

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Derivation

1. Initial program 53.3%

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

2. Applied egg-rr 54.3%

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

   [Start] 53.3	\[ \sqrt[3]{x + 1} - \sqrt[3]{x} \]
   flip3-- [=>] 53.4	\[ \color{blue}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}} \]
   div-inv [=>] 53.4	\[ \color{blue}{\left({\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}} \]
   rem-cube-cbrt [=>] 53.2	\[ \left(\color{blue}{\left(x + 1\right)} - {\left(\sqrt[3]{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
   rem-cube-cbrt [=>] 54.3	\[ \left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
   pow2 [=>] 54.3	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{2}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
   distribute-rgt-out [=>] 54.3	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}} \]
   +-commutative [<=] 54.3	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \color{blue}{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

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] 54.3	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate-*r/ [=>] 54.3	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   *-rgt-identity [=>] 54.3	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 54.3	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate--l+ [=>] 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 73.6%

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

   [Start] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   pow1/3 [=>] 73.6	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left({\left(1 + x\right)}^{0.3333333333333333}\right)}}^{2}\right)} \]
   pow-to-exp [=>] 73.6	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(e^{\log \left(1 + x\right) \cdot 0.3333333333333333}\right)}}^{2}\right)} \]
   log1p-def [=>] 73.6	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\color{blue}{\mathsf{log1p}\left(x\right)} \cdot 0.3333333333333333}\right)}^{2}\right)} \]

5. Simplified 73.8%

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

   [Start] 73.6	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\mathsf{log1p}\left(x\right) \cdot 0.3333333333333333}\right)}^{2}\right)} \]
   exp-prod [=>] 73.7	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left({\left(e^{\mathsf{log1p}\left(x\right)}\right)}^{0.3333333333333333}\right)}}^{2}\right)} \]
   unpow1/3 [=>] 73.8	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(\sqrt[3]{e^{\mathsf{log1p}\left(x\right)}}\right)}}^{2}\right)} \]

6. Taylor expanded in x around -inf 0.0%

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

7. Simplified 99.2%

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

   [Start] 0.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{e^{-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1} + \frac{e^{-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1}}{x}}\right)}^{2}\right)} \]
   exp-sum [=>] 0.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\color{blue}{e^{-1 \cdot \log \left(\frac{-1}{x}\right)} \cdot e^{\log -1}} + \frac{e^{-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1}}{x}}\right)}^{2}\right)} \]
   rem-exp-log [=>] 0.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{e^{-1 \cdot \log \left(\frac{-1}{x}\right)} \cdot \color{blue}{-1} + \frac{e^{-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1}}{x}}\right)}^{2}\right)} \]
   fma-def [=>] 0.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(e^{-1 \cdot \log \left(\frac{-1}{x}\right)}, -1, \frac{e^{-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1}}{x}\right)}}\right)}^{2}\right)} \]
   *-commutative [=>] 0.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\mathsf{fma}\left(e^{\color{blue}{\log \left(\frac{-1}{x}\right) \cdot -1}}, -1, \frac{e^{-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1}}{x}\right)}\right)}^{2}\right)} \]
   exp-to-pow [=>] 0.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\mathsf{fma}\left(\color{blue}{{\left(\frac{-1}{x}\right)}^{-1}}, -1, \frac{e^{-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1}}{x}\right)}\right)}^{2}\right)} \]
   exp-sum [=>] 0.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\mathsf{fma}\left({\left(\frac{-1}{x}\right)}^{-1}, -1, \frac{\color{blue}{e^{-1 \cdot \log \left(\frac{-1}{x}\right)} \cdot e^{\log -1}}}{x}\right)}\right)}^{2}\right)} \]
   rem-exp-log [=>] 47.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\mathsf{fma}\left({\left(\frac{-1}{x}\right)}^{-1}, -1, \frac{e^{-1 \cdot \log \left(\frac{-1}{x}\right)} \cdot \color{blue}{-1}}{x}\right)}\right)}^{2}\right)} \]
   associate-*l/ [<=] 47.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\mathsf{fma}\left({\left(\frac{-1}{x}\right)}^{-1}, -1, \color{blue}{\frac{e^{-1 \cdot \log \left(\frac{-1}{x}\right)}}{x} \cdot -1}\right)}\right)}^{2}\right)} \]
   *-commutative [=>] 47.0	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\mathsf{fma}\left({\left(\frac{-1}{x}\right)}^{-1}, -1, \frac{e^{\color{blue}{\log \left(\frac{-1}{x}\right) \cdot -1}}}{x} \cdot -1\right)}\right)}^{2}\right)} \]
   exp-to-pow [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\mathsf{fma}\left({\left(\frac{-1}{x}\right)}^{-1}, -1, \frac{\color{blue}{{\left(\frac{-1}{x}\right)}^{-1}}}{x} \cdot -1\right)}\right)}^{2}\right)} \]

8. Final simplification 99.2%

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

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	79.4%
Cost	33232

\[\begin{array}{l} t_0 := \sqrt[3]{x \cdot x}\\ t_1 := \sqrt[3]{1 + x}\\ t_2 := \sqrt[3]{x} + t_1\\ t_3 := \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_2, 1\right)}\\ t_4 := \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_2, t_0\right)}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+15}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq 10000000000000:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{t_1}^{2} + \left(t_0 + \sqrt[3]{x \cdot \left(1 + x\right)}\right)}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Accuracy	87.7%
Cost	33032

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

Alternative 4
Accuracy	99.2%
Cost	32896

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

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	61.8%
Cost	27145

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -9 \cdot 10^{+15} \lor \neg \left(x \leq 9 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{t_0}^{2} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{x \cdot \left(1 + x\right)}\right)}\\ \end{array} \]

Alternative 7
Accuracy	58.8%
Cost	26176

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

Alternative 8
Accuracy	53.3%
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 2023137 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))