2cbrt (problem 3.3.4)

Average Error: 30.0 → 0.6

Time: 10.6s

Precision: binary64

Cost: 52480

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Derivation

1. Initial program 30.0

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

2. Applied egg-rr 30.0

   \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right)} \]

3. Applied egg-rr 30.6

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

4. Simplified 12.7

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

   [Start] 30.6	\[ \frac{{\left(1 + x\right)}^{2} - x \cdot x}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right) \cdot \left(1 + \left(x + x\right)\right)} \]
   unpow2 [=>] 30.6	\[ \frac{\color{blue}{\left(1 + x\right) \cdot \left(1 + x\right)} - x \cdot x}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right) \cdot \left(1 + \left(x + x\right)\right)} \]
   difference-of-squares [=>] 29.2	\[ \frac{\color{blue}{\left(\left(1 + x\right) + x\right) \cdot \left(\left(1 + x\right) - x\right)}}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right) \cdot \left(1 + \left(x + x\right)\right)} \]
   associate-+r+ [<=] 29.2	\[ \frac{\color{blue}{\left(1 + \left(x + x\right)\right)} \cdot \left(\left(1 + x\right) - x\right)}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right) \cdot \left(1 + \left(x + x\right)\right)} \]
   associate-+r- [<=] 12.7	\[ \frac{\left(1 + \left(x + x\right)\right) \cdot \color{blue}{\left(1 + \left(x - x\right)\right)}}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right) \cdot \left(1 + \left(x + x\right)\right)} \]
   +-commutative [=>] 12.7	\[ \frac{\left(1 + \left(x + x\right)\right) \cdot \color{blue}{\left(\left(x - x\right) + 1\right)}}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right) \cdot \left(1 + \left(x + x\right)\right)} \]
   +-inverses [=>] 12.7	\[ \frac{\left(1 + \left(x + x\right)\right) \cdot \left(\color{blue}{0} + 1\right)}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right) \cdot \left(1 + \left(x + x\right)\right)} \]
   metadata-eval [=>] 12.7	\[ \frac{\left(1 + \left(x + x\right)\right) \cdot \color{blue}{1}}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right) \cdot \left(1 + \left(x + x\right)\right)} \]
   +-commutative [=>] 12.7	\[ \frac{\color{blue}{\left(\left(x + x\right) + 1\right)} \cdot 1}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right) \cdot \left(1 + \left(x + x\right)\right)} \]
   count-2 [=>] 12.7	\[ \frac{\left(\color{blue}{2 \cdot x} + 1\right) \cdot 1}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right) \cdot \left(1 + \left(x + x\right)\right)} \]
   fma-def [=>] 12.7	\[ \frac{\color{blue}{\mathsf{fma}\left(2, x, 1\right)} \cdot 1}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\right) \cdot \left(1 + \left(x + x\right)\right)} \]
   +-commutative [=>] 12.7	\[ \frac{\mathsf{fma}\left(2, x, 1\right) \cdot 1}{\color{blue}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \cdot \left(1 + \left(x + x\right)\right)} \]
   fma-def [=>] 12.7	\[ \frac{\mathsf{fma}\left(2, x, 1\right) \cdot 1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \cdot \left(1 + \left(x + x\right)\right)} \]
   +-commutative [=>] 12.7	\[ \frac{\mathsf{fma}\left(2, x, 1\right) \cdot 1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right) \cdot \color{blue}{\left(\left(x + x\right) + 1\right)}} \]
   count-2 [=>] 12.7	\[ \frac{\mathsf{fma}\left(2, x, 1\right) \cdot 1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right) \cdot \left(\color{blue}{2 \cdot x} + 1\right)} \]
   fma-def [=>] 12.7	\[ \frac{\mathsf{fma}\left(2, x, 1\right) \cdot 1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right) \cdot \color{blue}{\mathsf{fma}\left(2, x, 1\right)}} \]

5. Applied egg-rr 0.6

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

6. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.6
Cost	46208

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

Alternative 2
Error	0.6
Cost	46208

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

Alternative 3
Error	0.5
Cost	39168

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

Alternative 4
Error	7.4
Cost	33160

\[\begin{array}{l} t_0 := \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + t_0}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_0 + \sqrt[3]{{\left(x + 1\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 + e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}}\\ \end{array} \]

Alternative 5
Error	0.5
Cost	33024

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

Alternative 6
Error	14.0
Cost	32964

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

Alternative 7
Error	0.5
Cost	32896

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

Alternative 8
Error	25.2
Cost	26632

\[\begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \]

Alternative 9
Error	25.2
Cost	26441

\[\begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+14} \lor \neg \left(x \leq 9.2 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \end{array} \]

Alternative 10
Error	25.1
Cost	26185

\[\begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+15} \lor \neg \left(x \leq 1.46 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(t_0 - \sqrt[3]{x}\right)}\\ \end{array} \]

Alternative 11
Error	25.1
Cost	20169

\[\begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+15} \lor \neg \left(x \leq 1.46 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \sqrt[3]{x}\\ \end{array} \]

Alternative 12
Error	30.0
Cost	13120

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

Alternative 13
Error	61.7
Cost	64

\[0 \]

Alternative 14
Error	32.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023046 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))