2cbrt (problem 3.3.4)

Average Error: 29.5 → 0.3

Time: 8.8s

Precision: binary64

Cost: 52928

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := {t_0}^{2}\\ \frac{1}{t_1 + \frac{\sqrt[3]{x}}{t_1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} - t_0\right)} \cdot \left(x + \left(1 + x\right)\right)} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (pow t_0 2.0)))
   (/
    1.0
    (+
     t_1
     (* (/ (cbrt x) (+ t_1 (* (cbrt x) (- (cbrt x) t_0)))) (+ x (+ 1.0 x)))))))

C

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

↓

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = pow(t_0, 2.0);
	return 1.0 / (t_1 + ((cbrt(x) / (t_1 + (cbrt(x) * (cbrt(x) - t_0)))) * (x + (1.0 + x))));
}

Java

public static double code(double x) {
	return Math.cbrt((x + 1.0)) - Math.cbrt(x);
}

↓

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double t_1 = Math.pow(t_0, 2.0);
	return 1.0 / (t_1 + ((Math.cbrt(x) / (t_1 + (Math.cbrt(x) * (Math.cbrt(x) - t_0)))) * (x + (1.0 + x))));
}

Julia

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

↓

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = t_0 ^ 2.0
	return Float64(1.0 / Float64(t_1 + Float64(Float64(cbrt(x) / Float64(t_1 + Float64(cbrt(x) * Float64(cbrt(x) - t_0)))) * Float64(x + Float64(1.0 + x)))))
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]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, N[(1.0 / N[(t$95$1 + N[(N[(N[Power[x, 1/3], $MachinePrecision] / N[(t$95$1 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 29.5

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

2. Applied egg-rr 28.8

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

3. Taylor expanded in x around 0 0.5

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

4. Applied egg-rr 0.3

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

5. Simplified 0.3

   \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \color{blue}{\frac{\sqrt[3]{x}}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right)} \cdot \left(x + \left(1 + x\right)\right)}} \]
   Proof
   (/.f64 1 (+.f64 (pow.f64 (cbrt.f64 (+.f64 x 1)) 2) (*.f64 (/.f64 (cbrt.f64 x) (+.f64 (pow.f64 (cbrt.f64 (+.f64 1 x)) 2) (*.f64 (cbrt.f64 x) (-.f64 (cbrt.f64 x) (cbrt.f64 (+.f64 1 x)))))) (+.f64 x (+.f64 1 x))))): 0 points increase in error, 0 points decrease in error
   (/.f64 1 (+.f64 (pow.f64 (cbrt.f64 (+.f64 x 1)) 2) (*.f64 (/.f64 (cbrt.f64 x) (+.f64 (pow.f64 (cbrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1))) 2) (*.f64 (cbrt.f64 x) (-.f64 (cbrt.f64 x) (cbrt.f64 (+.f64 1 x)))))) (+.f64 x (+.f64 1 x))))): 0 points increase in error, 5 points decrease in error
   (/.f64 1 (+.f64 (pow.f64 (cbrt.f64 (+.f64 x 1)) 2) (*.f64 (/.f64 (cbrt.f64 x) (+.f64 (pow.f64 (cbrt.f64 (+.f64 x 1)) 2) (*.f64 (cbrt.f64 x) (-.f64 (cbrt.f64 x) (cbrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1))))))) (+.f64 x (+.f64 1 x))))): 4 points increase in error, 0 points decrease in error
   (/.f64 1 (+.f64 (pow.f64 (cbrt.f64 (+.f64 x 1)) 2) (*.f64 (/.f64 (cbrt.f64 x) (+.f64 (pow.f64 (cbrt.f64 (+.f64 x 1)) 2) (*.f64 (cbrt.f64 x) (-.f64 (cbrt.f64 x) (cbrt.f64 (+.f64 x 1)))))) (+.f64 x (Rewrite<= +-commutative_binary64 (+.f64 x 1)))))): 0 points increase in error, 4 points decrease in error
   (/.f64 1 (+.f64 (pow.f64 (cbrt.f64 (+.f64 x 1)) 2) (Rewrite<= associate-/r/_binary64 (/.f64 (cbrt.f64 x) (/.f64 (+.f64 (pow.f64 (cbrt.f64 (+.f64 x 1)) 2) (*.f64 (cbrt.f64 x) (-.f64 (cbrt.f64 x) (cbrt.f64 (+.f64 x 1))))) (+.f64 x (+.f64 x 1))))))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.5
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
Error	7.3
Cost	33096

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_1, 1\right)}\\ \mathbf{elif}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot \left(1 + x\right)} + \sqrt[3]{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot t_1 + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \]

Alternative 3
Error	0.5
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 4
Error	13.9
Cost	26756

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

Alternative 5
Error	24.7
Cost	26441

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+15} \lor \neg \left(x \leq 2.12 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1 - x \cdot x}{1 - x}} - \sqrt[3]{x}\\ \end{array} \]

Alternative 6
Error	27.1
Cost	26244

\[\begin{array}{l} \mathbf{if}\;x \leq 2.12 \cdot 10^{+15}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + {x}^{0.3333333333333333}, 1\right)}\\ \end{array} \]

Alternative 7
Error	29.5
Cost	25920

\[e^{\log \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)} \]

Alternative 8
Error	29.5
Cost	13120

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

Alternative 9
Error	61.7
Cost	64

\[0 \]

Alternative 10
Error	31.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))