2cbrt (problem 3.3.4)

Average Error: 29.9 → 0.7

Time: 8.9s

Precision: binary64

Cost: 33160

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x}\right)}\\ \mathbf{if}\;x \leq -1.9801969054324128 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.226820418126073 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x)))
        (t_1 (/ 1.0 (+ (pow t_0 2.0) (* (cbrt x) (+ (cbrt x) (cbrt x)))))))
   (if (<= x -1.9801969054324128e+157)
     t_1
     (if (<= x 2.226820418126073e+146)
       (/ 1.0 (+ (* (cbrt x) (+ t_0 (cbrt x))) (cbrt (pow (+ 1.0 x) 2.0))))
       t_1))))

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 = 1.0 / (pow(t_0, 2.0) + (cbrt(x) * (cbrt(x) + cbrt(x))));
	double tmp;
	if (x <= -1.9801969054324128e+157) {
		tmp = t_1;
	} else if (x <= 2.226820418126073e+146) {
		tmp = 1.0 / ((cbrt(x) * (t_0 + cbrt(x))) + cbrt(pow((1.0 + x), 2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

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 = 1.0 / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (Math.cbrt(x) + Math.cbrt(x))));
	double tmp;
	if (x <= -1.9801969054324128e+157) {
		tmp = t_1;
	} else if (x <= 2.226820418126073e+146) {
		tmp = 1.0 / ((Math.cbrt(x) * (t_0 + Math.cbrt(x))) + Math.cbrt(Math.pow((1.0 + x), 2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

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 = Float64(1.0 / Float64((t_0 ^ 2.0) + Float64(cbrt(x) * Float64(cbrt(x) + cbrt(x)))))
	tmp = 0.0
	if (x <= -1.9801969054324128e+157)
		tmp = t_1;
	elseif (x <= 2.226820418126073e+146)
		tmp = Float64(1.0 / Float64(Float64(cbrt(x) * Float64(t_0 + cbrt(x))) + cbrt((Float64(1.0 + x) ^ 2.0))));
	else
		tmp = t_1;
	end
	return tmp
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[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.9801969054324128e+157], t$95$1, If[LessEqual[x, 2.226820418126073e+146], N[(1.0 / N[(N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9801969054324128e157 or 2.226820418126073e146 < x

   1. Initial program 61.0

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

   2. Applied egg-rr 61.0

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

   3. Taylor expanded in x around 0 1.0

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

   4. Applied egg-rr 1.1

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

   5. Taylor expanded in x around inf 34.0

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

   6. Simplified 1.0

      \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\color{blue}{\sqrt[3]{x}} + \sqrt[3]{x}\right)} \]
      Proof
      (cbrt.f64 x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unpow1/3_binary64 (pow.f64 x 1/3)): 122 points increase in error, 132 points decrease in error

   if -1.9801969054324128e157 < x < 2.226820418126073e146

   1. Initial program 19.2

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

   2. Applied egg-rr 18.3

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

   3. Taylor expanded in x around 0 0.4

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

   4. Applied egg-rr 0.4

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\left(\sqrt[3]{x + 1}\right)}^{4}}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   5. Applied egg-rr 0.6

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{{\left(x + 1\right)}^{2}}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9801969054324128 \cdot 10^{+157}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x}\right)}\\ \mathbf{elif}\;x \leq 2.226820418126073 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	45956

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right) + {\left(1 + x\right)}^{0.6666666666666666}}\\ \end{array} \]

Alternative 2
Error	25.1
Cost	39236

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({t_1}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternative 3
Error	25.1
Cost	33092

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

Alternative 4
Error	0.5
Cost	32896

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

Alternative 5
Error	14.2
Cost	26628

\[\begin{array}{l} t_0 := \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\\ \mathbf{if}\;x \leq -4279664386679.3286:\\ \;\;\;\;\frac{1}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 + {\left(1 + x\right)}^{0.6666666666666666}}\\ \end{array} \]

Alternative 6
Error	26.1
Cost	26368

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

Alternative 7
Error	29.9
Cost	13120

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

Alternative 8
Error	61.7
Cost	64

\[0 \]

Alternative 9
Error	31.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022317 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))