2cbrt (problem 3.3.4)

Percentage Accurate: 6.7% → 98.3%

Time: 21.9s

Alternatives: 7

Speedup: 2.0×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

\[\begin{array}{l} \\ \sqrt[3]{x + 1} - \sqrt[3]{x} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

Initial Program: 6.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{x + 1} - \sqrt[3]{x} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 10^{-10}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{{\left(x + 1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{x \cdot \left(x + 1\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 1e-10)
   (* 0.3333333333333333 (pow (cbrt x) -2.0))
   (/
    (- (+ x 1.0) x)
    (+
     (pow (+ x 1.0) 0.6666666666666666)
     (+ (pow x 0.6666666666666666) (cbrt (* x (+ x 1.0))))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((x + 1.0)) - cbrt(x)) <= 1e-10) {
		tmp = 0.3333333333333333 * pow(cbrt(x), -2.0);
	} else {
		tmp = ((x + 1.0) - x) / (pow((x + 1.0), 0.6666666666666666) + (pow(x, 0.6666666666666666) + cbrt((x * (x + 1.0)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((Math.cbrt((x + 1.0)) - Math.cbrt(x)) <= 1e-10) {
		tmp = 0.3333333333333333 * Math.pow(Math.cbrt(x), -2.0);
	} else {
		tmp = ((x + 1.0) - x) / (Math.pow((x + 1.0), 0.6666666666666666) + (Math.pow(x, 0.6666666666666666) + Math.cbrt((x * (x + 1.0)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(x + 1.0)) - cbrt(x)) <= 1e-10)
		tmp = Float64(0.3333333333333333 * (cbrt(x) ^ -2.0));
	else
		tmp = Float64(Float64(Float64(x + 1.0) - x) / Float64((Float64(x + 1.0) ^ 0.6666666666666666) + Float64((x ^ 0.6666666666666666) + cbrt(Float64(x * Float64(x + 1.0))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 1e-10], N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision] + N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[Power[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 1.00000000000000004e-10

   1. Initial program 4.3%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 50.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 50.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   6. Step-by-step derivation

      1. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{x \cdot x}}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\sqrt[3]{\color{blue}{x \cdot x}}}\right)\right) \]

      3. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\sqrt[3]{x \cdot x}\right)}^{\color{blue}{-1}}\right)\right) \]

      4. cbrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{-1}\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{-1}\right)\right) \]

      6. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\sqrt[3]{x}\right)}^{\color{blue}{\left(2 \cdot -1\right)}}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\left(\sqrt[3]{x}\right), \color{blue}{\left(2 \cdot -1\right)}\right)\right) \]

      8. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(x\right), \left(\color{blue}{2} \cdot -1\right)\right)\right) \]

      9. metadata-eval 98.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(x\right), -2\right)\right) \]

   7. Applied egg-rr 98.4%

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

   if 1.00000000000000004e-10 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 64.6%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow1/3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left({x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      2. sqr-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot \color{blue}{{x}^{\left(\frac{\frac{1}{3}}{2}\right)}}\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left({\left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}^{\color{blue}{2}}\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{pow.f64}\left(\left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \color{blue}{2}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{\frac{1}{3}}{2}\right)\right), 2\right)\right) \]

      6. metadata-eval 62.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{6}\right), 2\right)\right) \]

   4. Applied egg-rr 62.8%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow N/A

         \[\leadsto \sqrt[3]{x + 1} - {x}^{\color{blue}{\left(\frac{1}{6} \cdot 2\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt[3]{x + 1} - {x}^{\frac{1}{3}} \]

      3. pow1/3 N/A

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

      4. flip3-- N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}\right), \color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)}\right) \]

      6. rem-cube-cbrt N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) - {\left(\sqrt[3]{x}\right)}^{3}\right), \left(\color{blue}{\sqrt[3]{x + 1}} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      7. rem-cube-cbrt N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) - x\right), \left(\sqrt[3]{x + 1} \cdot \color{blue}{\sqrt[3]{x + 1}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x + 1\right), x\right), \left(\color{blue}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + x\right), x\right), \left(\color{blue}{\sqrt[3]{x + 1}} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \left(\color{blue}{\sqrt[3]{x + 1}} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right), \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\right)\right) \]

      12. pow1/3 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(\left({\left(x + 1\right)}^{\frac{1}{3}} \cdot \sqrt[3]{x + 1}\right), \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      13. pow1/3 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(\left({\left(x + 1\right)}^{\frac{1}{3}} \cdot {\left(x + 1\right)}^{\frac{1}{3}}\right), \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{x}} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      14. pow-prod-up N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{3} + \frac{1}{3}\right)}\right), \left(\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      15. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \left(\frac{1}{3} + \frac{1}{3}\right)\right), \left(\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(1 + x\right), \left(\frac{1}{3} + \frac{1}{3}\right)\right), \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \left(\frac{1}{3} + \frac{1}{3}\right)\right), \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{2}{3}\right), \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{x}} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

   6. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{{\left(1 + x\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{x \cdot \left(1 + x\right)}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 10^{-10}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{{\left(x + 1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{x \cdot \left(x + 1\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.3333333333333333 (pow (cbrt x) -2.0)))

C

double code(double x) {
	return 0.3333333333333333 * pow(cbrt(x), -2.0);
}

Java

public static double code(double x) {
	return 0.3333333333333333 * Math.pow(Math.cbrt(x), -2.0);
}

Julia

function code(x)
	return Float64(0.3333333333333333 * (cbrt(x) ^ -2.0))
end

Wolfram

code[x_] := N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

   9. *-lowering-*.f64 50.3%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

5. Simplified 50.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
6. Step-by-step derivation

   1. cbrt-div N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{x \cdot x}}}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\sqrt[3]{\color{blue}{x \cdot x}}}\right)\right) \]

   3. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\sqrt[3]{x \cdot x}\right)}^{\color{blue}{-1}}\right)\right) \]

   4. cbrt-prod N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{-1}\right)\right) \]

   5. pow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{-1}\right)\right) \]

   6. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\sqrt[3]{x}\right)}^{\color{blue}{\left(2 \cdot -1\right)}}\right)\right) \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\left(\sqrt[3]{x}\right), \color{blue}{\left(2 \cdot -1\right)}\right)\right) \]

   8. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(x\right), \left(\color{blue}{2} \cdot -1\right)\right)\right) \]

   9. metadata-eval 96.7%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(x\right), -2\right)\right) \]

7. Applied egg-rr 96.7%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{-2}} \]
8. Add Preprocessing

Alternative 3: 92.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\frac{-1}{\frac{-1}{x}}\right)}^{-0.6666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 0.3333333333333333 (cbrt (* x x)))
   (* 0.3333333333333333 (pow (/ -1.0 (/ -1.0 x)) -0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 / cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 * pow((-1.0 / (-1.0 / x)), -0.6666666666666666);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 / Math.cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 * Math.pow((-1.0 / (-1.0 / x)), -0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(0.3333333333333333 / cbrt(Float64(x * x)));
	else
		tmp = Float64(0.3333333333333333 * (Float64(-1.0 / Float64(-1.0 / x)) ^ -0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(0.3333333333333333 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[Power[N[(-1.0 / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], -0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.5%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 95.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 95.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   6. Step-by-step derivation

      1. cbrt-div N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{x \cdot x}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1}{\sqrt[3]{\color{blue}{x \cdot x}}} \]

      3. cbrt-prod N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1}{\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{x}}} \]

      4. un-div-inv N/A

         \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

      6. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

      7. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}\right)\right) \]

      10. metadata-eval 88.6%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \frac{2}{3}\right)\right) \]

   7. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\frac{1}{3} + \color{blue}{\frac{1}{3}}\right)}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \color{blue}{{x}^{\frac{1}{3}}}\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({\left(x \cdot x\right)}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      4. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{x \cdot x}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(x \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 95.3%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right) \]

   9. Applied egg-rr 95.3%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\sqrt[3]{x \cdot x}}} \]

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 4.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 4.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{x \cdot x}} \cdot \color{blue}{\frac{1}{3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{x \cdot x}}\right), \color{blue}{\frac{1}{3}}\right) \]

      3. pow1/3 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{x \cdot x}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({\left(x \cdot x\right)}^{-1}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot x\right)}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{2}\right)}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

      7. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)}\right), \frac{1}{3}\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)\right), \frac{1}{3}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \frac{-1}{3}\right)\right), \frac{1}{3}\right) \]

      10. metadata-eval 89.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-2}{3}\right), \frac{1}{3}\right) \]

   7. Applied egg-rr 89.1%

      \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
   8. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\log x \cdot \frac{-2}{3}}\right), \frac{1}{3}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\frac{-2}{3} \cdot \log x}\right), \frac{1}{3}\right) \]

      3. exp-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(e^{\frac{-2}{3}}\right)}^{\log x}\right), \frac{1}{3}\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(e^{\frac{-2}{3}}\right), \log x\right), \frac{1}{3}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\frac{-2}{3}\right), \log x\right), \frac{1}{3}\right) \]

      6. log-lowering-log.f64 90.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\frac{-2}{3}\right), \mathsf{log.f64}\left(x\right)\right), \frac{1}{3}\right) \]

   9. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\log x}} \cdot 0.3333333333333333 \]

   10. Taylor expanded in x around -inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(e^{\frac{-2}{3} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}, \frac{1}{3}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{\left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right) \cdot \frac{-2}{3}}\right), \frac{1}{3}\right) \]

       2. exp-prod N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(e^{\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)}\right)}^{\frac{-2}{3}}\right), \frac{1}{3}\right) \]

       3. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(e^{\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)}\right), \frac{-2}{3}\right), \frac{1}{3}\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(e^{\log -1 + \left(\mathsf{neg}\left(\log \left(\frac{-1}{x}\right)\right)\right)}\right), \frac{-2}{3}\right), \frac{1}{3}\right) \]

       5. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(e^{\log -1 - \log \left(\frac{-1}{x}\right)}\right), \frac{-2}{3}\right), \frac{1}{3}\right) \]

       6. exp-diff N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{e^{\log -1}}{e^{\log \left(\frac{-1}{x}\right)}}\right), \frac{-2}{3}\right), \frac{1}{3}\right) \]

       7. rem-exp-log N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{-1}{e^{\log \left(\frac{-1}{x}\right)}}\right), \frac{-2}{3}\right), \frac{1}{3}\right) \]

       8. rem-exp-log N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{-1}{\frac{-1}{x}}\right), \frac{-2}{3}\right), \frac{1}{3}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(-1, \left(\frac{-1}{x}\right)\right), \frac{-2}{3}\right), \frac{1}{3}\right) \]

       10. /-lowering-/.f64 89.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(-1, x\right)\right), \frac{-2}{3}\right), \frac{1}{3}\right) \]

   12. Simplified 89.1%

       \[\leadsto \color{blue}{{\left(\frac{-1}{\frac{-1}{x}}\right)}^{-0.6666666666666666}} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\frac{-1}{\frac{-1}{x}}\right)}^{-0.6666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.6666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 0.3333333333333333 (cbrt (* x x)))
   (* 0.3333333333333333 (pow x -0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 / cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 * pow(x, -0.6666666666666666);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 / Math.cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 * Math.pow(x, -0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(0.3333333333333333 / cbrt(Float64(x * x)));
	else
		tmp = Float64(0.3333333333333333 * (x ^ -0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(0.3333333333333333 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[Power[x, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.5%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 95.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 95.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   6. Step-by-step derivation

      1. cbrt-div N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{x \cdot x}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1}{\sqrt[3]{\color{blue}{x \cdot x}}} \]

      3. cbrt-prod N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1}{\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{x}}} \]

      4. un-div-inv N/A

         \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

      6. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

      7. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}\right)\right) \]

      10. metadata-eval 88.6%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \frac{2}{3}\right)\right) \]

   7. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\frac{1}{3} + \color{blue}{\frac{1}{3}}\right)}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \color{blue}{{x}^{\frac{1}{3}}}\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({\left(x \cdot x\right)}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      4. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{x \cdot x}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(x \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 95.3%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right) \]

   9. Applied egg-rr 95.3%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\sqrt[3]{x \cdot x}}} \]

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 4.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 4.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{x \cdot x}} \cdot \color{blue}{\frac{1}{3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{x \cdot x}}\right), \color{blue}{\frac{1}{3}}\right) \]

      3. pow1/3 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{x \cdot x}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({\left(x \cdot x\right)}^{-1}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot x\right)}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{2}\right)}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

      7. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)}\right), \frac{1}{3}\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)\right), \frac{1}{3}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \frac{-1}{3}\right)\right), \frac{1}{3}\right) \]

      10. metadata-eval 89.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-2}{3}\right), \frac{1}{3}\right) \]

   7. Applied egg-rr 89.1%

      \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.6666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{{x}^{0.6666666666666666}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 0.3333333333333333 (pow x 0.6666666666666666)))

C

double code(double x) {
	return 0.3333333333333333 / pow(x, 0.6666666666666666);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3333333333333333d0 / (x ** 0.6666666666666666d0)
end function

Java

public static double code(double x) {
	return 0.3333333333333333 / Math.pow(x, 0.6666666666666666);
}

Python

def code(x):
	return 0.3333333333333333 / math.pow(x, 0.6666666666666666)

Julia

function code(x)
	return Float64(0.3333333333333333 / (x ^ 0.6666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333 / (x ^ 0.6666666666666666);
end

Wolfram

code[x_] := N[(0.3333333333333333 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

   9. *-lowering-*.f64 50.3%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

5. Simplified 50.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
6. Step-by-step derivation

   1. cbrt-div N/A

      \[\leadsto \frac{1}{3} \cdot \frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{x \cdot x}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\sqrt[3]{\color{blue}{x \cdot x}}} \]

   3. cbrt-prod N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{x}}} \]

   4. un-div-inv N/A

      \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

   6. pow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

   7. pow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

   8. pow-prod-up N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}\right)\right) \]

   10. metadata-eval 88.9%

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \frac{2}{3}\right)\right) \]

7. Applied egg-rr 88.9%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
8. Add Preprocessing

Alternative 6: 89.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot {x}^{-0.6666666666666666} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.3333333333333333 (pow x -0.6666666666666666)))

C

double code(double x) {
	return 0.3333333333333333 * pow(x, -0.6666666666666666);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3333333333333333d0 * (x ** (-0.6666666666666666d0))
end function

Java

public static double code(double x) {
	return 0.3333333333333333 * Math.pow(x, -0.6666666666666666);
}

Python

def code(x):
	return 0.3333333333333333 * math.pow(x, -0.6666666666666666)

Julia

function code(x)
	return Float64(0.3333333333333333 * (x ^ -0.6666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333 * (x ^ -0.6666666666666666);
end

Wolfram

code[x_] := N[(0.3333333333333333 * N[Power[x, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

   9. *-lowering-*.f64 50.3%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

5. Simplified 50.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt[3]{\frac{1}{x \cdot x}} \cdot \color{blue}{\frac{1}{3}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{x \cdot x}}\right), \color{blue}{\frac{1}{3}}\right) \]

   3. pow1/3 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{x \cdot x}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

   4. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left({\left(x \cdot x\right)}^{-1}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

   5. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot x\right)}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

   6. pow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{2}\right)}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

   7. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)}\right), \frac{1}{3}\right) \]

   8. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)\right), \frac{1}{3}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \frac{-1}{3}\right)\right), \frac{1}{3}\right) \]

   10. metadata-eval 88.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-2}{3}\right), \frac{1}{3}\right) \]

7. Applied egg-rr 88.9%

   \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]

8. Final simplification 88.9%

   \[\leadsto 0.3333333333333333 \cdot {x}^{-0.6666666666666666} \]
9. Add Preprocessing

Alternative 7: 1.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 - \sqrt[3]{x} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[(1.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - \sqrt[3]{x}} \]
4. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt[3]{x}\right)}\right) \]

   2. cbrt-lowering-cbrt.f64 1.8%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{cbrt.f64}\left(x\right)\right) \]

5. Simplified 1.8%

   \[\leadsto \color{blue}{1 - \sqrt[3]{x}} \]
6. Add Preprocessing

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \frac{1}{\left(t\_0 \cdot t\_0 + \sqrt[3]{x} \cdot t\_0\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Julia

function code(x)
	t_0 = cbrt(Float64(x + 1.0))
	return Float64(1.0 / Float64(Float64(Float64(t_0 * t_0) + Float64(cbrt(x) * t_0)) + Float64(cbrt(x) * cbrt(x))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024141 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (cbrt (+ x 1)) (cbrt (+ x 1))) (* (cbrt x) (cbrt (+ x 1))) (* (cbrt x) (cbrt x)))))

  (- (cbrt (+ x 1.0)) (cbrt x)))