2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.0%

Time: 3.2s

Alternatives: 5

Speedup: N/A×

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: 7.0% 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: 98.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{{x}^{-1}}\\ t_1 := {\left(x - -1\right)}^{0.3333333333333333}\\ \mathbf{if}\;x \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{\mathsf{fma}\left(t\_1, t\_1, {x}^{0.3333333333333333} \cdot \mathsf{fma}\left({x}^{0.3333333333333333}, 1, t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (pow x -1.0))) (t_1 (pow (- x -1.0) 0.3333333333333333)))
   (if (<= x 2e+14)
     (/
      (- (- x -1.0) x)
      (fma
       t_1
       t_1
       (*
        (pow x 0.3333333333333333)
        (fma (pow x 0.3333333333333333) 1.0 t_1))))
     (* (* t_0 t_0) 0.3333333333333333))))

C

double code(double x) {
	double t_0 = cbrt(pow(x, -1.0));
	double t_1 = pow((x - -1.0), 0.3333333333333333);
	double tmp;
	if (x <= 2e+14) {
		tmp = ((x - -1.0) - x) / fma(t_1, t_1, (pow(x, 0.3333333333333333) * fma(pow(x, 0.3333333333333333), 1.0, t_1)));
	} else {
		tmp = (t_0 * t_0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt((x ^ -1.0))
	t_1 = Float64(x - -1.0) ^ 0.3333333333333333
	tmp = 0.0
	if (x <= 2e+14)
		tmp = Float64(Float64(Float64(x - -1.0) - x) / fma(t_1, t_1, Float64((x ^ 0.3333333333333333) * fma((x ^ 0.3333333333333333), 1.0, t_1))));
	else
		tmp = Float64(Float64(t_0 * t_0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[x, -1.0], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x - -1.0), $MachinePrecision], 0.3333333333333333], $MachinePrecision]}, If[LessEqual[x, 2e+14], N[(N[(N[(x - -1.0), $MachinePrecision] - x), $MachinePrecision] / N[(t$95$1 * t$95$1 + N[(N[Power[x, 0.3333333333333333], $MachinePrecision] * N[(N[Power[x, 0.3333333333333333], $MachinePrecision] * 1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e14

   1. Initial program 52.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. rem-cube-cbrt N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\frac{\left(x - -1\right) - x}{\mathsf{fma}\left({\left(x - -1\right)}^{0.3333333333333333}, {\left(x - -1\right)}^{0.3333333333333333}, {x}^{0.3333333333333333} \cdot \mathsf{fma}\left({x}^{0.3333333333333333}, 1, {\left(x - -1\right)}^{0.3333333333333333}\right)\right)}} \]

   if 2e14 < x

   1. Initial program 4.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \color{blue}{\frac{1}{3}} \]

      3. pow1/3 N/A

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

      4. lower-pow.f64 N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 45.5

         \[\leadsto {\left({\left(x \cdot x\right)}^{-1}\right)}^{0.3333333333333333} \cdot 0.3333333333333333 \]

   5. Applied rewrites 45.5%

      \[\leadsto \color{blue}{{\left({\left(x \cdot x\right)}^{-1}\right)}^{0.3333333333333333} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow1/3 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. pow2 N/A

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

      7. inv-pow N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3} \]

      8. pow-flip N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 51.6

          \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   7. Applied rewrites 51.6%

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]
   8. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3} \]

      2. lift-pow.f64 N/A

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

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

      4. pow-flip N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3} \]

      5. inv-pow N/A

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

      6. pow-to-exp N/A

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

      7. *-commutative N/A

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

      8. count-2-rev N/A

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

      9. log-prod N/A

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

      10. exp-prod N/A

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

      11. unpow1/3 N/A

          \[\leadsto {\left(e^{\log \left(x \cdot x\right) \cdot -1}\right)}^{\frac{1}{3}} \cdot \frac{1}{3} \]

      12. pow-exp N/A

          \[\leadsto e^{\left(\log \left(x \cdot x\right) \cdot -1\right) \cdot \frac{1}{3}} \cdot \frac{1}{3} \]

      13. lower-exp.f64 N/A

          \[\leadsto e^{\left(\log \left(x \cdot x\right) \cdot -1\right) \cdot \frac{1}{3}} \cdot \frac{1}{3} \]

      14. lower-*.f64 N/A

          \[\leadsto e^{\left(\log \left(x \cdot x\right) \cdot -1\right) \cdot \frac{1}{3}} \cdot \frac{1}{3} \]

      15. lower-*.f64 N/A

          \[\leadsto e^{\left(\log \left(x \cdot x\right) \cdot -1\right) \cdot \frac{1}{3}} \cdot \frac{1}{3} \]

      16. log-prod N/A

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

      17. count-2-rev N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lift-log.f64 90.7

          \[\leadsto e^{\left(\left(\log x \cdot 2\right) \cdot -1\right) \cdot 0.3333333333333333} \cdot 0.3333333333333333 \]

   9. Applied rewrites 90.7%

      \[\leadsto e^{\left(\left(\log x \cdot 2\right) \cdot -1\right) \cdot 0.3333333333333333} \cdot 0.3333333333333333 \]
   10. Step-by-step derivation

       1. lift-exp.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-log.f64 N/A

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

       6. exp-prod N/A

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

       7. *-commutative N/A

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

       8. count-2-rev N/A

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

       9. exp-prod N/A

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

       10. count-2-rev N/A

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

       11. *-commutative N/A

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

       12. pow-to-exp N/A

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

       13. pow2 N/A

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

       14. pow1/3 N/A

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

       15. unpow-prod-down N/A

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

       16. lift-pow.f64 N/A

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

       17. lift-pow.f64 N/A

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

       18. cbrt-prod N/A

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

       19. lower-*.f64 N/A

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

       20. lower-cbrt.f64 N/A

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

       21. lower-cbrt.f64 98.1

           \[\leadsto \left(\sqrt[3]{{x}^{-1}} \cdot \sqrt[3]{{x}^{-1}}\right) \cdot 0.3333333333333333 \]

   11. Applied rewrites 98.1%

       \[\leadsto \left(\sqrt[3]{{x}^{-1}} \cdot \sqrt[3]{{x}^{-1}}\right) \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{{x}^{-1}}\\ \left(t\_0 \cdot t\_0\right) \cdot 0.3333333333333333 \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (pow x -1.0)))) (* (* t_0 t_0) 0.3333333333333333)))

C

double code(double x) {
	double t_0 = cbrt(pow(x, -1.0));
	return (t_0 * t_0) * 0.3333333333333333;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(Math.pow(x, -1.0));
	return (t_0 * t_0) * 0.3333333333333333;
}

Julia

function code(x)
	t_0 = cbrt((x ^ -1.0))
	return Float64(Float64(t_0 * t_0) * 0.3333333333333333)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[x, -1.0], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Initial program 7.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. *-commutative N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \color{blue}{\frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \color{blue}{\frac{1}{3}} \]

   3. pow1/3 N/A

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

   4. lower-pow.f64 N/A

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

   5. inv-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 46.7

      \[\leadsto {\left({\left(x \cdot x\right)}^{-1}\right)}^{0.3333333333333333} \cdot 0.3333333333333333 \]

5. Applied rewrites 46.7%

   \[\leadsto \color{blue}{{\left({\left(x \cdot x\right)}^{-1}\right)}^{0.3333333333333333} \cdot 0.3333333333333333} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow1/3 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. pow2 N/A

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

   7. inv-pow N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3} \]

   8. pow-flip N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 52.5

       \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

7. Applied rewrites 52.5%

   \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]
8. Step-by-step derivation

   1. lift-cbrt.f64 N/A

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3} \]

   2. lift-pow.f64 N/A

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3} \]

   3. metadata-eval N/A

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

   4. pow-flip N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3} \]

   5. inv-pow N/A

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

   6. pow-to-exp N/A

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

   7. *-commutative N/A

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

   8. count-2-rev N/A

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

   9. log-prod N/A

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

   10. exp-prod N/A

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

   11. unpow1/3 N/A

       \[\leadsto {\left(e^{\log \left(x \cdot x\right) \cdot -1}\right)}^{\frac{1}{3}} \cdot \frac{1}{3} \]

   12. pow-exp N/A

       \[\leadsto e^{\left(\log \left(x \cdot x\right) \cdot -1\right) \cdot \frac{1}{3}} \cdot \frac{1}{3} \]

   13. lower-exp.f64 N/A

       \[\leadsto e^{\left(\log \left(x \cdot x\right) \cdot -1\right) \cdot \frac{1}{3}} \cdot \frac{1}{3} \]

   14. lower-*.f64 N/A

       \[\leadsto e^{\left(\log \left(x \cdot x\right) \cdot -1\right) \cdot \frac{1}{3}} \cdot \frac{1}{3} \]

   15. lower-*.f64 N/A

       \[\leadsto e^{\left(\log \left(x \cdot x\right) \cdot -1\right) \cdot \frac{1}{3}} \cdot \frac{1}{3} \]

   16. log-prod N/A

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

   17. count-2-rev N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lift-log.f64 89.3

       \[\leadsto e^{\left(\left(\log x \cdot 2\right) \cdot -1\right) \cdot 0.3333333333333333} \cdot 0.3333333333333333 \]

9. Applied rewrites 89.3%

   \[\leadsto e^{\left(\left(\log x \cdot 2\right) \cdot -1\right) \cdot 0.3333333333333333} \cdot 0.3333333333333333 \]
10. Step-by-step derivation

    1. lift-exp.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-log.f64 N/A

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

    6. exp-prod N/A

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

    7. *-commutative N/A

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

    8. count-2-rev N/A

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

    9. exp-prod N/A

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

    10. count-2-rev N/A

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

    11. *-commutative N/A

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

    12. pow-to-exp N/A

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

    13. pow2 N/A

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

    14. pow1/3 N/A

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

    15. unpow-prod-down N/A

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

    16. lift-pow.f64 N/A

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

    17. lift-pow.f64 N/A

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

    18. cbrt-prod N/A

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

    19. lower-*.f64 N/A

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

    20. lower-cbrt.f64 N/A

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

    21. lower-cbrt.f64 96.2

        \[\leadsto \left(\sqrt[3]{{x}^{-1}} \cdot \sqrt[3]{{x}^{-1}}\right) \cdot 0.3333333333333333 \]

11. Applied rewrites 96.2%

    \[\leadsto \left(\sqrt[3]{{x}^{-1}} \cdot \sqrt[3]{{x}^{-1}}\right) \cdot 0.3333333333333333 \]
12. Add Preprocessing

Alternative 3: 92.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(\log x \cdot 2\right) \cdot -1\right) \cdot 0.3333333333333333} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75e+155)
   (* (cbrt (pow x -2.0)) 0.3333333333333333)
   (*
    (exp (* (* (* (log x) 2.0) -1.0) 0.3333333333333333))
    0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.75e+155) {
		tmp = cbrt(pow(x, -2.0)) * 0.3333333333333333;
	} else {
		tmp = exp((((log(x) * 2.0) * -1.0) * 0.3333333333333333)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.75e+155) {
		tmp = Math.cbrt(Math.pow(x, -2.0)) * 0.3333333333333333;
	} else {
		tmp = Math.exp((((Math.log(x) * 2.0) * -1.0) * 0.3333333333333333)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75e+155)
		tmp = Float64(cbrt((x ^ -2.0)) * 0.3333333333333333);
	else
		tmp = Float64(exp(Float64(Float64(Float64(log(x) * 2.0) * -1.0) * 0.3333333333333333)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.75e+155], N[(N[Power[N[Power[x, -2.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Exp[N[(N[(N[(N[Log[x], $MachinePrecision] * 2.0), $MachinePrecision] * -1.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.74999999999999992e155

   1. Initial program 9.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \color{blue}{\frac{1}{3}} \]

      3. pow1/3 N/A

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

      4. lower-pow.f64 N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 87.4

         \[\leadsto {\left({\left(x \cdot x\right)}^{-1}\right)}^{0.3333333333333333} \cdot 0.3333333333333333 \]

   5. Applied rewrites 87.4%

      \[\leadsto \color{blue}{{\left({\left(x \cdot x\right)}^{-1}\right)}^{0.3333333333333333} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow1/3 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. pow2 N/A

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

      7. inv-pow N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3} \]

      8. pow-flip N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 95.0

          \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   7. Applied rewrites 95.0%

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   if 1.74999999999999992e155 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \color{blue}{\frac{1}{3}} \]

      3. pow1/3 N/A

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

      4. lower-pow.f64 N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 4.8

         \[\leadsto {\left({\left(x \cdot x\right)}^{-1}\right)}^{0.3333333333333333} \cdot 0.3333333333333333 \]

   5. Applied rewrites 4.8%

      \[\leadsto \color{blue}{{\left({\left(x \cdot x\right)}^{-1}\right)}^{0.3333333333333333} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow1/3 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. pow2 N/A

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

      7. inv-pow N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3} \]

      8. pow-flip N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 8.7

          \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   7. Applied rewrites 8.7%

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]
   8. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3} \]

      2. lift-pow.f64 N/A

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

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

      4. pow-flip N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3} \]

      5. inv-pow N/A

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

      6. pow-to-exp N/A

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

      7. *-commutative N/A

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

      8. count-2-rev N/A

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

      9. log-prod N/A

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

      10. exp-prod N/A

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

      11. unpow1/3 N/A

          \[\leadsto {\left(e^{\log \left(x \cdot x\right) \cdot -1}\right)}^{\frac{1}{3}} \cdot \frac{1}{3} \]

      12. pow-exp N/A

          \[\leadsto e^{\left(\log \left(x \cdot x\right) \cdot -1\right) \cdot \frac{1}{3}} \cdot \frac{1}{3} \]

      13. lower-exp.f64 N/A

          \[\leadsto e^{\left(\log \left(x \cdot x\right) \cdot -1\right) \cdot \frac{1}{3}} \cdot \frac{1}{3} \]

      14. lower-*.f64 N/A

          \[\leadsto e^{\left(\log \left(x \cdot x\right) \cdot -1\right) \cdot \frac{1}{3}} \cdot \frac{1}{3} \]

      15. lower-*.f64 N/A

          \[\leadsto e^{\left(\log \left(x \cdot x\right) \cdot -1\right) \cdot \frac{1}{3}} \cdot \frac{1}{3} \]

      16. log-prod N/A

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

      17. count-2-rev N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lift-log.f64 89.2

          \[\leadsto e^{\left(\left(\log x \cdot 2\right) \cdot -1\right) \cdot 0.3333333333333333} \cdot 0.3333333333333333 \]

   9. Applied rewrites 89.2%

      \[\leadsto e^{\left(\left(\log x \cdot 2\right) \cdot -1\right) \cdot 0.3333333333333333} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 51.1% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. *-commutative N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \color{blue}{\frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \color{blue}{\frac{1}{3}} \]

   3. pow1/3 N/A

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

   4. lower-pow.f64 N/A

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

   5. inv-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 46.7

      \[\leadsto {\left({\left(x \cdot x\right)}^{-1}\right)}^{0.3333333333333333} \cdot 0.3333333333333333 \]

5. Applied rewrites 46.7%

   \[\leadsto \color{blue}{{\left({\left(x \cdot x\right)}^{-1}\right)}^{0.3333333333333333} \cdot 0.3333333333333333} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow1/3 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. pow2 N/A

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

   7. inv-pow N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3} \]

   8. pow-flip N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 52.5

       \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

7. Applied rewrites 52.5%

   \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 5: 51.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{x}^{-1} \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cbrt (* (pow x -1.0) (* (pow x -0.5) (pow x -0.5)))) 0.3333333333333333))

C

double code(double x) {
	return cbrt((pow(x, -1.0) * (pow(x, -0.5) * pow(x, -0.5)))) * 0.3333333333333333;
}

Java

public static double code(double x) {
	return Math.cbrt((Math.pow(x, -1.0) * (Math.pow(x, -0.5) * Math.pow(x, -0.5)))) * 0.3333333333333333;
}

Julia

function code(x)
	return Float64(cbrt(Float64((x ^ -1.0) * Float64((x ^ -0.5) * (x ^ -0.5)))) * 0.3333333333333333)
end

Wolfram

code[x_] := N[(N[Power[N[(N[Power[x, -1.0], $MachinePrecision] * N[(N[Power[x, -0.5], $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 7.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. *-commutative N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \color{blue}{\frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \color{blue}{\frac{1}{3}} \]

   3. pow1/3 N/A

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

   4. lower-pow.f64 N/A

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

   5. inv-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 46.7

      \[\leadsto {\left({\left(x \cdot x\right)}^{-1}\right)}^{0.3333333333333333} \cdot 0.3333333333333333 \]

5. Applied rewrites 46.7%

   \[\leadsto \color{blue}{{\left({\left(x \cdot x\right)}^{-1}\right)}^{0.3333333333333333} \cdot 0.3333333333333333} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow1/3 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. pow2 N/A

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

   7. inv-pow N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3} \]

   8. pow-flip N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 52.5

       \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

7. Applied rewrites 52.5%

   \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3} \]

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. inv-pow N/A

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

   7. lower-*.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 52.5

       \[\leadsto \sqrt[3]{{x}^{-1} \cdot {x}^{-1}} \cdot 0.3333333333333333 \]

9. Applied rewrites 52.5%

   \[\leadsto \sqrt[3]{{x}^{-1} \cdot {x}^{-1}} \cdot 0.3333333333333333 \]
10. Step-by-step derivation

    1. lift-pow.f64 N/A

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

    2. sqr-pow N/A

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

    3. lower-*.f64 N/A

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

    4. lower-pow.f64 N/A

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

    5. metadata-eval N/A

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

    6. lower-pow.f64 N/A

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

    7. metadata-eval 52.5

       \[\leadsto \sqrt[3]{{x}^{-1} \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.3333333333333333 \]

11. Applied rewrites 52.5%

    \[\leadsto \sqrt[3]{{x}^{-1} \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.3333333333333333 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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