2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 98.5%

Time: 11.3s

Alternatives: 6

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: 6.9% 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: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.023148148148148147}{{x}^{1.6666666666666667}} + \left({x}^{-0.6666666666666666} \cdot 0.027777777777777776 + \left(0.3333333333333333 \cdot \sqrt[3]{x} + \left(\frac{0.08487654320987655}{{x}^{1.6666666666666667}} + {x}^{-0.6666666666666666} \cdot -0.1388888888888889\right)\right)\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+
   (/ -0.023148148148148147 (pow x 1.6666666666666667))
   (+
    (* (pow x -0.6666666666666666) 0.027777777777777776)
    (+
     (* 0.3333333333333333 (cbrt x))
     (+
      (/ 0.08487654320987655 (pow x 1.6666666666666667))
      (* (pow x -0.6666666666666666) -0.1388888888888889)))))
  x))

C

double code(double x) {
	return ((-0.023148148148148147 / pow(x, 1.6666666666666667)) + ((pow(x, -0.6666666666666666) * 0.027777777777777776) + ((0.3333333333333333 * cbrt(x)) + ((0.08487654320987655 / pow(x, 1.6666666666666667)) + (pow(x, -0.6666666666666666) * -0.1388888888888889))))) / x;
}

Java

public static double code(double x) {
	return ((-0.023148148148148147 / Math.pow(x, 1.6666666666666667)) + ((Math.pow(x, -0.6666666666666666) * 0.027777777777777776) + ((0.3333333333333333 * Math.cbrt(x)) + ((0.08487654320987655 / Math.pow(x, 1.6666666666666667)) + (Math.pow(x, -0.6666666666666666) * -0.1388888888888889))))) / x;
}

Julia

function code(x)
	return Float64(Float64(Float64(-0.023148148148148147 / (x ^ 1.6666666666666667)) + Float64(Float64((x ^ -0.6666666666666666) * 0.027777777777777776) + Float64(Float64(0.3333333333333333 * cbrt(x)) + Float64(Float64(0.08487654320987655 / (x ^ 1.6666666666666667)) + Float64((x ^ -0.6666666666666666) * -0.1388888888888889))))) / x)
end

Wolfram

code[x_] := N[(N[(N[(-0.023148148148148147 / N[Power[x, 1.6666666666666667], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.027777777777777776), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[(0.08487654320987655 / N[Power[x, 1.6666666666666667], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * -0.1388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

   1. pow1/3 N/A

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

   2. sqr-pow N/A

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

   3. pow1/3 N/A

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

   4. sqr-pow N/A

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

   5. difference-of-squares N/A

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

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

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

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

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

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

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

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

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

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \left({\left(x + \color{blue}{1}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \left(\frac{\frac{1}{3}}{2}\right)\right)\right), \left({\left(x + 1\right)}^{\color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{\color{blue}{2}}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({\color{blue}{x}}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   18. metadata-eval 6.4%

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

4. Applied egg-rr 6.4%

   \[\leadsto \color{blue}{\left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left({\left(x + 1\right)}^{0.16666666666666666} - {x}^{0.16666666666666666}\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\frac{-5}{216} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \left(\frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\frac{55}{648} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \frac{1}{3} \cdot \sqrt[3]{x}\right)\right)\right)}{x}} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\frac{-5}{216} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \left(\frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\frac{55}{648} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \frac{1}{3} \cdot \sqrt[3]{x}\right)\right)\right)\right), \color{blue}{x}\right) \]

7. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.027777777777777776 + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot -0.023148148148148147\right) + \left(\left(0.3333333333333333 \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.08487654320987655\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot -0.1388888888888889\right)}{x}} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-5}{216} + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{36}\right) + \left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{55}{648}\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{-5}{36}\right)\right), x\right) \]

   2. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-5}{216} + \left(\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{36} + \left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{55}{648}\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{-5}{36}\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-5}{216}\right), \left(\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{36} + \left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{55}{648}\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{-5}{36}\right)\right)\right), x\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-5}{216} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right), \left(\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{36} + \left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{55}{648}\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{-5}{36}\right)\right)\right), x\right) \]

   5. cbrt-div N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-5}{216} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{{x}^{5}}}\right), \left(\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{36} + \left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{55}{648}\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{-5}{36}\right)\right)\right), x\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-5}{216} \cdot \frac{1}{\sqrt[3]{{x}^{5}}}\right), \left(\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{36} + \left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{55}{648}\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{-5}{36}\right)\right)\right), x\right) \]

   7. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-5}{216}}{\sqrt[3]{{x}^{5}}}\right), \left(\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{36} + \left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{55}{648}\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{-5}{36}\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-5}{216}, \left(\sqrt[3]{{x}^{5}}\right)\right), \left(\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{36} + \left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{55}{648}\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{-5}{36}\right)\right)\right), x\right) \]

   9. pow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-5}{216}, \left({\left({x}^{5}\right)}^{\frac{1}{3}}\right)\right), \left(\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{36} + \left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{55}{648}\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{-5}{36}\right)\right)\right), x\right) \]

   10. pow-pow N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-5}{216}, \left({x}^{\left(5 \cdot \frac{1}{3}\right)}\right)\right), \left(\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{36} + \left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{55}{648}\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{-5}{36}\right)\right)\right), x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-5}{216}, \mathsf{pow.f64}\left(x, \left(5 \cdot \frac{1}{3}\right)\right)\right), \left(\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{36} + \left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{55}{648}\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{-5}{36}\right)\right)\right), x\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-5}{216}, \mathsf{pow.f64}\left(x, \frac{5}{3}\right)\right), \left(\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{36} + \left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{55}{648}\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{-5}{36}\right)\right)\right), x\right) \]

9. Applied egg-rr 98.9%

   \[\leadsto \frac{\color{blue}{\frac{-0.023148148148148147}{{x}^{1.6666666666666667}} + \left({x}^{-0.6666666666666666} \cdot 0.027777777777777776 + \left(0.3333333333333333 \cdot \sqrt[3]{x} + \left(\frac{0.08487654320987655}{{x}^{1.6666666666666667}} + {x}^{-0.6666666666666666} \cdot -0.1388888888888889\right)\right)\right)}}{x} \]
10. Add Preprocessing

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333 \cdot \sqrt[3]{x} + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot -0.1111111111111111}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+
   (* 0.3333333333333333 (cbrt x))
   (* (cbrt (/ (/ 1.0 x) x)) -0.1111111111111111))
  x))

C

double code(double x) {
	return ((0.3333333333333333 * cbrt(x)) + (cbrt(((1.0 / x) / x)) * -0.1111111111111111)) / x;
}

Java

public static double code(double x) {
	return ((0.3333333333333333 * Math.cbrt(x)) + (Math.cbrt(((1.0 / x) / x)) * -0.1111111111111111)) / x;
}

Julia

function code(x)
	return Float64(Float64(Float64(0.3333333333333333 * cbrt(x)) + Float64(cbrt(Float64(Float64(1.0 / x) / x)) * -0.1111111111111111)) / x)
end

Wolfram

code[x_] := N[(N[(N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

   1. pow1/3 N/A

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

   2. sqr-pow N/A

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

   3. pow1/3 N/A

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

   4. sqr-pow N/A

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

   5. difference-of-squares N/A

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

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

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

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

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

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

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

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

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

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \left({\left(x + \color{blue}{1}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \left(\frac{\frac{1}{3}}{2}\right)\right)\right), \left({\left(x + 1\right)}^{\color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{\color{blue}{2}}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({\color{blue}{x}}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   18. metadata-eval 6.4%

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

4. Applied egg-rr 6.4%

   \[\leadsto \color{blue}{\left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left({\left(x + 1\right)}^{0.16666666666666666} - {x}^{0.16666666666666666}\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{x}\right)}{x}} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{x}\right)\right), \color{blue}{x}\right) \]

   2. associate-+r+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) + \frac{1}{3} \cdot \sqrt[3]{x}\right), x\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{3} \cdot \sqrt[3]{x}\right), \left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{x}\right)\right), \left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)\right), x\right) \]

   7. distribute-rgt-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \left(\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \left(\frac{-5}{36} + \frac{1}{36}\right)\right)\right), x\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \left(\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{-1}{9}\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right), \frac{-1}{9}\right)\right), x\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{x \cdot x}}\right), \frac{-1}{9}\right)\right), x\right) \]

   11. associate-/r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}\right), \frac{-1}{9}\right)\right), x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\frac{1}{x}}{x}\right)\right), \frac{-1}{9}\right)\right), x\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{x}\right), x\right)\right), \frac{-1}{9}\right)\right), x\right) \]

   14. /-lowering-/.f64 98.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), x\right)\right), \frac{-1}{9}\right)\right), x\right) \]

7. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \sqrt[3]{x} + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot -0.1111111111111111}{x}} \]
8. Add Preprocessing

Alternative 3: 92.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.34e+154)
   (/ 0.3333333333333333 (cbrt (* x x)))
   (/ 0.3333333333333333 (pow x 0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.34e+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 <= 1.34e+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 <= 1.34e+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, 1.34e+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 < 1.34000000000000001e154

   1. Initial program 8.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-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 95.7%

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

   5. Simplified 95.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   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 89.3%

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

   7. Applied egg-rr 89.3%

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

      1. metadata-eval N/A

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

      2. pow-pow N/A

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

      3. pow2 N/A

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

      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 96.0%

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

   9. Applied egg-rr 96.0%

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

   if 1.34000000000000001e154 < 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. 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 89.2%

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

   7. Applied egg-rr 89.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* 0.3333333333333333 (cbrt x)) x))

C

double code(double x) {
	return (0.3333333333333333 * cbrt(x)) / x;
}

Java

public static double code(double x) {
	return (0.3333333333333333 * Math.cbrt(x)) / x;
}

Julia

function code(x)
	return Float64(Float64(0.3333333333333333 * cbrt(x)) / x)
end

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. pow1/3 N/A

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

   2. sqr-pow N/A

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

   3. pow1/3 N/A

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

   4. sqr-pow N/A

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

   5. difference-of-squares N/A

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

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

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

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

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

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

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

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

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

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \left({\left(x + \color{blue}{1}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \left(\frac{\frac{1}{3}}{2}\right)\right)\right), \left({\left(x + 1\right)}^{\color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{\color{blue}{2}}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({\color{blue}{x}}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   18. metadata-eval 6.4%

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

4. Applied egg-rr 6.4%

   \[\leadsto \color{blue}{\left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left({\left(x + 1\right)}^{0.16666666666666666} - {x}^{0.16666666666666666}\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\frac{-5}{216} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \left(\frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\frac{55}{648} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \frac{1}{3} \cdot \sqrt[3]{x}\right)\right)\right)}{x}} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\frac{-5}{216} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \left(\frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\frac{55}{648} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \frac{1}{3} \cdot \sqrt[3]{x}\right)\right)\right)\right), \color{blue}{x}\right) \]

7. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.027777777777777776 + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot -0.023148148148148147\right) + \left(\left(0.3333333333333333 \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.08487654320987655\right) + \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot -0.1388888888888889\right)}{x}} \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{3} \cdot \sqrt[3]{x}\right)}, x\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

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

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

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

   3. cbrt-lowering-cbrt.f64 97.5%

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

10. Simplified 97.5%

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

11. Final simplification 97.5%

    \[\leadsto \frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x} \]
12. Add Preprocessing

Alternative 5: 88.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 49.8%

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

5. Simplified 49.8%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
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 89.2%

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

7. Applied egg-rr 89.2%

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

Alternative 6: 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 6.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 2024155 
(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)))