2cbrt (problem 3.3.4)

Percentage Accurate: 6.8% → 97.4%

Time: 3.3s

Alternatives: 12

Speedup: 1.7×

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: 6.8% 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: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {\left(\frac{1}{\sqrt[3]{x}}\right)}^{2} \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (cbrt (pow x -5.0))
  -0.1111111111111111
  (* (pow (/ 1.0 (cbrt x)) 2.0) 0.3333333333333333)))

C

double code(double x) {
	return fma(cbrt(pow(x, -5.0)), -0.1111111111111111, (pow((1.0 / cbrt(x)), 2.0) * 0.3333333333333333));
}

Julia

function code(x)
	return fma(cbrt((x ^ -5.0)), -0.1111111111111111, Float64((Float64(1.0 / cbrt(x)) ^ 2.0) * 0.3333333333333333))
end

Wolfram

code[x_] := N[(N[Power[N[Power[x, -5.0], $MachinePrecision], 1/3], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.4%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{\color{blue}{{x}^{2}}} \]

5. Applied rewrites 26.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   3. lower-cbrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   4. pow-flip N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   7. pow-flip N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{\left(\mathsf{neg}\left(2\right)\right)}}\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

   9. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

   10. lift-cbrt.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

   12. lift-*.f64 53.0

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]

8. Applied rewrites 53.0%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]
9. Step-by-step derivation

   1. lift-cbrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

   2. pow1/3 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

   4. pow-pow N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

   6. metadata-eval 89.6

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

10. Applied rewrites 89.6%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
11. Step-by-step derivation

    1. lift-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

    3. pow-pow N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

    4. lift-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

    5. sqr-pow N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \left({\left({x}^{-2}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot {\left({x}^{-2}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right) \cdot \frac{1}{3}\right) \]

    6. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({\left({x}^{-2}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}^{2} \cdot \frac{1}{3}\right) \]

    7. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({\left({x}^{-2}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}^{2} \cdot \frac{1}{3}\right) \]

    8. lift-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({\left({x}^{-2}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}^{2} \cdot \frac{1}{3}\right) \]

    9. pow-pow N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{\left(-2 \cdot \frac{\frac{1}{3}}{2}\right)}\right)}^{2} \cdot \frac{1}{3}\right) \]

    10. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{\left(-2 \cdot \frac{1}{6}\right)}\right)}^{2} \cdot \frac{1}{3}\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{\frac{-1}{3}}\right)}^{2} \cdot \frac{1}{3}\right) \]

    12. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)}^{2} \cdot \frac{1}{3}\right) \]

    13. pow-neg N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left(\frac{1}{{x}^{\frac{1}{3}}}\right)}^{2} \cdot \frac{1}{3}\right) \]

    14. pow1/3 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left(\frac{1}{\sqrt[3]{x}}\right)}^{2} \cdot \frac{1}{3}\right) \]

    15. lower-/.f64 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left(\frac{1}{\sqrt[3]{x}}\right)}^{2} \cdot \frac{1}{3}\right) \]

    16. lift-cbrt.f64 97.3

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {\left(\frac{1}{\sqrt[3]{x}}\right)}^{2} \cdot 0.3333333333333333\right) \]

12. Applied rewrites 97.3%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {\left(\frac{1}{\sqrt[3]{x}}\right)}^{2} \cdot 0.3333333333333333\right) \]
13. Add Preprocessing

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (cbrt (pow x -5.0))
  -0.1111111111111111
  (* (/ 1.0 (pow (cbrt x) 2.0)) 0.3333333333333333)))

C

double code(double x) {
	return fma(cbrt(pow(x, -5.0)), -0.1111111111111111, ((1.0 / pow(cbrt(x), 2.0)) * 0.3333333333333333));
}

Julia

function code(x)
	return fma(cbrt((x ^ -5.0)), -0.1111111111111111, Float64(Float64(1.0 / (cbrt(x) ^ 2.0)) * 0.3333333333333333))
end

Wolfram

code[x_] := N[(N[Power[N[Power[x, -5.0], $MachinePrecision], 1/3], $MachinePrecision] * -0.1111111111111111 + N[(N[(1.0 / N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.4%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{\color{blue}{{x}^{2}}} \]

5. Applied rewrites 26.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   3. lower-cbrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   4. pow-flip N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   7. pow-flip N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{\left(\mathsf{neg}\left(2\right)\right)}}\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

   9. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

   10. lift-cbrt.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

   12. lift-*.f64 53.0

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]

8. Applied rewrites 53.0%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]
9. Step-by-step derivation

   1. lift-cbrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

   2. pow1/3 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

   4. pow-pow N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

   6. metadata-eval 89.6

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

10. Applied rewrites 89.6%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
11. Step-by-step derivation

    1. lift-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

    3. pow-pow N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{\left(\mathsf{neg}\left(2\right)\right)}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

    5. pow-flip N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

    6. pow1/3 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}\right) \]

    7. cbrt-div N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{\sqrt[3]{1}}{\sqrt[3]{{x}^{2}}} \cdot \frac{1}{3}\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{\sqrt[3]{{x}^{2}}} \cdot \frac{1}{3}\right) \]

    9. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{\sqrt[3]{x \cdot x}} \cdot \frac{1}{3}\right) \]

    10. cbrt-prod N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{1}{3}\right) \]

    11. lower-/.f64 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{1}{3}\right) \]

    12. pow2 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{1}{3}\right) \]

    13. lower-pow.f64 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{1}{3}\right) \]

    14. lift-cbrt.f64 97.3

        \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333\right) \]

12. Applied rewrites 97.3%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333\right) \]
13. Add Preprocessing

Alternative 3: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{{x}^{-5}}\\ \mathbf{if}\;x \leq 2.6 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -0.1111111111111111, e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (pow x -5.0))))
   (if (<= x 2.6e+155)
     (fma t_0 -0.1111111111111111 (* (cbrt (pow x -2.0)) 0.3333333333333333))
     (fma
      t_0
      -0.1111111111111111
      (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333)))))

C

double code(double x) {
	double t_0 = cbrt(pow(x, -5.0));
	double tmp;
	if (x <= 2.6e+155) {
		tmp = fma(t_0, -0.1111111111111111, (cbrt(pow(x, -2.0)) * 0.3333333333333333));
	} else {
		tmp = fma(t_0, -0.1111111111111111, (exp((log(x) * -0.6666666666666666)) * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt((x ^ -5.0))
	tmp = 0.0
	if (x <= 2.6e+155)
		tmp = fma(t_0, -0.1111111111111111, Float64(cbrt((x ^ -2.0)) * 0.3333333333333333));
	else
		tmp = fma(t_0, -0.1111111111111111, Float64(exp(Float64(log(x) * -0.6666666666666666)) * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[x, -5.0], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 2.6e+155], N[(t$95$0 * -0.1111111111111111 + N[(N[Power[N[Power[x, -2.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * -0.1111111111111111 + N[(N[Exp[N[(N[Log[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000002e155

   1. Initial program 7.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{\left(\mathsf{neg}\left(2\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      10. lift-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      12. lift-*.f64 96.6

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]

   8. Applied rewrites 96.6%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]

   if 2.6000000000000002e155 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{\left(\mathsf{neg}\left(2\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      10. lift-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      12. lift-*.f64 6.5

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]

   8. Applied rewrites 6.5%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]
   9. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      2. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      4. pow-pow N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

      6. metadata-eval 89.1

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

   10. Applied rewrites 89.1%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
   11. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       2. pow-to-exp N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       3. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       5. lift-log.f64 89.2

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\right) \]

   12. Applied rewrites 89.2%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.3e+155)
   (fma
    (cbrt (pow x -5.0))
    -0.1111111111111111
    (* (cbrt (pow x -2.0)) 0.3333333333333333))
   (fma
    (pow x -1.6666666666666667)
    -0.1111111111111111
    (* (pow x -0.6666666666666666) 0.3333333333333333))))

C

double code(double x) {
	double tmp;
	if (x <= 3.3e+155) {
		tmp = fma(cbrt(pow(x, -5.0)), -0.1111111111111111, (cbrt(pow(x, -2.0)) * 0.3333333333333333));
	} else {
		tmp = fma(pow(x, -1.6666666666666667), -0.1111111111111111, (pow(x, -0.6666666666666666) * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.3e+155)
		tmp = fma(cbrt((x ^ -5.0)), -0.1111111111111111, Float64(cbrt((x ^ -2.0)) * 0.3333333333333333));
	else
		tmp = fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64((x ^ -0.6666666666666666) * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.3e+155], N[(N[Power[N[Power[x, -5.0], $MachinePrecision], 1/3], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[N[Power[x, -2.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2999999999999999e155

   1. Initial program 7.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{\left(\mathsf{neg}\left(2\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      10. lift-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      12. lift-*.f64 96.6

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]

   8. Applied rewrites 96.6%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]

   if 3.2999999999999999e155 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{\left(\mathsf{neg}\left(2\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      10. lift-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      12. lift-*.f64 6.5

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]

   8. Applied rewrites 6.5%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]
   9. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      2. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      4. pow-pow N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

      6. metadata-eval 89.1

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

   10. Applied rewrites 89.1%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
   11. Step-by-step derivation

       1. lift-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       2. pow1/3 N/A

          \[\leadsto \mathsf{fma}\left({\left({x}^{-5}\right)}^{\frac{1}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       3. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({\left({x}^{-5}\right)}^{\frac{1}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       4. pow-pow N/A

          \[\leadsto \mathsf{fma}\left({x}^{\left(-5 \cdot \frac{1}{3}\right)}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       7. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       8. metadata-eval 89.1

          \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

   12. Applied rewrites 89.1%

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \cdot \left(x \cdot x\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/
    (*
     (fma
      (pow x -1.6666666666666667)
      -0.1111111111111111
      (* (cbrt (pow x -2.0)) 0.3333333333333333))
     (* x x))
    (* x x))
   (fma
    (pow x -1.6666666666666667)
    -0.1111111111111111
    (* (pow x -0.6666666666666666) 0.3333333333333333))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (fma(pow(x, -1.6666666666666667), -0.1111111111111111, (cbrt(pow(x, -2.0)) * 0.3333333333333333)) * (x * x)) / (x * x);
	} else {
		tmp = fma(pow(x, -1.6666666666666667), -0.1111111111111111, (pow(x, -0.6666666666666666) * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64(cbrt((x ^ -2.0)) * 0.3333333333333333)) * Float64(x * x)) / Float64(x * x));
	else
		tmp = fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64((x ^ -0.6666666666666666) * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[N[Power[x, -2.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{{x}^{2} \cdot \left(\frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)}{\color{blue}{x} \cdot x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \cdot {x}^{2}}{x \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \cdot {x}^{2}}{x \cdot x} \]

   8. Applied rewrites 96.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \cdot \left(x \cdot x\right)}{\color{blue}{x} \cdot x} \]
   9. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)}{x \cdot x} \]

      2. pow1/3 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left({x}^{-5}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)}{x \cdot x} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left({x}^{-5}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)}{x \cdot x} \]

      4. pow-pow N/A

         \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(-5 \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)}{x \cdot x} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)}{x \cdot x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)}{x \cdot x} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)}{x \cdot x} \]

      8. metadata-eval 96.7

         \[\leadsto \frac{\mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \cdot \left(x \cdot x\right)}{x \cdot x} \]

   10. Applied rewrites 96.7%

       \[\leadsto \frac{\mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \cdot \left(x \cdot x\right)}{x \cdot x} \]

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{\left(\mathsf{neg}\left(2\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      10. lift-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      12. lift-*.f64 7.9

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]

   8. Applied rewrites 7.9%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]
   9. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      2. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      4. pow-pow N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

      6. metadata-eval 89.1

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

   10. Applied rewrites 89.1%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
   11. Step-by-step derivation

       1. lift-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       2. pow1/3 N/A

          \[\leadsto \mathsf{fma}\left({\left({x}^{-5}\right)}^{\frac{1}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       3. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({\left({x}^{-5}\right)}^{\frac{1}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       4. pow-pow N/A

          \[\leadsto \mathsf{fma}\left({x}^{\left(-5 \cdot \frac{1}{3}\right)}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       7. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       8. metadata-eval 89.1

          \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

   12. Applied rewrites 89.1%

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (fma
    (cbrt (pow x -5.0))
    -0.1111111111111111
    (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333))
   (fma
    (pow x -1.6666666666666667)
    -0.1111111111111111
    (* (pow x -0.6666666666666666) 0.3333333333333333))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = fma(cbrt(pow(x, -5.0)), -0.1111111111111111, (cbrt((1.0 / (x * x))) * 0.3333333333333333));
	} else {
		tmp = fma(pow(x, -1.6666666666666667), -0.1111111111111111, (pow(x, -0.6666666666666666) * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = fma(cbrt((x ^ -5.0)), -0.1111111111111111, Float64(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333));
	else
		tmp = fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64((x ^ -0.6666666666666666) * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[Power[N[Power[x, -5.0], $MachinePrecision], 1/3], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{\left(\mathsf{neg}\left(2\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      10. lift-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      12. lift-*.f64 96.7

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]

   8. Applied rewrites 96.7%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{\left(\mathsf{neg}\left(2\right)\right)}} \cdot \frac{1}{3}\right) \]

      3. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3}\right) \]

      6. lower-*.f64 96.7

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\right) \]

   10. Applied rewrites 96.7%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\right) \]

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{\left(\mathsf{neg}\left(2\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      10. lift-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      12. lift-*.f64 7.9

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]

   8. Applied rewrites 7.9%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]
   9. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      2. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      4. pow-pow N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

      6. metadata-eval 89.1

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

   10. Applied rewrites 89.1%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
   11. Step-by-step derivation

       1. lift-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       2. pow1/3 N/A

          \[\leadsto \mathsf{fma}\left({\left({x}^{-5}\right)}^{\frac{1}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       3. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({\left({x}^{-5}\right)}^{\frac{1}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       4. pow-pow N/A

          \[\leadsto \mathsf{fma}\left({x}^{\left(-5 \cdot \frac{1}{3}\right)}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       7. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       8. metadata-eval 89.1

          \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

   12. Applied rewrites 89.1%

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* (/ 1.0 (cbrt (* x x))) 0.3333333333333333)
   (fma
    (pow x -1.6666666666666667)
    -0.1111111111111111
    (* (pow x -0.6666666666666666) 0.3333333333333333))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (1.0 / cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = fma(pow(x, -1.6666666666666667), -0.1111111111111111, (pow(x, -0.6666666666666666) * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(1.0 / cbrt(Float64(x * x))) * 0.3333333333333333);
	else
		tmp = fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64((x ^ -0.6666666666666666) * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(1.0 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 95.4

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

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   6. 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. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 95.6

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

   7. Applied rewrites 95.6%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\mathsf{neg}\left(5\right)\right)}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{\left(\mathsf{neg}\left(2\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      10. lift-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{{x}^{-2}}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      12. lift-*.f64 7.9

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]

   8. Applied rewrites 7.9%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]
   9. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3}\right) \]

      2. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      4. pow-pow N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

      6. metadata-eval 89.1

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

   10. Applied rewrites 89.1%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
   11. Step-by-step derivation

       1. lift-cbrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       2. pow1/3 N/A

          \[\leadsto \mathsf{fma}\left({\left({x}^{-5}\right)}^{\frac{1}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       3. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({\left({x}^{-5}\right)}^{\frac{1}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       4. pow-pow N/A

          \[\leadsto \mathsf{fma}\left({x}^{\left(-5 \cdot \frac{1}{3}\right)}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       7. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

       8. metadata-eval 89.1

          \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

   12. Applied rewrites 89.1%

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 51.0% accurate, 1.0× 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 6.4%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 52.4

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

5. Applied rewrites 52.4%

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

Alternative 9: 49.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.4%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 52.4

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

5. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
6. 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. cbrt-div N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 50.9

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

7. Applied rewrites 50.9%

   \[\leadsto \frac{1}{\sqrt[3]{x \cdot x}} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 10: 49.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.4%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 52.4

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

5. Applied rewrites 52.4%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-flip N/A

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

   4. lower-/.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 50.8

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

7. Applied rewrites 50.8%

   \[\leadsto \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 11: 4.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333, x, -\sqrt[3]{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma 0.3333333333333333 x (- (cbrt x))))

C

double code(double x) {
	return fma(0.3333333333333333, x, -cbrt(x));
}

Julia

function code(x)
	return fma(0.3333333333333333, x, Float64(-cbrt(x)))
end

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lift-cbrt.f64 4.2

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, x, 1 - \sqrt[3]{x}\right) \]

5. Applied rewrites 4.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, x, 1 - \sqrt[3]{x}\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\frac{1}{3}, x, -1 \cdot \sqrt[3]{x}\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, x, \mathsf{neg}\left(\sqrt[3]{x}\right)\right) \]

   2. lower-neg.f64 N/A

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

   3. lift-cbrt.f64 4.2

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, x, -\sqrt[3]{x}\right) \]

8. Applied rewrites 4.2%

   \[\leadsto \mathsf{fma}\left(0.3333333333333333, x, -\sqrt[3]{x}\right) \]
9. Add Preprocessing

Alternative 12: 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 6.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.8%

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

Developer Target 1: 98.4% 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 2025037 
(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)))