2cbrt (problem 3.3.4)

Percentage Accurate: 6.8% → 97.6%

Time: 2.7s

Alternatives: 7

Speedup: 1.4×

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: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \left({\left(\sqrt[3]{x}\right)}^{-1} \cdot \sqrt[3]{{x}^{-1}}\right) \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (pow x -1.6666666666666667)
  -0.1111111111111111
  (* (* (pow (cbrt x) -1.0) (cbrt (pow x -1.0))) 0.3333333333333333)))

C

double code(double x) {
	return fma(pow(x, -1.6666666666666667), -0.1111111111111111, ((pow(cbrt(x), -1.0) * cbrt(pow(x, -1.0))) * 0.3333333333333333));
}

Julia

function code(x)
	return fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64(Float64((cbrt(x) ^ -1.0) * cbrt((x ^ -1.0))) * 0.3333333333333333))
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{\color{blue}{{x}^{2}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{\color{blue}{x}}^{2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{\color{blue}{x}}^{2}} \]

   5. pow1/3 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\left({x}^{4}\right)}^{\frac{1}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   6. pow-pow N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(4 \cdot \frac{1}{3}\right)}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   7. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(4 \cdot \frac{1}{3}\right)}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   10. lift-cbrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

   12. lower-*.f64 45.2

       \[\leadsto \frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

4. Applied rewrites 45.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
6. 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. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. pow1/3 N/A

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

   14. pow-flip N/A

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

   15. pow-pow N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lift-pow.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-*.f64 89.9

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

7. Applied rewrites 89.9%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, \color{blue}{-0.1111111111111111}, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. metadata-eval N/A

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

   5. pow-prod-up N/A

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

   6. pow1/3 N/A

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

   7. pow1/3 N/A

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

   8. unpow-prod-down N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lift-cbrt.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lift-cbrt.f64 97.5

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \left({\left(\sqrt[3]{x}\right)}^{-1} \cdot {\left(\sqrt[3]{x}\right)}^{-1}\right) \cdot 0.3333333333333333\right) \]

9. Applied rewrites 97.5%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \left({\left(\sqrt[3]{x}\right)}^{-1} \cdot {\left(\sqrt[3]{x}\right)}^{-1}\right) \cdot 0.3333333333333333\right) \]
10. Step-by-step derivation

    1. lift-cbrt.f64 N/A

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

    2. lift-pow.f64 N/A

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

    3. unpow-1 N/A

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

    4. metadata-eval N/A

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

    5. cbrt-div N/A

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

    6. inv-pow N/A

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

    7. lift-pow.f64 N/A

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

    8. lift-cbrt.f64 97.6

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \left({\left(\sqrt[3]{x}\right)}^{-1} \cdot \sqrt[3]{{x}^{-1}}\right) \cdot 0.3333333333333333\right) \]

11. Applied rewrites 97.6%

    \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \left({\left(\sqrt[3]{x}\right)}^{-1} \cdot \sqrt[3]{{x}^{-1}}\right) \cdot 0.3333333333333333\right) \]
12. Add Preprocessing

Alternative 2: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{-1} \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (pow x -1.6666666666666667)
  -0.1111111111111111
  (* (pow (pow (cbrt x) 2.0) -1.0) 0.3333333333333333)))

C

double code(double x) {
	return fma(pow(x, -1.6666666666666667), -0.1111111111111111, (pow(pow(cbrt(x), 2.0), -1.0) * 0.3333333333333333));
}

Julia

function code(x)
	return fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64(((cbrt(x) ^ 2.0) ^ -1.0) * 0.3333333333333333))
end

Wolfram

code[x_] := N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision], -1.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{\color{blue}{{x}^{2}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{\color{blue}{x}}^{2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{\color{blue}{x}}^{2}} \]

   5. pow1/3 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\left({x}^{4}\right)}^{\frac{1}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   6. pow-pow N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(4 \cdot \frac{1}{3}\right)}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   7. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(4 \cdot \frac{1}{3}\right)}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   10. lift-cbrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

   12. lower-*.f64 45.2

       \[\leadsto \frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

4. Applied rewrites 45.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
6. 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. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. pow1/3 N/A

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

   14. pow-flip N/A

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

   15. pow-pow N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lift-pow.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-*.f64 89.9

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

7. Applied rewrites 89.9%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, \color{blue}{-0.1111111111111111}, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. metadata-eval N/A

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

   5. pow-prod-up N/A

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

   6. pow1/3 N/A

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

   7. pow1/3 N/A

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

   8. unpow-prod-down N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lift-cbrt.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lift-cbrt.f64 97.5

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \left({\left(\sqrt[3]{x}\right)}^{-1} \cdot {\left(\sqrt[3]{x}\right)}^{-1}\right) \cdot 0.3333333333333333\right) \]

9. Applied rewrites 97.5%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \left({\left(\sqrt[3]{x}\right)}^{-1} \cdot {\left(\sqrt[3]{x}\right)}^{-1}\right) \cdot 0.3333333333333333\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-cbrt.f64 N/A

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

    3. lift-pow.f64 N/A

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

    4. lift-cbrt.f64 N/A

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

    5. lift-pow.f64 N/A

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

    6. pow-prod-down N/A

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

    7. unpow2 N/A

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

    8. lower-pow.f64 N/A

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

    9. lift-pow.f64 N/A

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

    10. lift-cbrt.f64 97.6

        \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{-1} \cdot 0.3333333333333333\right) \]

11. Applied rewrites 97.6%

    \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{-1} \cdot 0.3333333333333333\right) \]
12. Add Preprocessing

Alternative 3: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {\left({\left(\sqrt[3]{x}\right)}^{-1}\right)}^{2} \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (pow x -1.6666666666666667)
  -0.1111111111111111
  (* (pow (pow (cbrt x) -1.0) 2.0) 0.3333333333333333)))

C

double code(double x) {
	return fma(pow(x, -1.6666666666666667), -0.1111111111111111, (pow(pow(cbrt(x), -1.0), 2.0) * 0.3333333333333333));
}

Julia

function code(x)
	return fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64(((cbrt(x) ^ -1.0) ^ 2.0) * 0.3333333333333333))
end

Wolfram

code[x_] := N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[N[Power[N[Power[x, 1/3], $MachinePrecision], -1.0], $MachinePrecision], 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{\color{blue}{{x}^{2}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{\color{blue}{x}}^{2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{\color{blue}{x}}^{2}} \]

   5. pow1/3 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\left({x}^{4}\right)}^{\frac{1}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   6. pow-pow N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(4 \cdot \frac{1}{3}\right)}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   7. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(4 \cdot \frac{1}{3}\right)}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   10. lift-cbrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

   12. lower-*.f64 45.2

       \[\leadsto \frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

4. Applied rewrites 45.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
6. 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. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. pow1/3 N/A

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

   14. pow-flip N/A

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

   15. pow-pow N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lift-pow.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-*.f64 89.9

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

7. Applied rewrites 89.9%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, \color{blue}{-0.1111111111111111}, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. metadata-eval N/A

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

   5. pow-prod-up N/A

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

   6. pow1/3 N/A

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

   7. pow1/3 N/A

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

   8. unpow-prod-down N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lift-cbrt.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lift-cbrt.f64 97.5

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \left({\left(\sqrt[3]{x}\right)}^{-1} \cdot {\left(\sqrt[3]{x}\right)}^{-1}\right) \cdot 0.3333333333333333\right) \]

9. Applied rewrites 97.5%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \left({\left(\sqrt[3]{x}\right)}^{-1} \cdot {\left(\sqrt[3]{x}\right)}^{-1}\right) \cdot 0.3333333333333333\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. pow2 N/A

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

    3. lower-pow.f64 97.5

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {\left({\left(\sqrt[3]{x}\right)}^{-1}\right)}^{2} \cdot 0.3333333333333333\right) \]

11. Applied rewrites 97.5%

    \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {\left({\left(\sqrt[3]{x}\right)}^{-1}\right)}^{2} \cdot 0.3333333333333333\right) \]
12. Add Preprocessing

Alternative 4: 96.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.pow(Math.cbrt(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, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 89.0

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

4. Applied rewrites 89.0%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. pow-flip N/A

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

   6. pow1/3 N/A

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

   7. cbrt-div N/A

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

   8. metadata-eval N/A

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

   9. inv-pow N/A

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

   10. pow2 N/A

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

   11. cbrt-prod N/A

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

   12. lower-pow.f64 N/A

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

   13. pow1/3 N/A

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

   14. pow1/3 N/A

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

   15. pow-prod-up N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

       \[\leadsto {\left({x}^{\left(\frac{\frac{4}{3}}{2}\right)}\right)}^{-1} \cdot \frac{1}{3} \]

   18. lower-pow.f64 N/A

       \[\leadsto {\left({x}^{\left(\frac{\frac{4}{3}}{2}\right)}\right)}^{-1} \cdot \frac{1}{3} \]

   19. metadata-eval 89.0

       \[\leadsto {\left({x}^{0.6666666666666666}\right)}^{-1} \cdot 0.3333333333333333 \]

6. Applied rewrites 89.0%

   \[\leadsto {\left({x}^{0.6666666666666666}\right)}^{-1} \cdot 0.3333333333333333 \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. pow1/3 N/A

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

   5. pow1/3 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-cbrt.f64 96.6

      \[\leadsto {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{-1} \cdot 0.3333333333333333 \]

8. Applied rewrites 96.6%

   \[\leadsto {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{-1} \cdot 0.3333333333333333 \]
9. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-pow N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. metadata-eval 96.6

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

10. Applied rewrites 96.6%

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

Alternative 5: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.1e+155)
   (* (cbrt (pow x -2.0)) 0.3333333333333333)
   (* (pow x -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 2.1e+155) {
		tmp = cbrt(pow(x, -2.0)) * 0.3333333333333333;
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.1e+155) {
		tmp = Math.cbrt(Math.pow(x, -2.0)) * 0.3333333333333333;
	} else {
		tmp = Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.1e+155)
		tmp = Float64(cbrt((x ^ -2.0)) * 0.3333333333333333);
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.1e+155], N[(N[Power[N[Power[x, -2.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1e155

   1. Initial program 8.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 88.8

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

   4. Applied rewrites 88.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. pow1/3 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. pow2 N/A

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

      14. pow-flip N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval 95.1

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

   6. Applied rewrites 95.1%

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

   if 2.1e155 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 89.1

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

   4. Applied rewrites 89.1%

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

Alternative 6: 89.0% accurate, 1.4× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 6.8%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 89.0

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

4. Applied rewrites 89.0%

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

Alternative 7: 1.8% accurate, 1.8× 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 6.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 1.8%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))
  (- (cbrt (+ x 1.0)) (cbrt x)))