2cbrt (problem 3.3.4)

Percentage Accurate: 7.2% → 98.5%

Time: 3.2s

Alternatives: 12

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.4× speedup

Math

\[\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, 0.06172839506172839 \cdot \frac{1}{{x}^{1.6666666666666667}}\right) - \left(\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + 0.0411522633744856 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)}{x} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (-
   (fma
    0.3333333333333333
    (cbrt x)
    (* 0.06172839506172839 (/ 1.0 (pow x 1.6666666666666667))))
   (+
    (/ 0.1111111111111111 (pow x 0.6666666666666666))
    (* 0.0411522633744856 (/ 1.0 (pow x 2.6666666666666665)))))
  x))

C

double code(double x) {
	return (fma(0.3333333333333333, cbrt(x), (0.06172839506172839 * (1.0 / pow(x, 1.6666666666666667)))) - ((0.1111111111111111 / pow(x, 0.6666666666666666)) + (0.0411522633744856 * (1.0 / pow(x, 2.6666666666666665))))) / x;
}

Julia

function code(x)
	return Float64(Float64(fma(0.3333333333333333, cbrt(x), Float64(0.06172839506172839 * Float64(1.0 / (x ^ 1.6666666666666667)))) - Float64(Float64(0.1111111111111111 / (x ^ 0.6666666666666666)) + Float64(0.0411522633744856 * Float64(1.0 / (x ^ 2.6666666666666665))))) / x)
end

Wolfram

code[x_] := N[(N[(N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision] + N[(0.06172839506172839 * N[(1.0 / N[Power[x, 1.6666666666666667], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.1111111111111111 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision] + N[(0.0411522633744856 * N[(1.0 / N[Power[x, 2.6666666666666665], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-to-mult N/A

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

   4. lower-unsound-*.f64 N/A

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

   5. lower-unsound--.f64 N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. metadata-eval 7.2

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

3. Applied rewrites 7.2%

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

4. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{1}{3} \cdot \sqrt[3]{x} + \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}{x}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{3} \cdot \sqrt[3]{x} + \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}{\color{blue}{x}} \]

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, 0.06172839506172839 \cdot \frac{1}{{x}^{1.6666666666666667}}\right) - \left(\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + 0.0411522633744856 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)}{x}} \]
7. Add Preprocessing

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\frac{1}{\frac{x}{\mathsf{fma}\left(\sqrt[3]{x}, 0.3333333333333333, {x}^{-1.6666666666666667} \cdot 0.06172839506172839\right) - \mathsf{fma}\left({x}^{-0.6666666666666666}, 0.1111111111111111, {x}^{-2.6666666666666665} \cdot 0.0411522633744856\right)}} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (/
   x
   (-
    (fma
     (cbrt x)
     0.3333333333333333
     (* (pow x -1.6666666666666667) 0.06172839506172839))
    (fma
     (pow x -0.6666666666666666)
     0.1111111111111111
     (* (pow x -2.6666666666666665) 0.0411522633744856))))))

C

double code(double x) {
	return 1.0 / (x / (fma(cbrt(x), 0.3333333333333333, (pow(x, -1.6666666666666667) * 0.06172839506172839)) - fma(pow(x, -0.6666666666666666), 0.1111111111111111, (pow(x, -2.6666666666666665) * 0.0411522633744856))));
}

Julia

function code(x)
	return Float64(1.0 / Float64(x / Float64(fma(cbrt(x), 0.3333333333333333, Float64((x ^ -1.6666666666666667) * 0.06172839506172839)) - fma((x ^ -0.6666666666666666), 0.1111111111111111, Float64((x ^ -2.6666666666666665) * 0.0411522633744856)))))
end

Wolfram

code[x_] := N[(1.0 / N[(x / N[(N[(N[Power[x, 1/3], $MachinePrecision] * 0.3333333333333333 + N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * 0.06172839506172839), $MachinePrecision]), $MachinePrecision] - N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.1111111111111111 + N[(N[Power[x, -2.6666666666666665], $MachinePrecision] * 0.0411522633744856), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-to-mult N/A

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

   4. lower-unsound-*.f64 N/A

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

   5. lower-unsound--.f64 N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. metadata-eval 7.2

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

3. Applied rewrites 7.2%

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

4. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{1}{3} \cdot \sqrt[3]{x} + \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}{x}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{3} \cdot \sqrt[3]{x} + \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}{\color{blue}{x}} \]

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, 0.06172839506172839 \cdot \frac{1}{{x}^{1.6666666666666667}}\right) - \left(\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + 0.0411522633744856 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)}{x}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}{\color{blue}{x}} \]

   2. div-flip N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}}} \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}}} \]

   4. lower-unsound-/.f64 98.5

      \[\leadsto \frac{1}{\frac{x}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, 0.06172839506172839 \cdot \frac{1}{{x}^{1.6666666666666667}}\right) - \left(\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + 0.0411522633744856 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)}}} \]

   5. lift-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{x}{\left(\frac{1}{3} \cdot \sqrt[3]{x} + \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\color{blue}{\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{1}{\frac{x}{\left(\sqrt[3]{x} \cdot \frac{1}{3} + \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\color{blue}{\frac{1}{9}}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}} \]

   7. lower-fma.f64 98.5

      \[\leadsto \frac{1}{\frac{x}{\mathsf{fma}\left(\sqrt[3]{x}, 0.3333333333333333, 0.06172839506172839 \cdot \frac{1}{{x}^{1.6666666666666667}}\right) - \left(\color{blue}{\frac{0.1111111111111111}{{x}^{0.6666666666666666}}} + 0.0411522633744856 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{x}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{1}{3}, \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\frac{x}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{1}{3}, \frac{1}{{x}^{\frac{5}{3}}} \cdot \frac{5}{81}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}} \]

   10. lower-*.f64 98.5

       \[\leadsto \frac{1}{\frac{x}{\mathsf{fma}\left(\sqrt[3]{x}, 0.3333333333333333, \frac{1}{{x}^{1.6666666666666667}} \cdot 0.06172839506172839\right) - \left(\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + 0.0411522633744856 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)}} \]

   11. lift-/.f64 N/A

       \[\leadsto \frac{1}{\frac{x}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{1}{3}, \frac{1}{{x}^{\frac{5}{3}}} \cdot \frac{5}{81}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}} \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{1}{\frac{x}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{1}{3}, \frac{1}{{x}^{\frac{5}{3}}} \cdot \frac{5}{81}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}} \]

   13. pow-flip N/A

       \[\leadsto \frac{1}{\frac{x}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{1}{3}, {x}^{\left(\mathsf{neg}\left(\frac{5}{3}\right)\right)} \cdot \frac{5}{81}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}} \]

   14. lower-pow.f64 N/A

       \[\leadsto \frac{1}{\frac{x}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{1}{3}, {x}^{\left(\mathsf{neg}\left(\frac{5}{3}\right)\right)} \cdot \frac{5}{81}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}} \]

   15. metadata-eval 98.5

       \[\leadsto \frac{1}{\frac{x}{\mathsf{fma}\left(\sqrt[3]{x}, 0.3333333333333333, {x}^{-1.6666666666666667} \cdot 0.06172839506172839\right) - \left(\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + 0.0411522633744856 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)}} \]

8. Applied rewrites 98.5%

   \[\leadsto \frac{1}{\color{blue}{\frac{x}{\mathsf{fma}\left(\sqrt[3]{x}, 0.3333333333333333, {x}^{-1.6666666666666667} \cdot 0.06172839506172839\right) - \mathsf{fma}\left({x}^{-0.6666666666666666}, 0.1111111111111111, {x}^{-2.6666666666666665} \cdot 0.0411522633744856\right)}}} \]
9. Add Preprocessing

Alternative 3: 98.5% accurate, 0.4× speedup

Math

\[\frac{1}{x} \cdot \left(\mathsf{fma}\left(\sqrt[3]{x}, 0.3333333333333333, {x}^{-1.6666666666666667} \cdot 0.06172839506172839\right) - \mathsf{fma}\left({x}^{-0.6666666666666666}, 0.1111111111111111, {x}^{-2.6666666666666665} \cdot 0.0411522633744856\right)\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 x)
  (-
   (fma
    (cbrt x)
    0.3333333333333333
    (* (pow x -1.6666666666666667) 0.06172839506172839))
   (fma
    (pow x -0.6666666666666666)
    0.1111111111111111
    (* (pow x -2.6666666666666665) 0.0411522633744856)))))

C

double code(double x) {
	return (1.0 / x) * (fma(cbrt(x), 0.3333333333333333, (pow(x, -1.6666666666666667) * 0.06172839506172839)) - fma(pow(x, -0.6666666666666666), 0.1111111111111111, (pow(x, -2.6666666666666665) * 0.0411522633744856)));
}

Julia

function code(x)
	return Float64(Float64(1.0 / x) * Float64(fma(cbrt(x), 0.3333333333333333, Float64((x ^ -1.6666666666666667) * 0.06172839506172839)) - fma((x ^ -0.6666666666666666), 0.1111111111111111, Float64((x ^ -2.6666666666666665) * 0.0411522633744856))))
end

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] * N[(N[(N[Power[x, 1/3], $MachinePrecision] * 0.3333333333333333 + N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * 0.06172839506172839), $MachinePrecision]), $MachinePrecision] - N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.1111111111111111 + N[(N[Power[x, -2.6666666666666665], $MachinePrecision] * 0.0411522633744856), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-to-mult N/A

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

   4. lower-unsound-*.f64 N/A

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

   5. lower-unsound--.f64 N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. metadata-eval 7.2

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

3. Applied rewrites 7.2%

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

4. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{1}{3} \cdot \sqrt[3]{x} + \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}{x}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{3} \cdot \sqrt[3]{x} + \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}{\color{blue}{x}} \]

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, 0.06172839506172839 \cdot \frac{1}{{x}^{1.6666666666666667}}\right) - \left(\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + 0.0411522633744856 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)}{x}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}{\color{blue}{x}} \]

   2. mult-flip N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot \color{blue}{\frac{1}{x}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right)} \]

   5. lower-/.f64 98.5

      \[\leadsto \frac{1}{x} \cdot \left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, 0.06172839506172839 \cdot \frac{1}{{x}^{1.6666666666666667}}\right)} - \left(\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + 0.0411522633744856 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)\right) \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{1}{x} \cdot \left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\color{blue}{\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{x} \cdot \left(\left(\sqrt[3]{x} \cdot \frac{1}{3} + \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\color{blue}{\frac{1}{9}}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \]

   8. lower-fma.f64 98.5

      \[\leadsto \frac{1}{x} \cdot \left(\mathsf{fma}\left(\sqrt[3]{x}, 0.3333333333333333, 0.06172839506172839 \cdot \frac{1}{{x}^{1.6666666666666667}}\right) - \left(\color{blue}{\frac{0.1111111111111111}{{x}^{0.6666666666666666}}} + 0.0411522633744856 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{x} \cdot \left(\mathsf{fma}\left(\sqrt[3]{x}, \frac{1}{3}, \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{x} \cdot \left(\mathsf{fma}\left(\sqrt[3]{x}, \frac{1}{3}, \frac{1}{{x}^{\frac{5}{3}}} \cdot \frac{5}{81}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \]

   11. lower-*.f64 98.5

       \[\leadsto \frac{1}{x} \cdot \left(\mathsf{fma}\left(\sqrt[3]{x}, 0.3333333333333333, \frac{1}{{x}^{1.6666666666666667}} \cdot 0.06172839506172839\right) - \left(\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + 0.0411522633744856 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)\right) \]

   12. lift-/.f64 N/A

       \[\leadsto \frac{1}{x} \cdot \left(\mathsf{fma}\left(\sqrt[3]{x}, \frac{1}{3}, \frac{1}{{x}^{\frac{5}{3}}} \cdot \frac{5}{81}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{1}{x} \cdot \left(\mathsf{fma}\left(\sqrt[3]{x}, \frac{1}{3}, \frac{1}{{x}^{\frac{5}{3}}} \cdot \frac{5}{81}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \]

   14. pow-flip N/A

       \[\leadsto \frac{1}{x} \cdot \left(\mathsf{fma}\left(\sqrt[3]{x}, \frac{1}{3}, {x}^{\left(\mathsf{neg}\left(\frac{5}{3}\right)\right)} \cdot \frac{5}{81}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{1}{x} \cdot \left(\mathsf{fma}\left(\sqrt[3]{x}, \frac{1}{3}, {x}^{\left(\mathsf{neg}\left(\frac{5}{3}\right)\right)} \cdot \frac{5}{81}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \]

   16. metadata-eval 98.5

       \[\leadsto \frac{1}{x} \cdot \left(\mathsf{fma}\left(\sqrt[3]{x}, 0.3333333333333333, {x}^{-1.6666666666666667} \cdot 0.06172839506172839\right) - \left(\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + 0.0411522633744856 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)\right) \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{1}{x} \cdot \left(\mathsf{fma}\left(\sqrt[3]{x}, \frac{1}{3}, {x}^{\frac{-5}{3}} \cdot \frac{5}{81}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \color{blue}{\frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}}\right)\right) \]

8. Applied rewrites 98.5%

   \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{x}, 0.3333333333333333, {x}^{-1.6666666666666667} \cdot 0.06172839506172839\right) - \mathsf{fma}\left({x}^{-0.6666666666666666}, 0.1111111111111111, {x}^{-2.6666666666666665} \cdot 0.0411522633744856\right)\right)} \]
9. Add Preprocessing

Alternative 4: 98.3% accurate, 0.6× speedup

Math

\[\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, \frac{0.06172839506172839}{{x}^{1.6666666666666667}}\right) - \frac{0.1111111111111111}{{x}^{0.6666666666666666}}}{x} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (-
   (fma
    0.3333333333333333
    (cbrt x)
    (/ 0.06172839506172839 (pow x 1.6666666666666667)))
   (/ 0.1111111111111111 (pow x 0.6666666666666666)))
  x))

C

double code(double x) {
	return (fma(0.3333333333333333, cbrt(x), (0.06172839506172839 / pow(x, 1.6666666666666667))) - (0.1111111111111111 / pow(x, 0.6666666666666666))) / x;
}

Julia

function code(x)
	return Float64(Float64(fma(0.3333333333333333, cbrt(x), Float64(0.06172839506172839 / (x ^ 1.6666666666666667))) - Float64(0.1111111111111111 / (x ^ 0.6666666666666666))) / x)
end

Wolfram

code[x_] := N[(N[(N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision] + N[(0.06172839506172839 / N[Power[x, 1.6666666666666667], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.1111111111111111 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-to-mult N/A

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

   4. lower-unsound-*.f64 N/A

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

   5. lower-unsound--.f64 N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. metadata-eval 7.2

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

3. Applied rewrites 7.2%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   4. lower-cbrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   8. lower-pow.f64 98.3

      \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, \frac{0.06172839506172839}{{x}^{1.6666666666666667}}\right) - \frac{0.1111111111111111}{{x}^{0.6666666666666666}}}{x} \]

6. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, \frac{0.06172839506172839}{{x}^{1.6666666666666667}}\right) - \frac{0.1111111111111111}{{x}^{0.6666666666666666}}}{x}} \]
7. Add Preprocessing

Alternative 5: 98.3% accurate, 0.7× speedup

Math

\[\mathsf{fma}\left(\frac{-0.3333333333333333 \cdot x}{x}, \frac{1}{x}, \frac{\mathsf{fma}\left(0.1111111111111111, x, -0.06172839506172839\right)}{\left(x \cdot x\right) \cdot x}\right) \cdot \left(-\sqrt[3]{x}\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (/ (* -0.3333333333333333 x) x)
   (/ 1.0 x)
   (/ (fma 0.1111111111111111 x -0.06172839506172839) (* (* x x) x)))
  (- (cbrt x))))

C

double code(double x) {
	return fma(((-0.3333333333333333 * x) / x), (1.0 / x), (fma(0.1111111111111111, x, -0.06172839506172839) / ((x * x) * x))) * -cbrt(x);
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(-0.3333333333333333 * x) / x), Float64(1.0 / x), Float64(fma(0.1111111111111111, x, -0.06172839506172839) / Float64(Float64(x * x) * x))) * Float64(-cbrt(x)))
end

Wolfram

code[x_] := N[(N[(N[(N[(-0.3333333333333333 * x), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(N[(0.1111111111111111 * x + -0.06172839506172839), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Power[x, 1/3], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. add-flip N/A

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

   5. sub-to-mult N/A

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

   6. lower-unsound-*.f64 N/A

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

   7. lower-unsound--.f64 N/A

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

   8. lower-unsound-/.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. cbrt-neg-rev N/A

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

   11. lift-+.f64 N/A

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

   12. add-flip N/A

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

   13. sub-negate-rev N/A

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

   14. lower-cbrt.f64 N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-neg.f64 7.2

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

3. Applied rewrites 7.2%

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

5. Applied rewrites 7.2%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 34.4

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

8. Applied rewrites 34.4%

   \[\leadsto \color{blue}{\frac{\left(-0.06172839506172839 + 0.1111111111111111 \cdot x\right) - 0.3333333333333333 \cdot {x}^{2}}{{x}^{3}}} \cdot \left(-\sqrt[3]{x}\right) \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

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

   4. +-commutative N/A

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

   5. div-add N/A

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

10. Applied rewrites 98.3%

    \[\leadsto \mathsf{fma}\left(\frac{-0.3333333333333333 \cdot x}{x}, \color{blue}{\frac{1}{x}}, \frac{\mathsf{fma}\left(0.1111111111111111, x, -0.06172839506172839\right)}{\left(x \cdot x\right) \cdot x}\right) \cdot \left(-\sqrt[3]{x}\right) \]
11. Add Preprocessing

Alternative 6: 98.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma
   (pow x -0.6666666666666666)
   -0.1111111111111111
   (* 0.3333333333333333 (cbrt x)))
  x))

C

double code(double x) {
	return fma(pow(x, -0.6666666666666666), -0.1111111111111111, (0.3333333333333333 * cbrt(x))) / x;
}

Julia

function code(x)
	return Float64(fma((x ^ -0.6666666666666666), -0.1111111111111111, Float64(0.3333333333333333 * cbrt(x))) / x)
end

Wolfram

code[x_] := N[(N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * -0.1111111111111111 + N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-to-mult N/A

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

   4. lower-unsound-*.f64 N/A

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

   5. lower-unsound--.f64 N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. metadata-eval 7.2

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

3. Applied rewrites 7.2%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 98.0

      \[\leadsto \frac{0.3333333333333333 \cdot \sqrt[3]{x} - 0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}}{x} \]

6. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \sqrt[3]{x} - 0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}}{x}} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. pow-flip N/A

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

   11. metadata-eval N/A

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

   12. lift-pow.f64 N/A

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

   13. metadata-eval 98.0

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

8. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{fma}\left({x}^{-0.6666666666666666}, -0.1111111111111111, 0.3333333333333333 \cdot \sqrt[3]{x}\right)}{x} \]
9. Add Preprocessing

Alternative 7: 98.0% accurate, 1.1× speedup

Math

\[\frac{0.1111111111111111 \cdot \frac{1}{x} - 0.3333333333333333}{x} \cdot \left(-\sqrt[3]{x}\right) \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. add-flip N/A

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

   5. sub-to-mult N/A

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

   6. lower-unsound-*.f64 N/A

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

   7. lower-unsound--.f64 N/A

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

   8. lower-unsound-/.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. cbrt-neg-rev N/A

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

   11. lift-+.f64 N/A

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

   12. add-flip N/A

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

   13. sub-negate-rev N/A

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

   14. lower-cbrt.f64 N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-neg.f64 7.2

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

3. Applied rewrites 7.2%

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

5. Applied rewrites 7.2%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 34.4

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

8. Applied rewrites 34.4%

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

9. Taylor expanded in x around inf

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

    1. lower-/.f64 N/A

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

    2. lower--.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-/.f64 98.0

       \[\leadsto \frac{0.1111111111111111 \cdot \frac{1}{x} - 0.3333333333333333}{x} \cdot \left(-\sqrt[3]{x}\right) \]

11. Applied rewrites 98.0%

    \[\leadsto \frac{0.1111111111111111 \cdot \frac{1}{x} - 0.3333333333333333}{\color{blue}{x}} \cdot \left(-\sqrt[3]{x}\right) \]
12. Add Preprocessing

Alternative 8: 96.9% accurate, 1.6× speedup

Math

\[\frac{-0.3333333333333333}{x} \cdot \left(-\sqrt[3]{x}\right) \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. add-flip N/A

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

   5. sub-to-mult N/A

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

   6. lower-unsound-*.f64 N/A

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

   7. lower-unsound--.f64 N/A

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

   8. lower-unsound-/.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. cbrt-neg-rev N/A

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

   11. lift-+.f64 N/A

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

   12. add-flip N/A

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

   13. sub-negate-rev N/A

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

   14. lower-cbrt.f64 N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-neg.f64 7.2

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

3. Applied rewrites 7.2%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 96.9

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

6. Applied rewrites 96.9%

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

Alternative 9: 92.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 7.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. cbrt-unprod N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 50.7

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

   6. Applied rewrites 50.7%

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

   if 1.35000000000000003e154 < x

   1. Initial program 7.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-flip N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 88.7

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

   6. Applied rewrites 88.7%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-unsound-exp.f64 N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound-log.f64 89.0

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

   8. Applied rewrites 89.0%

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

Alternative 10: 92.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 7.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. cbrt-unprod N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 50.7

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

   6. Applied rewrites 50.7%

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

   if 1.35000000000000003e154 < x

   1. Initial program 7.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-flip N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 88.7

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

   6. Applied rewrites 88.7%

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

Alternative 11: 88.7% accurate, 1.9× speedup

Math

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

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 7.2%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.7

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

4. Applied rewrites 88.7%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow-flip N/A

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

   7. lower-pow.f64 N/A

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

   8. metadata-eval 88.7

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

6. Applied rewrites 88.7%

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

Alternative 12: 6.3% accurate, 36.6× speedup

Math

\[1 \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

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 = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 7.2%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-to-mult N/A

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

   4. lower-unsound-*.f64 N/A

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

   5. lower-unsound--.f64 N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. metadata-eval 7.2

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

3. Applied rewrites 7.2%

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

4. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{1}{3} \cdot \sqrt[3]{x} + \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}{x}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{3} \cdot \sqrt[3]{x} + \frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) - \left(\frac{\frac{1}{9}}{{x}^{\frac{2}{3}}} + \frac{10}{243} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)}{\color{blue}{x}} \]

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, 0.06172839506172839 \cdot \frac{1}{{x}^{1.6666666666666667}}\right) - \left(\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + 0.0411522633744856 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)}{x}} \]

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
8. Step-by-step derivation

9. Applied rewrites 6.3%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Developer Target 1: 98.4% accurate, 0.3× speedup

Math

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

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 2025166 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))
  :herbie-expected 5/2

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

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