2cbrt (problem 3.3.4)

Percentage Accurate: 6.8% → 97.8%

Time: 4.0s

Alternatives: 11

Speedup: 1.9×

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: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \mathsf{fma}\left({x}^{-2.6666666666666665}, 0.06172839506172839, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{x} \cdot 1}}{\sqrt[3]{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.55e+14)
   (fma
    (pow x -1.6666666666666667)
    -0.1111111111111111
    (fma
     (pow x -2.6666666666666665)
     0.06172839506172839
     (* (pow x -0.6666666666666666) 0.3333333333333333)))
   (* (/ (cbrt (* (/ 1.0 x) 1.0)) (cbrt x)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 2.55e+14) {
		tmp = fma(pow(x, -1.6666666666666667), -0.1111111111111111, fma(pow(x, -2.6666666666666665), 0.06172839506172839, (pow(x, -0.6666666666666666) * 0.3333333333333333)));
	} else {
		tmp = (cbrt(((1.0 / x) * 1.0)) / cbrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.55e+14)
		tmp = fma((x ^ -1.6666666666666667), -0.1111111111111111, fma((x ^ -2.6666666666666665), 0.06172839506172839, Float64((x ^ -0.6666666666666666) * 0.3333333333333333)));
	else
		tmp = Float64(Float64(cbrt(Float64(Float64(1.0 / x) * 1.0)) / cbrt(x)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.55e+14], N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[x, -2.6666666666666665], $MachinePrecision] * 0.06172839506172839 + N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] * 1.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.55e14

   1. Initial program 58.9%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. pow1/3 N/A

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

      4. lower-pow.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 55.9

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

   3. Applied rewrites 55.9%

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

   4. Taylor expanded in x around inf

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

      1. pow-to-exp N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.1111111111111111, \sqrt[3]{x}, \mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, {x}^{-0.6666666666666666} \cdot 0.06172839506172839\right)\right)}{x \cdot x}} \]

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lift-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 85.5%

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

   if 2.55e14 < x

   1. Initial program 4.3%

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

   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 90.3

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

   4. Applied rewrites 90.3%

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

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 45.7

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

   6. Applied rewrites 45.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. pow1/3 N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. cbrt-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower-cbrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-cbrt.f64 98.4

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

   8. Applied rewrites 98.4%

      \[\leadsto \frac{\sqrt[3]{\frac{1}{x} \cdot 1}}{\sqrt[3]{x}} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. pow-prod-up N/A

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

   4. pow-prod-down N/A

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

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. inv-pow N/A

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

   8. pow1/3 N/A

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

   9. associate-/r* N/A

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

   10. cbrt-div N/A

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

   11. lower-/.f64 N/A

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

   12. lower-cbrt.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-cbrt.f64 97.6

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

9. Applied rewrites 97.6%

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

Alternative 3: 96.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

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

   3. pow-prod-up N/A

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

   4. pow-prod-down N/A

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

   5. pow2 N/A

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

   6. lower-pow.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 46.4

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

6. Applied rewrites 46.4%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. metadata-eval N/A

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

   5. pow-pow N/A

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

   6. inv-pow N/A

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

   7. pow1/3 N/A

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

   8. metadata-eval N/A

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

   9. pow2 N/A

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

   10. frac-times N/A

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

   11. lift-/.f64 N/A

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

   12. associate-*r/ N/A

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

   13. cbrt-div N/A

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

   14. lower-/.f64 N/A

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

   15. lower-cbrt.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lift-cbrt.f64 96.7

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

8. Applied rewrites 96.7%

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

Alternative 4: 96.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

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

   3. pow-prod-up N/A

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

   4. pow-prod-down N/A

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

   5. pow2 N/A

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

   6. lower-pow.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 46.4

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

6. Applied rewrites 46.4%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. metadata-eval N/A

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

   5. pow-pow N/A

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

   6. inv-pow N/A

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

   7. pow1/3 N/A

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

   8. metadata-eval N/A

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

   9. pow2 N/A

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

   10. frac-times N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. sqr-neg-rev N/A

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

   14. cbrt-prod N/A

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

   15. lower-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. distribute-neg-frac N/A

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

   18. metadata-eval N/A

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

   19. lower-cbrt.f64 N/A

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

   20. lower-/.f64 N/A

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

   21. lift-/.f64 N/A

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

   22. distribute-neg-frac N/A

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

   23. metadata-eval N/A

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

   24. lower-cbrt.f64 N/A

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

   25. lower-/.f64 96.3

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

8. Applied rewrites 96.3%

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

Alternative 5: 92.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (-1.0 / Math.cbrt(-(x * x))) * 0.3333333333333333;
	} else {
		tmp = Math.pow(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(Float64(-1.0 / cbrt(Float64(-Float64(x * x)))) * 0.3333333333333333);
	else
		tmp = Float64((exp(log(x)) ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.9%

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

   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. pow1/3 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. cbrt-div N/A

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

      10. metadata-eval N/A

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

      11. rem-cbrt-cube N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 95.3

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

   6. Applied rewrites 95.3%

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

   if 1.35000000000000003e154 < 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%

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

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 4.7

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

   6. Applied rewrites 4.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-prod-down N/A

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

      4. pow-prod-up N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-log.f64 89.8

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

   8. Applied rewrites 89.8%

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

Alternative 6: 92.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{\sqrt[3]{-x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* (/ -1.0 (cbrt (- (* x x)))) 0.3333333333333333)
   (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (-1.0 / cbrt(-(x * x))) * 0.3333333333333333;
	} else {
		tmp = exp((log(x) * -0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (-1.0 / Math.cbrt(-(x * x))) * 0.3333333333333333;
	} 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(Float64(-1.0 / cbrt(Float64(-Float64(x * x)))) * 0.3333333333333333);
	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[(N[(-1.0 / N[Power[(-N[(x * x), $MachinePrecision]), 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $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 8.9%

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

   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. pow1/3 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. cbrt-div N/A

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

      10. metadata-eval N/A

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

      11. rem-cbrt-cube N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 95.3

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

   6. Applied rewrites 95.3%

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

   if 1.35000000000000003e154 < 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%

      \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
   5. 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-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 89.5

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

   6. Applied rewrites 89.5%

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

Alternative 7: 92.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.35 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.35e+155)
   (* (cbrt (/ (/ 1.0 x) x)) 0.3333333333333333)
   (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 3.35e+155) {
		tmp = cbrt(((1.0 / x) / x)) * 0.3333333333333333;
	} else {
		tmp = exp((log(x) * -0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.35e+155) {
		tmp = Math.cbrt(((1.0 / x) / x)) * 0.3333333333333333;
	} else {
		tmp = Math.exp((Math.log(x) * -0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.35e+155)
		tmp = Float64(cbrt(Float64(Float64(1.0 / x) / x)) * 0.3333333333333333);
	else
		tmp = Float64(exp(Float64(log(x) * -0.6666666666666666)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.35e+155], N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $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 < 3.35e155

   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. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 94.3

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

   6. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 95.1

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

   8. Applied rewrites 95.1%

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

   if 3.35e155 < 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%

      \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
   5. 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-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 89.4

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

   6. Applied rewrites 89.4%

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

Alternative 8: 92.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333)
   (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} 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 = Math.cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} 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(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333);
	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[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $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 8.9%

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

   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. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 95.1

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

   6. Applied rewrites 95.1%

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

   if 1.35000000000000003e154 < 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%

      \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
   5. 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-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 89.5

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

   6. Applied rewrites 89.5%

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

Alternative 9: 89.2% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333))

C

double code(double x) {
	return exp((log(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 = exp((log(x) * (-0.6666666666666666d0))) * 0.3333333333333333d0
end function

Java

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

Python

def code(x):
	return math.exp((math.log(x) * -0.6666666666666666)) * 0.3333333333333333

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Exp[N[(N[Log[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]], $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. pow-to-exp N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-log.f64 89.2

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

6. Applied rewrites 89.2%

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

Alternative 10: 89.0% accurate, 1.9× 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 11: 1.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 - \sqrt[3]{x} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

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

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ x 1.0))))
   (/ 1.0 (+ (+ (* t_0 t_0) (* (cbrt x) t_0)) (* (cbrt x) (cbrt x))))))

C

double code(double x) {
	double t_0 = cbrt((x + 1.0));
	return 1.0 / (((t_0 * t_0) + (cbrt(x) * t_0)) + (cbrt(x) * cbrt(x)));
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((x + 1.0));
	return 1.0 / (((t_0 * t_0) + (Math.cbrt(x) * t_0)) + (Math.cbrt(x) * Math.cbrt(x)));
}

Julia

function code(x)
	t_0 = cbrt(Float64(x + 1.0))
	return Float64(1.0 / Float64(Float64(Float64(t_0 * t_0) + Float64(cbrt(x) * t_0)) + Float64(cbrt(x) * cbrt(x))))
end

Wolfram

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

Reproduce

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

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

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