2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.4%

Time: 6.6s

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: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(x, {\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}^{3}, {\left(1 + x\right)}^{2}\right)} \cdot \mathsf{fma}\left(t\_0, t\_0 - e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}, e^{\mathsf{log1p}\left(x\right) \cdot 1.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{-1}} \cdot 0.3333333333333333}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (cbrt x) (cbrt x) (cbrt (fma x x x)))))
   (if (<= x 5e+14)
     (*
      (/
       (- (+ 1.0 x) x)
       (fma x (pow (+ (cbrt (+ 1.0 x)) (cbrt x)) 3.0) (pow (+ 1.0 x) 2.0)))
      (fma
       t_0
       (- t_0 (exp (* (log1p x) 0.6666666666666666)))
       (exp (* (log1p x) 1.3333333333333333))))
     (/ (* (cbrt (pow x -1.0)) 0.3333333333333333) (cbrt x)))))

C

double code(double x) {
	double t_0 = fma(cbrt(x), cbrt(x), cbrt(fma(x, x, x)));
	double tmp;
	if (x <= 5e+14) {
		tmp = (((1.0 + x) - x) / fma(x, pow((cbrt((1.0 + x)) + cbrt(x)), 3.0), pow((1.0 + x), 2.0))) * fma(t_0, (t_0 - exp((log1p(x) * 0.6666666666666666))), exp((log1p(x) * 1.3333333333333333)));
	} else {
		tmp = (cbrt(pow(x, -1.0)) * 0.3333333333333333) / cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(cbrt(x), cbrt(x), cbrt(fma(x, x, x)))
	tmp = 0.0
	if (x <= 5e+14)
		tmp = Float64(Float64(Float64(Float64(1.0 + x) - x) / fma(x, (Float64(cbrt(Float64(1.0 + x)) + cbrt(x)) ^ 3.0), (Float64(1.0 + x) ^ 2.0))) * fma(t_0, Float64(t_0 - exp(Float64(log1p(x) * 0.6666666666666666))), exp(Float64(log1p(x) * 1.3333333333333333))));
	else
		tmp = Float64(Float64(cbrt((x ^ -1.0)) * 0.3333333333333333) / cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 1/3], $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e+14], N[(N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(x * N[Power[N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(t$95$0 - N[Exp[N[(N[Log[1 + x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Exp[N[(N[Log[1 + x], $MachinePrecision] * 1.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[Power[x, -1.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5e14

   1. Initial program 61.2%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. pow1/3 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. metadata-eval 59.5

          \[\leadsto \sqrt[3]{x + 1} - {\left(\sqrt{x}\right)}^{\color{blue}{0.6666666666666666}} \]

   4. Applied rewrites 59.5%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left(\sqrt{x}\right)}^{0.6666666666666666}} \]

   6. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(x, {\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}^{3}, {\left(1 + x\right)}^{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right), \mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right) - e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}, e^{\mathsf{log1p}\left(x\right) \cdot 1.3333333333333333}\right)} \]

   if 5e14 < x

   1. Initial program 4.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(x, {\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}^{3}, {\left(1 + x\right)}^{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right), \mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right) - e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}, e^{\mathsf{log1p}\left(x\right) \cdot 1.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{-1}} \cdot 0.3333333333333333}{\sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{x}, e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{-1}} \cdot 0.3333333333333333}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.2e+14)
   (/
    (- (+ 1.0 x) x)
    (fma
     (+ (cbrt (+ 1.0 x)) (cbrt x))
     (cbrt x)
     (exp (* (log1p x) 0.6666666666666666))))
   (/ (* (cbrt (pow x -1.0)) 0.3333333333333333) (cbrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.2e+14) {
		tmp = ((1.0 + x) - x) / fma((cbrt((1.0 + x)) + cbrt(x)), cbrt(x), exp((log1p(x) * 0.6666666666666666)));
	} else {
		tmp = (cbrt(pow(x, -1.0)) * 0.3333333333333333) / cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2e+14)
		tmp = Float64(Float64(Float64(1.0 + x) - x) / fma(Float64(cbrt(Float64(1.0 + x)) + cbrt(x)), cbrt(x), exp(Float64(log1p(x) * 0.6666666666666666))));
	else
		tmp = Float64(Float64(cbrt((x ^ -1.0)) * 0.3333333333333333) / cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.2e+14], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + N[Exp[N[(N[Log[1 + x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[Power[x, -1.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2e14

   1. Initial program 64.6%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. pow1/3 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. metadata-eval 62.9

          \[\leadsto \sqrt[3]{x + 1} - {\left(\sqrt{x}\right)}^{\color{blue}{0.6666666666666666}} \]

   4. Applied rewrites 62.9%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left(\sqrt{x}\right)}^{0.6666666666666666}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-pow2 N/A

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

      4. metadata-eval N/A

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

      5. pow1/3 N/A

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

      6. lift-cbrt.f64 64.6

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

      7. lower--.f64 N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cbrt.f64 N/A

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

      11. rem-cube-cbrt N/A

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

      12. lift-cbrt.f64 N/A

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

      13. rem-cube-cbrt N/A

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

      14. lower--.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

   6. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{x}, e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}\right)}} \]

   if 3.2e14 < x

   1. Initial program 4.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 54.9

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

   5. Applied rewrites 54.9%

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{x}, e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{-1}} \cdot 0.3333333333333333}{\sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. sqr-neg-rev N/A

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

   5. associate-/r* N/A

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

   6. distribute-neg-frac2 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. distribute-neg-frac2 N/A

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

   13. associate-/r* N/A

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

   14. sqr-neg-rev N/A

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

   15. unpow2 N/A

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

   16. unpow2 N/A

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

   17. associate-/r* N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 96.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 96.6%

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

11. Final simplification 96.6%

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

Alternative 4: 93.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.16666666666666666} \cdot \frac{0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.1e+155)
   (* (cbrt (/ (pow x -1.0) x)) 0.3333333333333333)
   (* (pow x -0.16666666666666666) (/ 0.3333333333333333 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.1e+155) {
		tmp = cbrt((pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = pow(x, -0.16666666666666666) * (0.3333333333333333 / sqrt(x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.1e+155) {
		tmp = Math.cbrt((Math.pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = Math.pow(x, -0.16666666666666666) * (0.3333333333333333 / Math.sqrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.1e+155)
		tmp = Float64(cbrt(Float64((x ^ -1.0) / x)) * 0.3333333333333333);
	else
		tmp = Float64((x ^ -0.16666666666666666) * Float64(0.3333333333333333 / sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.1e+155], N[(N[Power[N[(N[Power[x, -1.0], $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[x, -0.16666666666666666], $MachinePrecision] * N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e155

   1. Initial program 9.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if 1.1000000000000001e155 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 8.9

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

   5. Applied rewrites 8.9%

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 92.1%

      \[\leadsto {x}^{-0.16666666666666666} \cdot \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.16666666666666666} \cdot \frac{0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.16666666666666666}}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.1e+155)
   (* (cbrt (/ (pow x -1.0) x)) 0.3333333333333333)
   (* (/ (pow x -0.16666666666666666) (sqrt x)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.1e+155) {
		tmp = cbrt((pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = (pow(x, -0.16666666666666666) / sqrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.1e+155) {
		tmp = Math.cbrt((Math.pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = (Math.pow(x, -0.16666666666666666) / Math.sqrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.1e+155)
		tmp = Float64(cbrt(Float64((x ^ -1.0) / x)) * 0.3333333333333333);
	else
		tmp = Float64(Float64((x ^ -0.16666666666666666) / sqrt(x)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.1e+155], N[(N[Power[N[(N[Power[x, -1.0], $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[Power[x, -0.16666666666666666], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e155

   1. Initial program 9.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if 1.1000000000000001e155 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 8.9

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

   5. Applied rewrites 8.9%

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

   7. Applied rewrites 89.0%

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

   9. Applied rewrites 92.1%

      \[\leadsto \frac{{x}^{-0.16666666666666666}}{\sqrt{x}} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.16666666666666666}}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{x}\right)}^{-1.3333333333333333} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.45e+155)
   (* (cbrt (/ (pow x -1.0) x)) 0.3333333333333333)
   (* (pow (sqrt x) -1.3333333333333333) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.45e+155) {
		tmp = cbrt((pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = pow(sqrt(x), -1.3333333333333333) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.45e+155) {
		tmp = Math.cbrt((Math.pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = Math.pow(Math.sqrt(x), -1.3333333333333333) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45e+155)
		tmp = Float64(cbrt(Float64((x ^ -1.0) / x)) * 0.3333333333333333);
	else
		tmp = Float64((sqrt(x) ^ -1.3333333333333333) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.45e+155], N[(N[Power[N[(N[Power[x, -1.0], $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[N[Sqrt[x], $MachinePrecision], -1.3333333333333333], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e155

   1. Initial program 9.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if 1.45e155 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 8.9

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

   5. Applied rewrites 8.9%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 89.0%

      \[\leadsto {\left(\sqrt{x}\right)}^{-1.3333333333333333} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{x}\right)}^{-1.3333333333333333} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{x}\right)}^{-1.3333333333333333} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* (cbrt (pow (* x x) -1.0)) 0.3333333333333333)
   (* (pow (sqrt x) -1.3333333333333333) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = cbrt(pow((x * x), -1.0)) * 0.3333333333333333;
	} else {
		tmp = pow(sqrt(x), -1.3333333333333333) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = Math.cbrt(Math.pow((x * x), -1.0)) * 0.3333333333333333;
	} else {
		tmp = Math.pow(Math.sqrt(x), -1.3333333333333333) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(cbrt((Float64(x * x) ^ -1.0)) * 0.3333333333333333);
	else
		tmp = Float64((sqrt(x) ^ -1.3333333333333333) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[Power[N[Power[N[(x * x), $MachinePrecision], -1.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[N[Sqrt[x], $MachinePrecision], -1.3333333333333333], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 94.8%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 8.9

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

   5. Applied rewrites 8.9%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 89.0%

      \[\leadsto {\left(\sqrt{x}\right)}^{-1.3333333333333333} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{x}\right)}^{-1.3333333333333333} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(x)
	return Float64((cbrt(x) ^ -2.0) * 0.3333333333333333)
end

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. sqr-neg-rev N/A

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

   5. associate-/r* N/A

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

   6. distribute-neg-frac2 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. distribute-neg-frac2 N/A

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

   13. associate-/r* N/A

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

   14. sqr-neg-rev N/A

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

   15. unpow2 N/A

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

   16. unpow2 N/A

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

   17. associate-/r* N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 96.4%

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

Alternative 9: 88.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{x}\right)}^{-1.3333333333333333} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow (sqrt x) -1.3333333333333333) 0.3333333333333333))

C

double code(double x) {
	return pow(sqrt(x), -1.3333333333333333) * 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 = (sqrt(x) ** (-1.3333333333333333d0)) * 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.pow(Math.sqrt(x), -1.3333333333333333) * 0.3333333333333333;
}

Python

def code(x):
	return math.pow(math.sqrt(x), -1.3333333333333333) * 0.3333333333333333

Julia

function code(x)
	return Float64((sqrt(x) ^ -1.3333333333333333) * 0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = (sqrt(x) ^ -1.3333333333333333) * 0.3333333333333333;
end

Wolfram

code[x_] := N[(N[Power[N[Sqrt[x], $MachinePrecision], -1.3333333333333333], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. sqr-neg-rev N/A

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

   5. associate-/r* N/A

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

   6. distribute-neg-frac2 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. distribute-neg-frac2 N/A

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

   13. associate-/r* N/A

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

   14. sqr-neg-rev N/A

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

   15. unpow2 N/A

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

   16. unpow2 N/A

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

   17. associate-/r* N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 96.4%

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

9. Applied rewrites 88.6%

   \[\leadsto {\left(\sqrt{x}\right)}^{-1.3333333333333333} \cdot 0.3333333333333333 \]
10. Add Preprocessing

Alternative 10: 88.8% 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 7.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. sqr-neg-rev N/A

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

   5. associate-/r* N/A

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

   6. distribute-neg-frac2 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. distribute-neg-frac2 N/A

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

   13. associate-/r* N/A

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

   14. sqr-neg-rev N/A

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

   15. unpow2 N/A

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

   16. unpow2 N/A

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

   17. associate-/r* N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 88.6%

   \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
8. 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 7.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.8%

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

Developer Target 1: 98.5% 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 2024357 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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