2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 99.0%

Time: 7.8s

Alternatives: 7

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: 7.1% 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: 99.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+14)
   (/
    (- (+ 1.0 x) x)
    (fma
     (cbrt x)
     (+ (cbrt (+ 1.0 x)) (cbrt x))
     (exp (* 0.6666666666666666 (log1p x)))))
   (/ (* 0.3333333333333333 (cbrt x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1e+14) {
		tmp = ((1.0 + x) - x) / fma(cbrt(x), (cbrt((1.0 + x)) + cbrt(x)), exp((0.6666666666666666 * log1p(x))));
	} else {
		tmp = (0.3333333333333333 * cbrt(x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+14)
		tmp = Float64(Float64(Float64(1.0 + x) - x) / fma(cbrt(x), Float64(cbrt(Float64(1.0 + x)) + cbrt(x)), exp(Float64(0.6666666666666666 * log1p(x)))));
	else
		tmp = Float64(Float64(0.3333333333333333 * cbrt(x)) / x);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1e14

   1. Initial program 60.2%

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

      1. lift-+.f64 N/A

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

      2. rem-cube-cbrt N/A

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

      3. lift-cbrt.f64 N/A

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

      4. sqr-pow N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow1/3 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. unpow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. pow1/3 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. unpow1/2 N/A

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

      19. lower-sqrt.f64 60.6

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

   4. Applied rewrites 60.6%

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

   6. Applied rewrites 97.6%

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

   if 1e14 < x

   1. Initial program 4.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 22.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   7. Applied rewrites 72.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.0%

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

4. Final simplification 98.9%

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

Alternative 2: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{-2}, 0.06172839506172839, \mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right) \cdot \sqrt[3]{x}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.3333333333333333 \cdot \sqrt[3]{x}\right) \cdot {x}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e+80)
   (/
    (/
     (fma
      (pow (cbrt x) -2.0)
      0.06172839506172839
      (* (fma 0.3333333333333333 x -0.1111111111111111) (cbrt x)))
     x)
    x)
   (* (* 0.3333333333333333 (cbrt x)) (pow x -1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e+80) {
		tmp = (fma(pow(cbrt(x), -2.0), 0.06172839506172839, (fma(0.3333333333333333, x, -0.1111111111111111) * cbrt(x))) / x) / x;
	} else {
		tmp = (0.3333333333333333 * cbrt(x)) * pow(x, -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e+80)
		tmp = Float64(Float64(fma((cbrt(x) ^ -2.0), 0.06172839506172839, Float64(fma(0.3333333333333333, x, -0.1111111111111111) * cbrt(x))) / x) / x);
	else
		tmp = Float64(Float64(0.3333333333333333 * cbrt(x)) * (x ^ -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e+80], N[(N[(N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.06172839506172839 + N[(N[(0.3333333333333333 * x + -0.1111111111111111), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.49999999999999967e80

   1. Initial program 15.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   7. Applied rewrites 97.8%

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

   9. Applied rewrites 98.0%

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

   if 5.49999999999999967e80 < x

   1. Initial program 4.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 0.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.0%

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

   12. Applied rewrites 99.0%

       \[\leadsto {x}^{-1} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot 0.3333333333333333\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{-2}, 0.06172839506172839, \mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right) \cdot \sqrt[3]{x}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.3333333333333333 \cdot \sqrt[3]{x}\right) \cdot {x}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{-2}, 0.06172839506172839, \mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right) \cdot \sqrt[3]{x}\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.3333333333333333 \cdot \sqrt[3]{x}\right) \cdot {x}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+80)
   (/
    (fma
     (pow (cbrt x) -2.0)
     0.06172839506172839
     (* (fma 0.3333333333333333 x -0.1111111111111111) (cbrt x)))
    (* x x))
   (* (* 0.3333333333333333 (cbrt x)) (pow x -1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 2e+80) {
		tmp = fma(pow(cbrt(x), -2.0), 0.06172839506172839, (fma(0.3333333333333333, x, -0.1111111111111111) * cbrt(x))) / (x * x);
	} else {
		tmp = (0.3333333333333333 * cbrt(x)) * pow(x, -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+80)
		tmp = Float64(fma((cbrt(x) ^ -2.0), 0.06172839506172839, Float64(fma(0.3333333333333333, x, -0.1111111111111111) * cbrt(x))) / Float64(x * x));
	else
		tmp = Float64(Float64(0.3333333333333333 * cbrt(x)) * (x ^ -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+80], N[(N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.06172839506172839 + N[(N[(0.3333333333333333 * x + -0.1111111111111111), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e80

   1. Initial program 15.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   7. Applied rewrites 97.8%

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

   9. Applied rewrites 97.9%

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

   if 2e80 < x

   1. Initial program 4.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 0.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.0%

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

   12. Applied rewrites 99.0%

       \[\leadsto {x}^{-1} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot 0.3333333333333333\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{-2}, 0.06172839506172839, \mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right) \cdot \sqrt[3]{x}\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.3333333333333333 \cdot \sqrt[3]{x}\right) \cdot {x}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 26.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

7. Applied rewrites 73.7%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 96.7%

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

Alternative 5: 88.7% 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

real(8) function code(x)
    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 7.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 53.1

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

5. Applied rewrites 53.1%

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

7. Applied rewrites 88.6%

   \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 6: 5.3% 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(Float64(-x)))
end

Wolfram

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

Derivation

1. Initial program 7.5%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lower-pow.f64 1.8

      \[\leadsto 1 - \color{blue}{{x}^{0.3333333333333333}} \]

7. Applied rewrites 1.8%

   \[\leadsto 1 - \color{blue}{{x}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

      \[\leadsto 1 - {x}^{\color{blue}{\left(2 \cdot \frac{1}{6}\right)}} \]

   3. pow-pow N/A

      \[\leadsto 1 - \color{blue}{{\left({x}^{2}\right)}^{\frac{1}{6}}} \]

   4. pow2 N/A

      \[\leadsto 1 - {\color{blue}{\left(x \cdot x\right)}}^{\frac{1}{6}} \]

   5. sqr-neg N/A

      \[\leadsto 1 - {\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}^{\frac{1}{6}} \]

   6. lift-neg.f64 N/A

      \[\leadsto 1 - {\left(\color{blue}{\left(-x\right)} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\frac{1}{6}} \]

   7. lift-neg.f64 N/A

      \[\leadsto 1 - {\left(\left(-x\right) \cdot \color{blue}{\left(-x\right)}\right)}^{\frac{1}{6}} \]

   8. pow-prod-down N/A

      \[\leadsto 1 - \color{blue}{{\left(-x\right)}^{\frac{1}{6}} \cdot {\left(-x\right)}^{\frac{1}{6}}} \]

   9. pow-prod-up N/A

      \[\leadsto 1 - \color{blue}{{\left(-x\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}} \]

   10. metadata-eval N/A

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

   11. pow1/3 N/A

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

   12. lift-cbrt.f64 5.4

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

9. Applied rewrites 5.4%

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

Alternative 7: 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 7.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 2024327 
(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)))