2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 97.9%

Time: 9.9s

Alternatives: 15

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: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)} + \sqrt[3]{\frac{1}{x}}\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (*
   x
   (+
    (cbrt (+ (/ 1.0 x) (/ 2.0 (* x x))))
    (+
     (cbrt (fma (pow x -0.5) (pow x -0.5) (/ 1.0 (* x x))))
     (cbrt (/ 1.0 x)))))))

C

double code(double x) {
	return 1.0 / (x * (cbrt(((1.0 / x) + (2.0 / (x * x)))) + (cbrt(fma(pow(x, -0.5), pow(x, -0.5), (1.0 / (x * x)))) + cbrt((1.0 / x)))));
}

Julia

function code(x)
	return Float64(1.0 / Float64(x * Float64(cbrt(Float64(Float64(1.0 / x) + Float64(2.0 / Float64(x * x)))) + Float64(cbrt(fma((x ^ -0.5), (x ^ -0.5), Float64(1.0 / Float64(x * x)))) + cbrt(Float64(1.0 / x))))))
end

Wolfram

code[x_] := N[(1.0 / N[(x * N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(N[Power[x, -0.5], $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision] + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.4%

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

4. Applied rewrites 8.9%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 N/A

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

7. Applied rewrites 98.1%

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

9. Applied rewrites 98.2%

   \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)} + \sqrt[3]{\frac{\color{blue}{1}}{x}}\right)\right)} \]
10. Add Preprocessing

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{x}}\\ \frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\frac{1}{x}} + \sqrt[3]{\mathsf{fma}\left(t\_0, t\_0, \frac{1}{x \cdot x}\right)}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt x))))
   (/
    1.0
    (*
     x
     (+
      (cbrt (+ (/ 1.0 x) (/ 2.0 (* x x))))
      (+ (cbrt (/ 1.0 x)) (cbrt (fma t_0 t_0 (/ 1.0 (* x x))))))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(x);
	return 1.0 / (x * (cbrt(((1.0 / x) + (2.0 / (x * x)))) + (cbrt((1.0 / x)) + cbrt(fma(t_0, t_0, (1.0 / (x * x)))))));
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(x))
	return Float64(1.0 / Float64(x * Float64(cbrt(Float64(Float64(1.0 / x) + Float64(2.0 / Float64(x * x)))) + Float64(cbrt(Float64(1.0 / x)) + cbrt(fma(t_0, t_0, Float64(1.0 / Float64(x * x))))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(x * N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * t$95$0 + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.1%

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

4. Applied rewrites 9.4%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 N/A

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

7. Applied rewrites 97.9%

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

9. Applied rewrites 97.9%

   \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)} + \sqrt[3]{\frac{\color{blue}{1}}{x}}\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 97.9%

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

12. Final simplification 97.9%

    \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\frac{1}{x}} + \sqrt[3]{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \frac{1}{\sqrt{x}}, \frac{1}{x \cdot x}\right)}\right)\right)} \]
13. 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 2024230 
(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)))