2cbrt (problem 3.3.4)

Percentage Accurate: 53.4% → 99.2%

Time: 8.3s

Alternatives: 7

Speedup: 1.0×

Specification

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: 53.4% 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: 99.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	return 1.0 / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (t_0 + Math.cbrt(x))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

   1. expm1-log1p-u 51.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right)} \]

4. Applied egg-rr 51.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right)} \]

6. Applied egg-rr 51.6%

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

7. Taylor expanded in x around 0 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

   1. pow1/3 49.0%

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

4. Applied egg-rr 49.0%

   \[\leadsto \color{blue}{{\left(x + 1\right)}^{0.3333333333333333}} - \sqrt[3]{x} \]
5. Step-by-step derivation

   1. pow1/3 51.2%

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

   2. metadata-eval 51.2%

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

   3. sub-neg 51.2%

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

   4. flip-- 50.3%

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

   5. metadata-eval 50.3%

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

   6. fma-neg 50.3%

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

   7. metadata-eval 50.3%

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

   8. cbrt-undiv 50.5%

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

   9. clear-num 50.4%

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

   10. clear-num 50.5%

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

   11. cbrt-undiv 50.4%

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

   12. metadata-eval 50.4%

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

   13. fma-neg 50.4%

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

   14. metadata-eval 50.4%

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

   15. flip-- 51.2%

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

   16. sub-neg 51.2%

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

   17. metadata-eval 51.2%

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

6. Applied egg-rr 51.2%

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

7. Final simplification 51.2%

   \[\leadsto \frac{1}{\frac{1}{\sqrt[3]{1 + x}}} - \sqrt[3]{x} \]
8. Add Preprocessing

Alternative 3: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

3. Final simplification 51.2%

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

Alternative 4: 50.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot 0.3333333333333333 - \sqrt[3]{x}\right) \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

   1. add-sqr-sqrt 23.4%

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

   2. pow2 23.4%

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

   3. pow1/3 23.8%

      \[\leadsto \sqrt[3]{x + 1} - {\left(\sqrt{\color{blue}{{x}^{0.3333333333333333}}}\right)}^{2} \]

   4. sqrt-pow1 23.8%

      \[\leadsto \sqrt[3]{x + 1} - {\color{blue}{\left({x}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{2} \]

   5. metadata-eval 23.8%

      \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

4. Applied egg-rr 23.8%

   \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]

5. Taylor expanded in x around 0 22.6%

   \[\leadsto \color{blue}{\left(1 + 0.3333333333333333 \cdot x\right) - {x}^{0.3333333333333333}} \]
6. Step-by-step derivation

   1. unpow1/3 49.0%

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

   2. associate--l+ 49.0%

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

   3. *-commutative 49.0%

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

7. Simplified 49.0%

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

8. Final simplification 49.0%

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

Alternative 5: 50.5% 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 51.2%

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

   1. add-sqr-sqrt 23.4%

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

   2. pow2 23.4%

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

   3. pow1/3 23.8%

      \[\leadsto \sqrt[3]{x + 1} - {\left(\sqrt{\color{blue}{{x}^{0.3333333333333333}}}\right)}^{2} \]

   4. sqrt-pow1 23.8%

      \[\leadsto \sqrt[3]{x + 1} - {\color{blue}{\left({x}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{2} \]

   5. metadata-eval 23.8%

      \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

4. Applied egg-rr 23.8%

   \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]

5. Taylor expanded in x around 0 21.5%

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

   1. unpow1/3 48.7%

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

7. Simplified 48.7%

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

8. Final simplification 48.7%

   \[\leadsto 1 - \sqrt[3]{x} \]
9. Add Preprocessing

Alternative 6: 3.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around inf 3.7%

   \[\leadsto \color{blue}{0} \]

4. Final simplification 3.7%

   \[\leadsto 0 \]
5. Add Preprocessing

Alternative 7: 49.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around 0 48.1%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 48.1%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024013 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))