2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 96.8%

Time: 20.1s

Alternatives: 7

Speedup: 1.0×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 6.9% 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: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{x \cdot \left(-0.05555555555555555 \cdot \left(\sqrt[3]{\frac{1}{{x}^{4}}} \cdot \frac{1}{\sqrt{0.3333333333333333}}\right) + \sqrt{0.3333333333333333} \cdot \sqrt[3]{\frac{1}{x}}\right)}{x}\right)}^{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (/
   (*
    x
    (+
     (*
      -0.05555555555555555
      (* (cbrt (/ 1.0 (pow x 4.0))) (/ 1.0 (sqrt 0.3333333333333333))))
     (* (sqrt 0.3333333333333333) (cbrt (/ 1.0 x)))))
   x)
  2.0))

C

double code(double x) {
	return pow(((x * ((-0.05555555555555555 * (cbrt((1.0 / pow(x, 4.0))) * (1.0 / sqrt(0.3333333333333333)))) + (sqrt(0.3333333333333333) * cbrt((1.0 / x))))) / x), 2.0);
}

Java

public static double code(double x) {
	return Math.pow(((x * ((-0.05555555555555555 * (Math.cbrt((1.0 / Math.pow(x, 4.0))) * (1.0 / Math.sqrt(0.3333333333333333)))) + (Math.sqrt(0.3333333333333333) * Math.cbrt((1.0 / x))))) / x), 2.0);
}

Julia

function code(x)
	return Float64(Float64(x * Float64(Float64(-0.05555555555555555 * Float64(cbrt(Float64(1.0 / (x ^ 4.0))) * Float64(1.0 / sqrt(0.3333333333333333)))) + Float64(sqrt(0.3333333333333333) * cbrt(Float64(1.0 / x))))) / x) ^ 2.0
end

Wolfram

code[x_] := N[Power[N[(N[(x * N[(N[(-0.05555555555555555 * N[(N[Power[N[(1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(1.0 / N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[0.3333333333333333], $MachinePrecision] * N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in x around inf 26.5%

   \[\leadsto \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 26.4%

      \[\leadsto \color{blue}{\sqrt{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \cdot \sqrt{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}}} \]

   2. pow2 26.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}}\right)}^{2}} \]

   3. sqrt-div 26.4%

      \[\leadsto {\color{blue}{\left(\frac{\sqrt{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}}{\sqrt{{x}^{2}}}\right)}}^{2} \]

   4. +-commutative 26.4%

      \[\leadsto {\left(\frac{\sqrt{\color{blue}{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}} + -0.1111111111111111 \cdot \sqrt[3]{x}}}}{\sqrt{{x}^{2}}}\right)}^{2} \]

   5. fma-define 26.4%

      \[\leadsto {\left(\frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{{x}^{4}}, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}}}{\sqrt{{x}^{2}}}\right)}^{2} \]

   6. *-commutative 26.4%

      \[\leadsto {\left(\frac{\sqrt{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{{x}^{4}}, \color{blue}{\sqrt[3]{x} \cdot -0.1111111111111111}\right)}}{\sqrt{{x}^{2}}}\right)}^{2} \]

   7. sqrt-pow1 27.5%

      \[\leadsto {\left(\frac{\sqrt{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{{x}^{4}}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}}{\color{blue}{{x}^{\left(\frac{2}{2}\right)}}}\right)}^{2} \]

   8. metadata-eval 27.5%

      \[\leadsto {\left(\frac{\sqrt{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{{x}^{4}}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}}{{x}^{\color{blue}{1}}}\right)}^{2} \]

   9. pow1 27.5%

      \[\leadsto {\left(\frac{\sqrt{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{{x}^{4}}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}}{\color{blue}{x}}\right)}^{2} \]

5. Applied egg-rr 27.5%

   \[\leadsto \color{blue}{{\left(\frac{\sqrt{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{{x}^{4}}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}}{x}\right)}^{2}} \]

6. Taylor expanded in x around inf 97.7%

   \[\leadsto {\left(\frac{\color{blue}{x \cdot \left(-0.05555555555555555 \cdot \left(\sqrt[3]{\frac{1}{{x}^{4}}} \cdot \frac{1}{\sqrt{0.3333333333333333}}\right) + \sqrt[3]{\frac{1}{x}} \cdot \sqrt{0.3333333333333333}\right)}}{x}\right)}^{2} \]

7. Final simplification 97.7%

   \[\leadsto {\left(\frac{x \cdot \left(-0.05555555555555555 \cdot \left(\sqrt[3]{\frac{1}{{x}^{4}}} \cdot \frac{1}{\sqrt{0.3333333333333333}}\right) + \sqrt{0.3333333333333333} \cdot \sqrt[3]{\frac{1}{x}}\right)}{x}\right)}^{2} \]
8. Add Preprocessing

Alternative 2: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (*
    (/ 1.0 x)
    (/
     (fma
      0.3333333333333333
      (pow (cbrt x) 4.0)
      (cbrt (* x -0.0013717421124828531)))
     x))
   (/
    1.0
    (fma
     (cbrt x)
     (+ (cbrt x) (cbrt (+ x 1.0)))
     (exp (* (log1p x) 0.6666666666666666))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (1.0 / x) * (fma(0.3333333333333333, pow(cbrt(x), 4.0), cbrt((x * -0.0013717421124828531))) / x);
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((x + 1.0))), exp((log1p(x) * 0.6666666666666666)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(1.0 / x) * Float64(fma(0.3333333333333333, (cbrt(x) ^ 4.0), cbrt(Float64(x * -0.0013717421124828531))) / x));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(x + 1.0))), exp(Float64(log1p(x) * 0.6666666666666666))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55e+231], N[(N[(1.0 / x), $MachinePrecision] * N[(N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision] + N[Power[N[(x * -0.0013717421124828531), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Exp[N[(N[Log[1 + x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf 33.1%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. pow1/3 30.7%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \color{blue}{{\left({x}^{4}\right)}^{0.3333333333333333}}}{{x}^{2}} \]

      2. pow-pow 62.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \color{blue}{{x}^{\left(4 \cdot 0.3333333333333333\right)}}}{{x}^{2}} \]

      3. metadata-eval 62.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{\color{blue}{1.3333333333333333}}}{{x}^{2}} \]

   5. Applied egg-rr 62.5%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \color{blue}{{x}^{1.3333333333333333}}}{{x}^{2}} \]

   7. Applied egg-rr 97.5%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}} \]

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 3.0%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      4. associate--l+ 91.3%

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

      5. +-inverses 91.3%

         \[\leadsto \frac{1 + \color{blue}{0}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

      6. metadata-eval 91.3%

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

      7. +-commutative 91.3%

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

      8. exp-prod 90.5%

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

   6. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}} \]
   7. Step-by-step derivation

      1. add-exp-log 90.8%

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

      2. log-pow 91.3%

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

      3. rem-log-exp 91.3%

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

   8. Applied egg-rr 91.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (*
    (/ 1.0 x)
    (/
     (fma
      0.3333333333333333
      (pow (cbrt x) 4.0)
      (cbrt (* x -0.0013717421124828531)))
     x))
   (/ 1.0 (fma (cbrt x) (+ (cbrt x) (cbrt (+ x 1.0))) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (1.0 / x) * (fma(0.3333333333333333, pow(cbrt(x), 4.0), cbrt((x * -0.0013717421124828531))) / x);
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((x + 1.0))), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(1.0 / x) * Float64(fma(0.3333333333333333, (cbrt(x) ^ 4.0), cbrt(Float64(x * -0.0013717421124828531))) / x));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(x + 1.0))), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55e+231], N[(N[(1.0 / x), $MachinePrecision] * N[(N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision] + N[Power[N[(x * -0.0013717421124828531), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf 33.1%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. pow1/3 30.7%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \color{blue}{{\left({x}^{4}\right)}^{0.3333333333333333}}}{{x}^{2}} \]

      2. pow-pow 62.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \color{blue}{{x}^{\left(4 \cdot 0.3333333333333333\right)}}}{{x}^{2}} \]

      3. metadata-eval 62.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{\color{blue}{1.3333333333333333}}}{{x}^{2}} \]

   5. Applied egg-rr 62.5%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \color{blue}{{x}^{1.3333333333333333}}}{{x}^{2}} \]

   7. Applied egg-rr 97.5%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}} \]

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 3.0%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      4. associate--l+ 91.3%

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

      5. +-inverses 91.3%

         \[\leadsto \frac{1 + \color{blue}{0}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

      6. metadata-eval 91.3%

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

      7. +-commutative 91.3%

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

      8. exp-prod 90.5%

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

   6. Simplified 90.5%

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

   7. Taylor expanded in x around 0 20.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{1}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/
    (+ (* (cbrt x) -0.1111111111111111) (* 0.3333333333333333 (* x (cbrt x))))
    (pow x 2.0))
   (/ 1.0 (fma (cbrt x) (+ (cbrt x) (cbrt (+ x 1.0))) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = ((cbrt(x) * -0.1111111111111111) + (0.3333333333333333 * (x * cbrt(x)))) / pow(x, 2.0);
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((x + 1.0))), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(Float64(cbrt(x) * -0.1111111111111111) + Float64(0.3333333333333333 * Float64(x * cbrt(x)))) / (x ^ 2.0));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(x + 1.0))), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision] + N[(0.3333333333333333 * N[(x * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 5.9%

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

   3. Taylor expanded in x around inf 48.8%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. pow1/3 45.3%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \color{blue}{{\left({x}^{4}\right)}^{0.3333333333333333}}}{{x}^{2}} \]

      2. pow-pow 90.1%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \color{blue}{{x}^{\left(4 \cdot 0.3333333333333333\right)}}}{{x}^{2}} \]

      3. metadata-eval 90.1%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{\color{blue}{1.3333333333333333}}}{{x}^{2}} \]

   5. Applied egg-rr 90.1%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \color{blue}{{x}^{1.3333333333333333}}}{{x}^{2}} \]
   6. Step-by-step derivation

      1. metadata-eval 90.1%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{\color{blue}{\left(0.3333333333333333 \cdot 4\right)}}}{{x}^{2}} \]

      2. pow-pow 90.1%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \color{blue}{{\left({x}^{0.3333333333333333}\right)}^{4}}}{{x}^{2}} \]

      3. pow1/3 97.3%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {\color{blue}{\left(\sqrt[3]{x}\right)}}^{4}}{{x}^{2}} \]

      4. metadata-eval 97.3%

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

      5. pow-sqr 97.3%

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

      6. unpow2 97.3%

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

      7. associate-*l* 97.2%

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

      8. unpow2 97.2%

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

      9. cube-mult 97.3%

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

      10. rem-cube-cbrt 98.6%

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

   7. Applied egg-rr 98.6%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

      1. flip3-- 4.7%

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

      2. div-inv 4.7%

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

      3. rem-cube-cbrt 3.2%

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

      4. rem-cube-cbrt 4.7%

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

      5. +-commutative 4.7%

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

      6. distribute-rgt-out 4.7%

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

      7. +-commutative 4.7%

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

      8. fma-define 4.7%

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

      9. add-exp-log 4.7%

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

   4. Applied egg-rr 4.7%

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

      1. associate-*r/ 4.7%

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

      2. *-rgt-identity 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 91.8%

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

      5. +-inverses 91.8%

         \[\leadsto \frac{1 + \color{blue}{0}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

      6. metadata-eval 91.8%

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

      7. +-commutative 91.8%

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

      8. exp-prod 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in x around 0 20.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{1}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.3333333333333333 (cbrt (/ 1.0 (pow x 2.0)))))

C

double code(double x) {
	return 0.3333333333333333 * cbrt((1.0 / pow(x, 2.0)));
}

Java

public static double code(double x) {
	return 0.3333333333333333 * Math.cbrt((1.0 / Math.pow(x, 2.0)));
}

Julia

function code(x)
	return Float64(0.3333333333333333 * cbrt(Float64(1.0 / (x ^ 2.0))))
end

Wolfram

code[x_] := N[(0.3333333333333333 * N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in x around inf 54.9%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]

4. Final simplification 54.9%

   \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} \]
5. Add Preprocessing

Alternative 6: 5.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in x around inf 4.1%

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

   1. pow1/3 5.9%

      \[\leadsto \sqrt[3]{x} - \color{blue}{{x}^{0.3333333333333333}} \]

5. Applied egg-rr 5.9%

   \[\leadsto \sqrt[3]{x} - \color{blue}{{x}^{0.3333333333333333}} \]

6. Final simplification 5.9%

   \[\leadsto \sqrt[3]{x} - {x}^{0.3333333333333333} \]
7. Add Preprocessing

Alternative 7: 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(x))
end

Wolfram

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

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in x around 0 1.8%

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

   1. sub-neg 1.8%

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

   2. rem-square-sqrt 0.0%

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

   3. fabs-sqr 0.0%

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

   4. rem-square-sqrt 5.4%

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

   5. fabs-neg 5.4%

      \[\leadsto 1 + \color{blue}{\left|\sqrt[3]{x}\right|} \]

   6. unpow1/3 5.4%

      \[\leadsto 1 + \left|\color{blue}{{x}^{0.3333333333333333}}\right| \]

   7. metadata-eval 5.4%

      \[\leadsto 1 + \left|{x}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}}\right| \]

   8. pow-sqr 5.4%

      \[\leadsto 1 + \left|\color{blue}{{x}^{0.16666666666666666} \cdot {x}^{0.16666666666666666}}\right| \]

   9. fabs-sqr 5.4%

      \[\leadsto 1 + \color{blue}{{x}^{0.16666666666666666} \cdot {x}^{0.16666666666666666}} \]

   10. pow-sqr 5.4%

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

   11. metadata-eval 5.4%

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

   12. unpow1/3 5.4%

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

5. Simplified 5.4%

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

6. Final simplification 5.4%

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

Developer target: 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 2024081 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (+ (* (cbrt (+ x 1.0)) (cbrt (+ x 1.0))) (* (cbrt x) (cbrt (+ x 1.0)))) (* (cbrt x) (cbrt x))))

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