2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 98.4%

Time: 18.8s

Alternatives: 18

Speedup: 2.0×

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: 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: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{0.6666666666666666} + {\left(x \cdot \left(x + 1\right)\right)}^{0.3333333333333333}\\ \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\left(x + 1\right) \cdot \left(x + 1\right) + {t\_0}^{3}} \cdot \left({\left(x + 1\right)}^{1.3333333333333333} + t\_0 \cdot \left(t\_0 - {\left(x + 1\right)}^{0.6666666666666666}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+
          (pow x 0.6666666666666666)
          (pow (* x (+ x 1.0)) 0.3333333333333333))))
   (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 5e-11)
     (* 0.3333333333333333 (/ (cbrt (/ -1.0 x)) (- 0.0 (cbrt x))))
     (*
      (/ (- (+ x 1.0) x) (+ (* (+ x 1.0) (+ x 1.0)) (pow t_0 3.0)))
      (+
       (pow (+ x 1.0) 1.3333333333333333)
       (* t_0 (- t_0 (pow (+ x 1.0) 0.6666666666666666))))))))

C

double code(double x) {
	double t_0 = pow(x, 0.6666666666666666) + pow((x * (x + 1.0)), 0.3333333333333333);
	double tmp;
	if ((cbrt((x + 1.0)) - cbrt(x)) <= 5e-11) {
		tmp = 0.3333333333333333 * (cbrt((-1.0 / x)) / (0.0 - cbrt(x)));
	} else {
		tmp = (((x + 1.0) - x) / (((x + 1.0) * (x + 1.0)) + pow(t_0, 3.0))) * (pow((x + 1.0), 1.3333333333333333) + (t_0 * (t_0 - pow((x + 1.0), 0.6666666666666666))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.pow(x, 0.6666666666666666) + Math.pow((x * (x + 1.0)), 0.3333333333333333);
	double tmp;
	if ((Math.cbrt((x + 1.0)) - Math.cbrt(x)) <= 5e-11) {
		tmp = 0.3333333333333333 * (Math.cbrt((-1.0 / x)) / (0.0 - Math.cbrt(x)));
	} else {
		tmp = (((x + 1.0) - x) / (((x + 1.0) * (x + 1.0)) + Math.pow(t_0, 3.0))) * (Math.pow((x + 1.0), 1.3333333333333333) + (t_0 * (t_0 - Math.pow((x + 1.0), 0.6666666666666666))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64((x ^ 0.6666666666666666) + (Float64(x * Float64(x + 1.0)) ^ 0.3333333333333333))
	tmp = 0.0
	if (Float64(cbrt(Float64(x + 1.0)) - cbrt(x)) <= 5e-11)
		tmp = Float64(0.3333333333333333 * Float64(cbrt(Float64(-1.0 / x)) / Float64(0.0 - cbrt(x))));
	else
		tmp = Float64(Float64(Float64(Float64(x + 1.0) - x) / Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) + (t_0 ^ 3.0))) * Float64((Float64(x + 1.0) ^ 1.3333333333333333) + Float64(t_0 * Float64(t_0 - (Float64(x + 1.0) ^ 0.6666666666666666)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[Power[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-11], N[(0.3333333333333333 * N[(N[Power[N[(-1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[(0.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(x + 1.0), $MachinePrecision], 1.3333333333333333], $MachinePrecision] + N[(t$95$0 * N[(t$95$0 - N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 5.00000000000000018e-11

   1. Initial program 4.4%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 52.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 52.8%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\frac{1}{x}}{x}}\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\mathsf{neg}\left(x\right)}}\right)\right) \]

      3. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(x\right)}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}\right), \color{blue}{\left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), \left(\sqrt[3]{\color{blue}{\mathsf{neg}\left(x\right)}}\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{-1}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      9. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(0 - x\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\log 1 - x\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(\log 1, x\right)\right)\right)\right) \]

      13. metadata-eval 98.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right)\right) \]

   7. Applied egg-rr 98.5%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{0 - x}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      2. rem-cube-cbrt N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left({\left(\sqrt[3]{x}\right)}^{3}\right)}\right)\right)\right) \]

      3. cube-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)}^{3}}\right)\right)\right) \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(0 - \sqrt[3]{x}\right)}^{3}}\right)\right)\right) \]

      5. rem-cbrt-cube N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(0 - \color{blue}{\sqrt[3]{x}}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      8. cbrt-lowering-cbrt.f64 98.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\mathsf{cbrt.f64}\left(x\right)\right)\right)\right) \]

   9. Applied egg-rr 98.5%

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

   if 5.00000000000000018e-11 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 62.9%

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

      1. pow1/3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left({x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left(e^{\log x \cdot \frac{1}{3}}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{exp.f64}\left(\left(\log x \cdot \frac{1}{3}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{exp.f64}\left(\left(\frac{1}{3} \cdot \log x\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \log x\right)\right)\right) \]

      6. log-lowering-log.f64 60.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   4. Applied egg-rr 60.8%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{e^{0.3333333333333333 \cdot \log x}} \]

   6. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\left(1 + x\right) \cdot \left(1 + x\right) + {\left({x}^{0.6666666666666666} + {\left(x \cdot \left(1 + x\right)\right)}^{0.3333333333333333}\right)}^{3}} \cdot \left({\left(1 + x\right)}^{1.3333333333333333} + \left({x}^{0.6666666666666666} + {\left(x \cdot \left(1 + x\right)\right)}^{0.3333333333333333}\right) \cdot \left(\left({x}^{0.6666666666666666} + {\left(x \cdot \left(1 + x\right)\right)}^{0.3333333333333333}\right) - {\left(1 + x\right)}^{0.6666666666666666}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\left(x + 1\right) \cdot \left(x + 1\right) + {\left({x}^{0.6666666666666666} + {\left(x \cdot \left(x + 1\right)\right)}^{0.3333333333333333}\right)}^{3}} \cdot \left({\left(x + 1\right)}^{1.3333333333333333} + \left({x}^{0.6666666666666666} + {\left(x \cdot \left(x + 1\right)\right)}^{0.3333333333333333}\right) \cdot \left(\left({x}^{0.6666666666666666} + {\left(x \cdot \left(x + 1\right)\right)}^{0.3333333333333333}\right) - {\left(x + 1\right)}^{0.6666666666666666}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\left({x}^{0.6666666666666666} + {\left(x \cdot \left(x + 1\right)\right)}^{0.3333333333333333}\right) + {\left(x + 1\right)}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 5e-11)
   (* 0.3333333333333333 (/ (cbrt (/ -1.0 x)) (- 0.0 (cbrt x))))
   (/
    (- (+ x 1.0) x)
    (+
     (+ (pow x 0.6666666666666666) (pow (* x (+ x 1.0)) 0.3333333333333333))
     (pow (+ x 1.0) 0.6666666666666666)))))

C

double code(double x) {
	double tmp;
	if ((cbrt((x + 1.0)) - cbrt(x)) <= 5e-11) {
		tmp = 0.3333333333333333 * (cbrt((-1.0 / x)) / (0.0 - cbrt(x)));
	} else {
		tmp = ((x + 1.0) - x) / ((pow(x, 0.6666666666666666) + pow((x * (x + 1.0)), 0.3333333333333333)) + pow((x + 1.0), 0.6666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((Math.cbrt((x + 1.0)) - Math.cbrt(x)) <= 5e-11) {
		tmp = 0.3333333333333333 * (Math.cbrt((-1.0 / x)) / (0.0 - Math.cbrt(x)));
	} else {
		tmp = ((x + 1.0) - x) / ((Math.pow(x, 0.6666666666666666) + Math.pow((x * (x + 1.0)), 0.3333333333333333)) + Math.pow((x + 1.0), 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(x + 1.0)) - cbrt(x)) <= 5e-11)
		tmp = Float64(0.3333333333333333 * Float64(cbrt(Float64(-1.0 / x)) / Float64(0.0 - cbrt(x))));
	else
		tmp = Float64(Float64(Float64(x + 1.0) - x) / Float64(Float64((x ^ 0.6666666666666666) + (Float64(x * Float64(x + 1.0)) ^ 0.3333333333333333)) + (Float64(x + 1.0) ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-11], N[(0.3333333333333333 * N[(N[Power[N[(-1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[(0.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[Power[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 5.00000000000000018e-11

   1. Initial program 4.4%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 52.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 52.8%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\frac{1}{x}}{x}}\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\mathsf{neg}\left(x\right)}}\right)\right) \]

      3. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(x\right)}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}\right), \color{blue}{\left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), \left(\sqrt[3]{\color{blue}{\mathsf{neg}\left(x\right)}}\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{-1}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      9. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(0 - x\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\log 1 - x\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(\log 1, x\right)\right)\right)\right) \]

      13. metadata-eval 98.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right)\right) \]

   7. Applied egg-rr 98.5%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{0 - x}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      2. rem-cube-cbrt N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left({\left(\sqrt[3]{x}\right)}^{3}\right)}\right)\right)\right) \]

      3. cube-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)}^{3}}\right)\right)\right) \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(0 - \sqrt[3]{x}\right)}^{3}}\right)\right)\right) \]

      5. rem-cbrt-cube N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(0 - \color{blue}{\sqrt[3]{x}}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      8. cbrt-lowering-cbrt.f64 98.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\mathsf{cbrt.f64}\left(x\right)\right)\right)\right) \]

   9. Applied egg-rr 98.5%

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

   if 5.00000000000000018e-11 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 62.9%

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

      1. pow1/3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left({x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left(e^{\log x \cdot \frac{1}{3}}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{exp.f64}\left(\left(\log x \cdot \frac{1}{3}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{exp.f64}\left(\left(\frac{1}{3} \cdot \log x\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \log x\right)\right)\right) \]

      6. log-lowering-log.f64 60.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   4. Applied egg-rr 60.8%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{e^{0.3333333333333333 \cdot \log x}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{x + 1} - e^{\log x \cdot \frac{1}{3}} \]

      2. pow-to-exp N/A

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

      3. pow1/3 N/A

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

      4. flip3-- N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}\right), \color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)}\right) \]

      6. rem-cube-cbrt N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) - {\left(\sqrt[3]{x}\right)}^{3}\right), \left(\color{blue}{\sqrt[3]{x + 1}} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      7. rem-cube-cbrt N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) - x\right), \left(\sqrt[3]{x + 1} \cdot \color{blue}{\sqrt[3]{x + 1}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x + 1\right), x\right), \left(\color{blue}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + x\right), x\right), \left(\color{blue}{\sqrt[3]{x + 1}} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \left(\color{blue}{\sqrt[3]{x + 1}} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right), \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\right)\right) \]

   6. Applied egg-rr 97.0%

      \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{{\left(1 + x\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + {\left(x \cdot \left(1 + x\right)\right)}^{0.3333333333333333}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\left({x}^{0.6666666666666666} + {\left(x \cdot \left(x + 1\right)\right)}^{0.3333333333333333}\right) + {\left(x + 1\right)}^{0.6666666666666666}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 2.7 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(x \cdot 0.3333333333333333\right) \cdot \sqrt[3]{x \cdot t\_0} + \sqrt[3]{x} \cdot \left(0.06172839506172839 + x \cdot -0.1111111111111111\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x 2.7e+75)
     (/
      (+
       (* (* x 0.3333333333333333) (cbrt (* x t_0)))
       (* (cbrt x) (+ 0.06172839506172839 (* x -0.1111111111111111))))
      t_0)
     (* 0.3333333333333333 (/ (cbrt (/ -1.0 x)) (- 0.0 (cbrt x)))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= 2.7e+75) {
		tmp = (((x * 0.3333333333333333) * cbrt((x * t_0))) + (cbrt(x) * (0.06172839506172839 + (x * -0.1111111111111111)))) / t_0;
	} else {
		tmp = 0.3333333333333333 * (cbrt((-1.0 / x)) / (0.0 - cbrt(x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= 2.7e+75) {
		tmp = (((x * 0.3333333333333333) * Math.cbrt((x * t_0))) + (Math.cbrt(x) * (0.06172839506172839 + (x * -0.1111111111111111)))) / t_0;
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt((-1.0 / x)) / (0.0 - Math.cbrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= 2.7e+75)
		tmp = Float64(Float64(Float64(Float64(x * 0.3333333333333333) * cbrt(Float64(x * t_0))) + Float64(cbrt(x) * Float64(0.06172839506172839 + Float64(x * -0.1111111111111111)))) / t_0);
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(Float64(-1.0 / x)) / Float64(0.0 - cbrt(x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.7e+75], N[(N[(N[(N[(x * 0.3333333333333333), $MachinePrecision] * N[Power[N[(x * t$95$0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(0.06172839506172839 + N[(x * -0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[N[(-1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[(0.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.69999999999999998e75

   1. Initial program 22.9%

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

      1. rem-cbrt-cube N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{3}}\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      2. unpow3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      3. pow1/3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot {\left(x + 1\right)}^{\frac{1}{3}}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right) \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   4. Applied egg-rr 26.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{0.8333333333333334}} \cdot \sqrt[3]{{\left(x + 1\right)}^{0.16666666666666666}}} - \sqrt[3]{x} \]
   5. Step-by-step derivation

      1. cbrt-unprod N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\frac{5}{6}} \cdot {\left(x + 1\right)}^{\frac{1}{6}}}\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{5}{6} + \frac{1}{6}\right)}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{1}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      4. unpow1 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{x + 1}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      5. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      7. cbrt-div N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}}\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sqrt[3]{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\sqrt[3]{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}\right)\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      10. cbrt-lowering-cbrt.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}}\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      12. flip3-+ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{x + 1}\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x + 1\right)\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      15. +-lowering-+.f64 24.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   6. Applied egg-rr 24.4%

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

   7. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right), \color{blue}{\left({x}^{2}\right)}\right) \]

   9. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(0.3333333333333333 \cdot \sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.06172839506172839\right)}{x \cdot x}} \]

   10. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{5}{81} \cdot \sqrt[3]{x} + x \cdot \left(\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right), \color{blue}{\left({x}^{3}\right)}\right) \]

   12. Simplified 94.0%

       \[\leadsto \color{blue}{\frac{\left(x \cdot 0.3333333333333333\right) \cdot \sqrt[3]{\left(x \cdot \left(x \cdot x\right)\right) \cdot x} + \sqrt[3]{x} \cdot \left(0.06172839506172839 + x \cdot -0.1111111111111111\right)}{x \cdot \left(x \cdot x\right)}} \]

   if 2.69999999999999998e75 < x

   1. Initial program 4.4%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 41.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 41.6%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\frac{1}{x}}{x}}\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\mathsf{neg}\left(x\right)}}\right)\right) \]

      3. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(x\right)}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}\right), \color{blue}{\left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), \left(\sqrt[3]{\color{blue}{\mathsf{neg}\left(x\right)}}\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{-1}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      9. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(0 - x\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\log 1 - x\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(\log 1, x\right)\right)\right)\right) \]

      13. metadata-eval 98.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right)\right) \]

   7. Applied egg-rr 98.5%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{0 - x}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      2. rem-cube-cbrt N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left({\left(\sqrt[3]{x}\right)}^{3}\right)}\right)\right)\right) \]

      3. cube-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)}^{3}}\right)\right)\right) \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(0 - \sqrt[3]{x}\right)}^{3}}\right)\right)\right) \]

      5. rem-cbrt-cube N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(0 - \color{blue}{\sqrt[3]{x}}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      8. cbrt-lowering-cbrt.f64 98.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\mathsf{cbrt.f64}\left(x\right)\right)\right)\right) \]

   9. Applied egg-rr 98.5%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(x \cdot 0.3333333333333333\right) \cdot \sqrt[3]{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \sqrt[3]{x} \cdot \left(0.06172839506172839 + x \cdot -0.1111111111111111\right)}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+75}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3e+75)
   (/
    (+
     (* (cbrt x) -0.1111111111111111)
     (* 0.3333333333333333 (cbrt (* (* x x) (* x x)))))
    (* x x))
   (* 0.3333333333333333 (/ (cbrt (/ -1.0 x)) (- 0.0 (cbrt x))))))

C

double code(double x) {
	double tmp;
	if (x <= 3e+75) {
		tmp = ((cbrt(x) * -0.1111111111111111) + (0.3333333333333333 * cbrt(((x * x) * (x * x))))) / (x * x);
	} else {
		tmp = 0.3333333333333333 * (cbrt((-1.0 / x)) / (0.0 - cbrt(x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 3e+75) {
		tmp = ((Math.cbrt(x) * -0.1111111111111111) + (0.3333333333333333 * Math.cbrt(((x * x) * (x * x))))) / (x * x);
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt((-1.0 / x)) / (0.0 - Math.cbrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3e+75)
		tmp = Float64(Float64(Float64(cbrt(x) * -0.1111111111111111) + Float64(0.3333333333333333 * cbrt(Float64(Float64(x * x) * Float64(x * x))))) / Float64(x * x));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(Float64(-1.0 / x)) / Float64(0.0 - cbrt(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3e+75], N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision] + N[(0.3333333333333333 * N[Power[N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[N[(-1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[(0.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3e75

   1. Initial program 22.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{9} \cdot \sqrt[3]{x}\right), \left(\frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\sqrt[3]{x} \cdot \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right), \left({x}^{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{x}\right), \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right), \left({x}^{2}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right), \left({x}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{{x}^{4}}\right)\right)\right), \left({x}^{2}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{{x}^{\left(2 \cdot 2\right)}}\right)\right)\right), \left({x}^{2}\right)\right) \]

      8. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{{x}^{2} \cdot {x}^{2}}\right)\right)\right), \left({x}^{2}\right)\right) \]

      9. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right), \left({x}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right)\right)\right)\right), \left({x}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right)\right)\right)\right), \left({x}^{2}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right)\right)\right)\right), \left({x}^{2}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right)\right)\right)\right), \left({x}^{2}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \left({x}^{2}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \left(x \cdot \color{blue}{x}\right)\right) \]

      16. *-lowering-*.f64 92.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 92.6%

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

   if 3e75 < x

   1. Initial program 4.4%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 41.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 41.6%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\frac{1}{x}}{x}}\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\mathsf{neg}\left(x\right)}}\right)\right) \]

      3. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(x\right)}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}\right), \color{blue}{\left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), \left(\sqrt[3]{\color{blue}{\mathsf{neg}\left(x\right)}}\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{-1}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      9. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(0 - x\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\log 1 - x\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(\log 1, x\right)\right)\right)\right) \]

      13. metadata-eval 98.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right)\right) \]

   7. Applied egg-rr 98.5%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{0 - x}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      2. rem-cube-cbrt N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left({\left(\sqrt[3]{x}\right)}^{3}\right)}\right)\right)\right) \]

      3. cube-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)}^{3}}\right)\right)\right) \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(0 - \sqrt[3]{x}\right)}^{3}}\right)\right)\right) \]

      5. rem-cbrt-cube N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(0 - \color{blue}{\sqrt[3]{x}}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      8. cbrt-lowering-cbrt.f64 98.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\mathsf{cbrt.f64}\left(x\right)\right)\right)\right) \]

   9. Applied egg-rr 98.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+75}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 46000000:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{1}{x + 1}}} + \frac{-1}{\sqrt[3]{\frac{x}{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 46000000.0)
   (+ (/ 1.0 (cbrt (/ 1.0 (+ x 1.0)))) (/ -1.0 (cbrt (/ x (* x x)))))
   (* 0.3333333333333333 (/ (cbrt (/ -1.0 x)) (- 0.0 (cbrt x))))))

C

double code(double x) {
	double tmp;
	if (x <= 46000000.0) {
		tmp = (1.0 / cbrt((1.0 / (x + 1.0)))) + (-1.0 / cbrt((x / (x * x))));
	} else {
		tmp = 0.3333333333333333 * (cbrt((-1.0 / x)) / (0.0 - cbrt(x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 46000000.0) {
		tmp = (1.0 / Math.cbrt((1.0 / (x + 1.0)))) + (-1.0 / Math.cbrt((x / (x * x))));
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt((-1.0 / x)) / (0.0 - Math.cbrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 46000000.0)
		tmp = Float64(Float64(1.0 / cbrt(Float64(1.0 / Float64(x + 1.0)))) + Float64(-1.0 / cbrt(Float64(x / Float64(x * x)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(Float64(-1.0 / x)) / Float64(0.0 - cbrt(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 46000000.0], N[(N[(1.0 / N[Power[N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[N[(-1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[(0.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.6e7

   1. Initial program 78.9%

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

      1. rem-cbrt-cube N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{3}}\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      2. unpow3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      3. pow1/3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot {\left(x + 1\right)}^{\frac{1}{3}}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right) \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   4. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{0.8333333333333334}} \cdot \sqrt[3]{{\left(x + 1\right)}^{0.16666666666666666}}} - \sqrt[3]{x} \]
   5. Step-by-step derivation

      1. cbrt-unprod N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\frac{5}{6}} \cdot {\left(x + 1\right)}^{\frac{1}{6}}}\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{5}{6} + \frac{1}{6}\right)}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{1}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      4. unpow1 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{x + 1}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      5. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      7. cbrt-div N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}}\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sqrt[3]{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\sqrt[3]{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}\right)\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      10. cbrt-lowering-cbrt.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}}\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      12. flip3-+ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{x + 1}\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x + 1\right)\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      15. +-lowering-+.f64 79.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   6. Applied egg-rr 79.2%

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

      1. rem-cube-cbrt N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{{\left(\sqrt[3]{x}\right)}^{3}}\right)\right) \]

      2. sqr-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{{\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}}\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}}\right)\right) \]

      4. sqr-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{{\left(\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)}^{\left(\frac{3}{2}\right)}}\right)\right) \]

      5. sub0-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{{\left(\left(0 - \sqrt[3]{x}\right) \cdot \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)}^{\left(\frac{3}{2}\right)}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{{\left(\left(0 - \sqrt[3]{x}\right) \cdot \left(0 - \sqrt[3]{x}\right)\right)}^{\left(\frac{3}{2}\right)}}\right)\right) \]

      7. pow-prod-down N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{{\left(0 - \sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - \sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}}\right)\right) \]

      8. sqr-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{{\left(0 - \sqrt[3]{x}\right)}^{3}}\right)\right) \]

      9. sub0-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)}^{3}}\right)\right) \]

      10. cube-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{\mathsf{neg}\left({\left(\sqrt[3]{x}\right)}^{3}\right)}\right)\right) \]

      11. rem-cube-cbrt N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right) \]

      12. sub0-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{0 - x}\right)\right) \]

      13. flip-- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{\frac{0 \cdot 0 - x \cdot x}{0 + x}}\right)\right) \]

      14. clear-num N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\sqrt[3]{\frac{1}{\frac{0 + x}{0 \cdot 0 - x \cdot x}}}\right)\right) \]

      15. cbrt-div N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{\frac{0 + x}{0 \cdot 0 - x \cdot x}}}}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \left(\frac{1}{\sqrt[3]{\color{blue}{\frac{0 + x}{0 \cdot 0 - x \cdot x}}}}\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt[3]{\frac{0 + x}{0 \cdot 0 - x \cdot x}}\right)}\right)\right) \]

      18. cbrt-lowering-cbrt.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{0 + x}{0 \cdot 0 - x \cdot x}\right)\right)\right)\right) \]

      19. +-lft-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{x}{0 \cdot 0 - x \cdot x}\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{x}{0 - x \cdot x}\right)\right)\right)\right) \]

      21. sub0-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{x}{\mathsf{neg}\left(x \cdot x\right)}\right)\right)\right)\right) \]

   8. Applied egg-rr 79.3%

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

   if 4.6e7 < x

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 53.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 53.6%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\frac{1}{x}}{x}}\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\mathsf{neg}\left(x\right)}}\right)\right) \]

      3. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(x\right)}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}\right), \color{blue}{\left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), \left(\sqrt[3]{\color{blue}{\mathsf{neg}\left(x\right)}}\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{-1}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      9. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(0 - x\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\log 1 - x\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(\log 1, x\right)\right)\right)\right) \]

      13. metadata-eval 97.7%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right)\right) \]

   7. Applied egg-rr 97.7%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{0 - x}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      2. rem-cube-cbrt N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left({\left(\sqrt[3]{x}\right)}^{3}\right)}\right)\right)\right) \]

      3. cube-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)}^{3}}\right)\right)\right) \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(0 - \sqrt[3]{x}\right)}^{3}}\right)\right)\right) \]

      5. rem-cbrt-cube N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(0 - \color{blue}{\sqrt[3]{x}}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      8. cbrt-lowering-cbrt.f64 97.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\mathsf{cbrt.f64}\left(x\right)\right)\right)\right) \]

   9. Applied egg-rr 97.7%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 46000000:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{1}{x + 1}}} + \frac{-1}{\sqrt[3]{\frac{x}{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 35000000:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{1}{x + 1}}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 35000000.0)
   (- (/ 1.0 (cbrt (/ 1.0 (+ x 1.0)))) (cbrt x))
   (* 0.3333333333333333 (/ (cbrt (/ -1.0 x)) (- 0.0 (cbrt x))))))

C

double code(double x) {
	double tmp;
	if (x <= 35000000.0) {
		tmp = (1.0 / cbrt((1.0 / (x + 1.0)))) - cbrt(x);
	} else {
		tmp = 0.3333333333333333 * (cbrt((-1.0 / x)) / (0.0 - cbrt(x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 35000000.0) {
		tmp = (1.0 / Math.cbrt((1.0 / (x + 1.0)))) - Math.cbrt(x);
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt((-1.0 / x)) / (0.0 - Math.cbrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 35000000.0)
		tmp = Float64(Float64(1.0 / cbrt(Float64(1.0 / Float64(x + 1.0)))) - cbrt(x));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(Float64(-1.0 / x)) / Float64(0.0 - cbrt(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 35000000.0], N[(N[(1.0 / N[Power[N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[N[(-1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[(0.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5e7

   1. Initial program 78.9%

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

      1. rem-cbrt-cube N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{3}}\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      2. unpow3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      3. pow1/3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot {\left(x + 1\right)}^{\frac{1}{3}}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right) \cdot {\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   4. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{0.8333333333333334}} \cdot \sqrt[3]{{\left(x + 1\right)}^{0.16666666666666666}}} - \sqrt[3]{x} \]
   5. Step-by-step derivation

      1. cbrt-unprod N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\frac{5}{6}} \cdot {\left(x + 1\right)}^{\frac{1}{6}}}\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{5}{6} + \frac{1}{6}\right)}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{{\left(x + 1\right)}^{1}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      4. unpow1 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{x + 1}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      5. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      7. cbrt-div N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}}\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sqrt[3]{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\sqrt[3]{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}\right)\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      10. cbrt-lowering-cbrt.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}}\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      12. flip3-+ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{x + 1}\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x + 1\right)\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      15. +-lowering-+.f64 79.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   6. Applied egg-rr 79.2%

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

   if 3.5e7 < x

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 53.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 53.6%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\frac{1}{x}}{x}}\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\mathsf{neg}\left(x\right)}}\right)\right) \]

      3. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(x\right)}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}\right), \color{blue}{\left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), \left(\sqrt[3]{\color{blue}{\mathsf{neg}\left(x\right)}}\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{-1}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      9. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(0 - x\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\log 1 - x\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(\log 1, x\right)\right)\right)\right) \]

      13. metadata-eval 97.7%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right)\right) \]

   7. Applied egg-rr 97.7%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{0 - x}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      2. rem-cube-cbrt N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left({\left(\sqrt[3]{x}\right)}^{3}\right)}\right)\right)\right) \]

      3. cube-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)}^{3}}\right)\right)\right) \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(0 - \sqrt[3]{x}\right)}^{3}}\right)\right)\right) \]

      5. rem-cbrt-cube N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(0 - \color{blue}{\sqrt[3]{x}}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      8. cbrt-lowering-cbrt.f64 97.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\mathsf{cbrt.f64}\left(x\right)\right)\right)\right) \]

   9. Applied egg-rr 97.7%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 35000000:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{1}{x + 1}}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 17000000:\\ \;\;\;\;{\left(x + 1\right)}^{0.3333333333333333} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 17000000.0)
   (- (pow (+ x 1.0) 0.3333333333333333) (cbrt x))
   (* 0.3333333333333333 (/ (cbrt (/ -1.0 x)) (- 0.0 (cbrt x))))))

C

double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = pow((x + 1.0), 0.3333333333333333) - cbrt(x);
	} else {
		tmp = 0.3333333333333333 * (cbrt((-1.0 / x)) / (0.0 - cbrt(x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = Math.pow((x + 1.0), 0.3333333333333333) - Math.cbrt(x);
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt((-1.0 / x)) / (0.0 - Math.cbrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 17000000.0)
		tmp = Float64((Float64(x + 1.0) ^ 0.3333333333333333) - cbrt(x));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(Float64(-1.0 / x)) / Float64(0.0 - cbrt(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 17000000.0], N[(N[Power[N[(x + 1.0), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[N[(-1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[(0.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e7

   1. Initial program 82.9%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{x + 1}\right), \color{blue}{\left(\sqrt[3]{x}\right)}\right) \]

      2. pow1/3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({\left(x + 1\right)}^{\frac{1}{3}}\right), \left(\sqrt[3]{\color{blue}{x}}\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \frac{1}{3}\right), \left(\sqrt[3]{\color{blue}{x}}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{3}\right), \left(\sqrt[3]{x}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 83.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{3}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   4. Applied egg-rr 83.1%

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

   if 1.7e7 < x

   1. Initial program 6.2%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 53.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 53.7%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\frac{1}{x}}{x}}\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\mathsf{neg}\left(x\right)}}\right)\right) \]

      3. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(x\right)}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}\right), \color{blue}{\left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), \left(\sqrt[3]{\color{blue}{\mathsf{neg}\left(x\right)}}\right)\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{-1}{x}\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

      9. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(0 - x\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\left(\log 1 - x\right)\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(\log 1, x\right)\right)\right)\right) \]

      13. metadata-eval 97.4%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{cbrt.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right)\right) \]

   7. Applied egg-rr 97.4%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{0 - x}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      2. rem-cube-cbrt N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{\mathsf{neg}\left({\left(\sqrt[3]{x}\right)}^{3}\right)}\right)\right)\right) \]

      3. cube-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)}^{3}}\right)\right)\right) \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\sqrt[3]{{\left(0 - \sqrt[3]{x}\right)}^{3}}\right)\right)\right) \]

      5. rem-cbrt-cube N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(0 - \color{blue}{\sqrt[3]{x}}\right)\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\left(\sqrt[3]{x}\right)\right)\right)\right) \]

      8. cbrt-lowering-cbrt.f64 97.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right), \mathsf{neg.f64}\left(\mathsf{cbrt.f64}\left(x\right)\right)\right)\right) \]

   9. Applied egg-rr 97.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 17000000:\\ \;\;\;\;{\left(x + 1\right)}^{0.3333333333333333} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{0 - \sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 17000000:\\ \;\;\;\;{\left(x + 1\right)}^{0.3333333333333333} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 17000000.0)
   (- (pow (+ x 1.0) 0.3333333333333333) (cbrt x))
   (/ (* 0.3333333333333333 (cbrt (/ 1.0 x))) (cbrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = pow((x + 1.0), 0.3333333333333333) - cbrt(x);
	} else {
		tmp = (0.3333333333333333 * cbrt((1.0 / x))) / cbrt(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = Math.pow((x + 1.0), 0.3333333333333333) - Math.cbrt(x);
	} else {
		tmp = (0.3333333333333333 * Math.cbrt((1.0 / x))) / Math.cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 17000000.0)
		tmp = Float64((Float64(x + 1.0) ^ 0.3333333333333333) - cbrt(x));
	else
		tmp = Float64(Float64(0.3333333333333333 * cbrt(Float64(1.0 / x))) / cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 17000000.0], N[(N[Power[N[(x + 1.0), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e7

   1. Initial program 82.9%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{x + 1}\right), \color{blue}{\left(\sqrt[3]{x}\right)}\right) \]

      2. pow1/3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({\left(x + 1\right)}^{\frac{1}{3}}\right), \left(\sqrt[3]{\color{blue}{x}}\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \frac{1}{3}\right), \left(\sqrt[3]{\color{blue}{x}}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{3}\right), \left(\sqrt[3]{x}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 83.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{3}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   4. Applied egg-rr 83.1%

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

   if 1.7e7 < x

   1. Initial program 6.2%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 53.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 53.7%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. un-div-inv N/A

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

      7. associate-/r* N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{\sqrt[3]{x}}\right), \color{blue}{\left(\sqrt[3]{x}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{x}\right)\right), \left(\sqrt[3]{\color{blue}{x}}\right)\right) \]

      10. cbrt-lowering-cbrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \left(\sqrt[3]{x}\right)\right) \]

      11. cbrt-lowering-cbrt.f64 97.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   7. Applied egg-rr 97.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{3} \cdot \sqrt[3]{\frac{1}{x}}\right)}, \mathsf{cbrt.f64}\left(x\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{\frac{1}{x}} \cdot \frac{1}{3}\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{x}}\right), \frac{1}{3}\right), \mathsf{cbrt.f64}\left(\color{blue}{x}\right)\right) \]

      3. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{x}\right)\right), \frac{1}{3}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      4. /-lowering-/.f64 97.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \frac{1}{3}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   10. Simplified 97.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 17000000:\\ \;\;\;\;{\left(x + 1\right)}^{0.3333333333333333} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 17000000:\\ \;\;\;\;{\left(x + 1\right)}^{0.3333333333333333} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 17000000.0)
   (- (pow (+ x 1.0) 0.3333333333333333) (cbrt x))
   (* 0.3333333333333333 (/ (/ 1.0 (cbrt x)) (cbrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = pow((x + 1.0), 0.3333333333333333) - cbrt(x);
	} else {
		tmp = 0.3333333333333333 * ((1.0 / cbrt(x)) / cbrt(x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = Math.pow((x + 1.0), 0.3333333333333333) - Math.cbrt(x);
	} else {
		tmp = 0.3333333333333333 * ((1.0 / Math.cbrt(x)) / Math.cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 17000000.0)
		tmp = Float64((Float64(x + 1.0) ^ 0.3333333333333333) - cbrt(x));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(1.0 / cbrt(x)) / cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 17000000.0], N[(N[Power[N[(x + 1.0), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e7

   1. Initial program 82.9%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{x + 1}\right), \color{blue}{\left(\sqrt[3]{x}\right)}\right) \]

      2. pow1/3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({\left(x + 1\right)}^{\frac{1}{3}}\right), \left(\sqrt[3]{\color{blue}{x}}\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \frac{1}{3}\right), \left(\sqrt[3]{\color{blue}{x}}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{3}\right), \left(\sqrt[3]{x}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 83.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{3}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   4. Applied egg-rr 83.1%

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

   if 1.7e7 < x

   1. Initial program 6.2%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 53.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 53.7%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\frac{1}{x}}{x}}\right)\right) \]

      2. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{\frac{1}{x}}}{\color{blue}{\sqrt[3]{x}}}\right)\right) \]

      3. pow1/3 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{{\left(\frac{1}{x}\right)}^{\frac{1}{3}}}{\sqrt[3]{\color{blue}{x}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right), \color{blue}{\left(\sqrt[3]{x}\right)}\right)\right) \]

      5. pow1/3 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\sqrt[3]{\frac{1}{x}}\right), \left(\sqrt[3]{\color{blue}{x}}\right)\right)\right) \]

      6. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\frac{\sqrt[3]{1}}{\sqrt[3]{x}}\right), \left(\sqrt[3]{\color{blue}{x}}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\frac{1}{\sqrt[3]{x}}\right), \left(\sqrt[3]{x}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\sqrt[3]{x}\right)\right), \left(\sqrt[3]{\color{blue}{x}}\right)\right)\right) \]

      9. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(x\right)\right), \left(\sqrt[3]{x}\right)\right)\right) \]

      10. cbrt-lowering-cbrt.f64 97.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(x\right)\right), \mathsf{cbrt.f64}\left(x\right)\right)\right) \]

   7. Applied egg-rr 97.3%

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

Alternative 10: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 17000000:\\ \;\;\;\;{\left(x + 1\right)}^{0.3333333333333333} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 17000000.0)
   (- (pow (+ x 1.0) 0.3333333333333333) (cbrt x))
   (* 0.3333333333333333 (pow (cbrt x) -2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = pow((x + 1.0), 0.3333333333333333) - cbrt(x);
	} else {
		tmp = 0.3333333333333333 * pow(cbrt(x), -2.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = Math.pow((x + 1.0), 0.3333333333333333) - Math.cbrt(x);
	} else {
		tmp = 0.3333333333333333 * Math.pow(Math.cbrt(x), -2.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 17000000.0)
		tmp = Float64((Float64(x + 1.0) ^ 0.3333333333333333) - cbrt(x));
	else
		tmp = Float64(0.3333333333333333 * (cbrt(x) ^ -2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 17000000.0], N[(N[Power[N[(x + 1.0), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e7

   1. Initial program 82.9%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt[3]{x + 1}\right), \color{blue}{\left(\sqrt[3]{x}\right)}\right) \]

      2. pow1/3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({\left(x + 1\right)}^{\frac{1}{3}}\right), \left(\sqrt[3]{\color{blue}{x}}\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \frac{1}{3}\right), \left(\sqrt[3]{\color{blue}{x}}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{3}\right), \left(\sqrt[3]{x}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 83.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{3}\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   4. Applied egg-rr 83.1%

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

   if 1.7e7 < x

   1. Initial program 6.2%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 53.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 53.7%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\frac{1}{x}}{x}}\right)\right) \]

      2. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{\frac{1}{x}}}{\color{blue}{\sqrt[3]{x}}}\right)\right) \]

      3. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{\sqrt[3]{\color{blue}{x}}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{x}}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}\right)\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\color{blue}{-1}}\right)\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{-1}\right)\right) \]

      8. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\sqrt[3]{x}\right)}^{\color{blue}{\left(2 \cdot -1\right)}}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\left(\sqrt[3]{x}\right), \color{blue}{\left(2 \cdot -1\right)}\right)\right) \]

      10. cbrt-lowering-cbrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(x\right), \left(\color{blue}{2} \cdot -1\right)\right)\right) \]

      11. metadata-eval 97.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(x\right), -2\right)\right) \]

   7. Applied egg-rr 97.3%

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

Alternative 11: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 49000000:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 49000000.0)
   (- (cbrt (+ x 1.0)) (cbrt x))
   (* 0.3333333333333333 (pow (cbrt x) -2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 49000000.0) {
		tmp = cbrt((x + 1.0)) - cbrt(x);
	} else {
		tmp = 0.3333333333333333 * pow(cbrt(x), -2.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 49000000.0) {
		tmp = Math.cbrt((x + 1.0)) - Math.cbrt(x);
	} else {
		tmp = 0.3333333333333333 * Math.pow(Math.cbrt(x), -2.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 49000000.0)
		tmp = Float64(cbrt(Float64(x + 1.0)) - cbrt(x));
	else
		tmp = Float64(0.3333333333333333 * (cbrt(x) ^ -2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 49000000.0], N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9e7

   1. Initial program 78.9%

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

   if 4.9e7 < x

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 53.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 53.6%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\frac{1}{x}}{x}}\right)\right) \]

      2. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{\frac{1}{x}}}{\color{blue}{\sqrt[3]{x}}}\right)\right) \]

      3. cbrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{\sqrt[3]{\color{blue}{x}}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{x}}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}\right)\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\color{blue}{-1}}\right)\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{-1}\right)\right) \]

      8. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\sqrt[3]{x}\right)}^{\color{blue}{\left(2 \cdot -1\right)}}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\left(\sqrt[3]{x}\right), \color{blue}{\left(2 \cdot -1\right)}\right)\right) \]

      10. cbrt-lowering-cbrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(x\right), \left(\color{blue}{2} \cdot -1\right)\right)\right) \]

      11. metadata-eval 97.7%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(x\right), -2\right)\right) \]

   7. Applied egg-rr 97.7%

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

Alternative 12: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

   9. *-lowering-*.f64 53.0%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

5. Simplified 53.0%

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

   1. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\frac{1}{x}}{x}}\right)\right) \]

   2. cbrt-div N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{\frac{1}{x}}}{\color{blue}{\sqrt[3]{x}}}\right)\right) \]

   3. cbrt-div N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{\sqrt[3]{\color{blue}{x}}}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{x}}\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}\right)\right) \]

   6. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\color{blue}{-1}}\right)\right) \]

   7. pow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{-1}\right)\right) \]

   8. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\sqrt[3]{x}\right)}^{\color{blue}{\left(2 \cdot -1\right)}}\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\left(\sqrt[3]{x}\right), \color{blue}{\left(2 \cdot -1\right)}\right)\right) \]

   10. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(x\right), \left(\color{blue}{2} \cdot -1\right)\right)\right) \]

   11. metadata-eval 95.0%

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(x\right), -2\right)\right) \]

7. Applied egg-rr 95.0%

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

Alternative 13: 92.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{3 \cdot \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{1}{{x}^{-0.6666666666666666}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 1.0 (* 3.0 (cbrt (* x x))))
   (/ 0.3333333333333333 (/ 1.0 (pow x -0.6666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (3.0 * cbrt((x * x)));
	} else {
		tmp = 0.3333333333333333 / (1.0 / pow(x, -0.6666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (3.0 * Math.cbrt((x * x)));
	} else {
		tmp = 0.3333333333333333 / (1.0 / Math.pow(x, -0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(1.0 / Float64(3.0 * cbrt(Float64(x * x))));
	else
		tmp = Float64(0.3333333333333333 / Float64(1.0 / (x ^ -0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(1.0 / N[(3.0 * N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(1.0 / N[Power[x, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 12.8%

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

      1. pow1/3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left({x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left(e^{\log x \cdot \frac{1}{3}}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{exp.f64}\left(\left(\log x \cdot \frac{1}{3}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{exp.f64}\left(\left(\frac{1}{3} \cdot \log x\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \log x\right)\right)\right) \]

      6. log-lowering-log.f64 14.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{log.f64}\left(x\right)\right)\right)\right) \]

   4. Applied egg-rr 14.4%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{e^{0.3333333333333333 \cdot \log x}} \]
   5. Step-by-step derivation

      1. pow1/3 N/A

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

      2. *-commutative N/A

         \[\leadsto {\left(x + 1\right)}^{\frac{1}{3}} - e^{\log x \cdot \frac{1}{3}} \]

      3. pow-to-exp N/A

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

      4. pow1/3 N/A

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

      5. remove-double-div N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{{\left(x + 1\right)}^{\frac{1}{3}} - \sqrt[3]{x}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left({\left(x + 1\right)}^{\frac{1}{3}} - \sqrt[3]{x}\right)}\right)\right) \]

      8. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\sqrt[3]{x + 1} - \sqrt[3]{\color{blue}{x}}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\sqrt[3]{x + 1}\right), \color{blue}{\left(\sqrt[3]{x}\right)}\right)\right)\right) \]

      10. cbrt-lowering-cbrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\left(x + 1\right)\right), \left(\sqrt[3]{\color{blue}{x}}\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\left(1 + x\right)\right), \left(\sqrt[3]{x}\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(1, x\right)\right), \left(\sqrt[3]{x}\right)\right)\right)\right) \]

      13. cbrt-lowering-cbrt.f64 12.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(1, x\right)\right), \mathsf{cbrt.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 12.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(3 \cdot \sqrt[3]{{x}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\sqrt[3]{{x}^{2}} \cdot \color{blue}{3}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt[3]{{x}^{2}}\right), \color{blue}{3}\right)\right) \]

      3. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left({x}^{2}\right)\right), 3\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(x \cdot x\right)\right), 3\right)\right) \]

      5. *-lowering-*.f64 92.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(x, x\right)\right), 3\right)\right) \]

   9. Simplified 92.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 4.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 4.8%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. un-div-inv N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

      8. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

      9. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      10. pow-prod-up N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}\right)\right) \]

      12. metadata-eval 89.1%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \frac{2}{3}\right)\right) \]

   7. Applied egg-rr 89.1%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\frac{1}{3} + \color{blue}{\frac{1}{3}}\right)}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \color{blue}{{x}^{\frac{1}{3}}}\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({\left(x \cdot x\right)}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      4. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{x \cdot x}\right)\right) \]

      5. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{x \cdot x}{1}}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{1}{\frac{1}{x \cdot x}}}\right)\right) \]

      7. cbrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{\frac{1}{x \cdot x}}}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{1}{\sqrt[3]{\color{blue}{\frac{1}{x \cdot x}}}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt[3]{\frac{1}{x \cdot x}}\right)}\right)\right) \]

      10. pow1/3 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x \cdot x}\right)}^{\color{blue}{\frac{1}{3}}}\right)\right)\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({\left({\left(x \cdot x\right)}^{-1}\right)}^{\frac{1}{3}}\right)\right)\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({\left({\left({x}^{2}\right)}^{-1}\right)}^{\frac{1}{3}}\right)\right)\right) \]

      13. pow-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({\left({x}^{\left(2 \cdot -1\right)}\right)}^{\frac{1}{3}}\right)\right)\right) \]

      14. pow-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({x}^{\color{blue}{\left(\left(2 \cdot -1\right) \cdot \frac{1}{3}\right)}}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({x}^{\left(-2 \cdot \frac{1}{3}\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({x}^{\frac{-2}{3}}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({x}^{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}\right)\right)\right) \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}\right)\right)\right) \]

      19. metadata-eval 89.1%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \frac{-2}{3}\right)\right)\right) \]

   9. Applied egg-rr 89.1%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{1}{{x}^{-0.6666666666666666}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{3 \cdot \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{1}{{x}^{-0.6666666666666666}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 92.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{1}{{x}^{-0.6666666666666666}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 0.3333333333333333 (cbrt (* x x)))
   (/ 0.3333333333333333 (/ 1.0 (pow x -0.6666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 / cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 / (1.0 / pow(x, -0.6666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 / Math.cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 / (1.0 / Math.pow(x, -0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(0.3333333333333333 / cbrt(Float64(x * x)));
	else
		tmp = Float64(0.3333333333333333 / Float64(1.0 / (x ^ -0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(0.3333333333333333 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(1.0 / N[Power[x, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 12.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 92.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 92.4%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. un-div-inv N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

      8. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

      9. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      10. pow-prod-up N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}\right)\right) \]

      12. metadata-eval 86.3%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \frac{2}{3}\right)\right) \]

   7. Applied egg-rr 86.3%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\frac{1}{3} + \color{blue}{\frac{1}{3}}\right)}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \color{blue}{{x}^{\frac{1}{3}}}\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({\left(x \cdot x\right)}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      4. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{x \cdot x}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(x \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 92.5%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right) \]

   9. Applied egg-rr 92.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 4.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 4.8%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. un-div-inv N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

      8. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

      9. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      10. pow-prod-up N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}\right)\right) \]

      12. metadata-eval 89.1%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \frac{2}{3}\right)\right) \]

   7. Applied egg-rr 89.1%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\frac{1}{3} + \color{blue}{\frac{1}{3}}\right)}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \color{blue}{{x}^{\frac{1}{3}}}\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({\left(x \cdot x\right)}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      4. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{x \cdot x}\right)\right) \]

      5. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{x \cdot x}{1}}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{1}{\frac{1}{x \cdot x}}}\right)\right) \]

      7. cbrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{\frac{1}{x \cdot x}}}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{1}{\sqrt[3]{\color{blue}{\frac{1}{x \cdot x}}}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt[3]{\frac{1}{x \cdot x}}\right)}\right)\right) \]

      10. pow1/3 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({\left(\frac{1}{x \cdot x}\right)}^{\color{blue}{\frac{1}{3}}}\right)\right)\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({\left({\left(x \cdot x\right)}^{-1}\right)}^{\frac{1}{3}}\right)\right)\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({\left({\left({x}^{2}\right)}^{-1}\right)}^{\frac{1}{3}}\right)\right)\right) \]

      13. pow-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({\left({x}^{\left(2 \cdot -1\right)}\right)}^{\frac{1}{3}}\right)\right)\right) \]

      14. pow-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({x}^{\color{blue}{\left(\left(2 \cdot -1\right) \cdot \frac{1}{3}\right)}}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({x}^{\left(-2 \cdot \frac{1}{3}\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({x}^{\frac{-2}{3}}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \left({x}^{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}\right)\right)\right) \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}\right)\right)\right) \]

      19. metadata-eval 89.1%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \frac{-2}{3}\right)\right)\right) \]

   9. Applied egg-rr 89.1%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{1}{{x}^{-0.6666666666666666}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 92.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.6666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 0.3333333333333333 (cbrt (* x x)))
   (* 0.3333333333333333 (pow x -0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 / cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 * pow(x, -0.6666666666666666);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 / Math.cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 * Math.pow(x, -0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(0.3333333333333333 / cbrt(Float64(x * x)));
	else
		tmp = Float64(0.3333333333333333 * (x ^ -0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(0.3333333333333333 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[Power[x, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 12.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 92.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 92.4%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. un-div-inv N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

      8. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

      9. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      10. pow-prod-up N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}\right)\right) \]

      12. metadata-eval 86.3%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \frac{2}{3}\right)\right) \]

   7. Applied egg-rr 86.3%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\frac{1}{3} + \color{blue}{\frac{1}{3}}\right)}\right)\right) \]

      2. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \color{blue}{{x}^{\frac{1}{3}}}\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({\left(x \cdot x\right)}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      4. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{x \cdot x}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(x \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 92.5%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right) \]

   9. Applied egg-rr 92.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 4.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 4.8%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{x \cdot x}}\right), \color{blue}{\frac{1}{3}}\right) \]

      3. pow1/3 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{x \cdot x}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({\left(x \cdot x\right)}^{-1}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot x\right)}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{2}\right)}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

      7. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)}\right), \frac{1}{3}\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)\right), \frac{1}{3}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \frac{-1}{3}\right)\right), \frac{1}{3}\right) \]

      10. metadata-eval 89.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-2}{3}\right), \frac{1}{3}\right) \]

   7. Applied egg-rr 89.1%

      \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.6666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 88.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{{x}^{0.6666666666666666}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 0.3333333333333333 (pow x 0.6666666666666666)))

C

double code(double x) {
	return 0.3333333333333333 / pow(x, 0.6666666666666666);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3333333333333333d0 / (x ** 0.6666666666666666d0)
end function

Java

public static double code(double x) {
	return 0.3333333333333333 / Math.pow(x, 0.6666666666666666);
}

Python

def code(x):
	return 0.3333333333333333 / math.pow(x, 0.6666666666666666)

Julia

function code(x)
	return Float64(0.3333333333333333 / (x ^ 0.6666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333 / (x ^ 0.6666666666666666);
end

Wolfram

code[x_] := N[(0.3333333333333333 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

   9. *-lowering-*.f64 53.0%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

5. Simplified 53.0%

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

   1. associate-/r* N/A

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

   2. cbrt-div N/A

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

   3. cbrt-div N/A

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

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. un-div-inv N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

   8. pow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

   9. pow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

   10. pow-prod-up N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right)\right) \]

   11. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{3} + \frac{1}{3}\right)}\right)\right) \]

   12. metadata-eval 87.5%

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \frac{2}{3}\right)\right) \]

7. Applied egg-rr 87.5%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
8. Add Preprocessing

Alternative 17: 88.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot {x}^{-0.6666666666666666} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.3333333333333333 (pow x -0.6666666666666666)))

C

double code(double x) {
	return 0.3333333333333333 * pow(x, -0.6666666666666666);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3333333333333333d0 * (x ** (-0.6666666666666666d0))
end function

Java

public static double code(double x) {
	return 0.3333333333333333 * Math.pow(x, -0.6666666666666666);
}

Python

def code(x):
	return 0.3333333333333333 * math.pow(x, -0.6666666666666666)

Julia

function code(x)
	return Float64(0.3333333333333333 * (x ^ -0.6666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333 * (x ^ -0.6666666666666666);
end

Wolfram

code[x_] := N[(0.3333333333333333 * N[Power[x, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

   9. *-lowering-*.f64 53.0%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

5. Simplified 53.0%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{x \cdot x}}\right), \color{blue}{\frac{1}{3}}\right) \]

   3. pow1/3 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{x \cdot x}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

   4. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left({\left(x \cdot x\right)}^{-1}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

   5. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot x\right)}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

   6. pow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{2}\right)}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

   7. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)}\right), \frac{1}{3}\right) \]

   8. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)\right), \frac{1}{3}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \frac{-1}{3}\right)\right), \frac{1}{3}\right) \]

   10. metadata-eval 87.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-2}{3}\right), \frac{1}{3}\right) \]

7. Applied egg-rr 87.5%

   \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]

8. Final simplification 87.5%

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

Alternative 18: 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 9.2%

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

3. Taylor expanded in x around 0

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt[3]{x}\right)}\right) \]

   2. cbrt-lowering-cbrt.f64 1.8%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{cbrt.f64}\left(x\right)\right) \]

5. Simplified 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 2024129 
(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)))