2cbrt (problem 3.3.4)

Percentage Accurate: 52.7% → 99.1%

Time: 10.9s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 52.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. flip3-- 56.0%

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

   2. div-inv 56.0%

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

   3. rem-cube-cbrt 56.3%

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

   4. rem-cube-cbrt 57.2%

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

   5. cbrt-unprod 57.2%

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

   6. pow2 57.2%

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

   7. distribute-rgt-out 57.2%

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

   8. +-commutative 57.2%

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

3. Applied egg-rr 57.2%

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

   1. associate-*r/ 57.2%

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

   2. *-rgt-identity 57.2%

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

   3. +-commutative 57.2%

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

   4. associate--l+ 77.7%

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

   5. +-inverses 77.7%

      \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   6. metadata-eval 77.7%

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

   7. +-commutative 77.7%

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

   8. fma-def 77.7%

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

   9. +-commutative 77.7%

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

   10. +-commutative 77.7%

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

5. Simplified 77.7%

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

   1. add-log-exp 55.3%

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

   2. *-un-lft-identity 55.3%

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

   3. log-prod 55.3%

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

   4. metadata-eval 55.3%

      \[\leadsto \frac{1}{\color{blue}{0} + \log \left(e^{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\right)} \]

   5. add-log-exp 77.7%

      \[\leadsto \frac{1}{0 + \color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}} \]

   6. fma-udef 77.7%

      \[\leadsto \frac{1}{0 + \color{blue}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}} \]

   7. pow1/3 38.4%

      \[\leadsto \frac{1}{0 + \left(\color{blue}{{x}^{0.3333333333333333}} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}\right)} \]

   8. *-commutative 38.4%

      \[\leadsto \frac{1}{0 + \left(\color{blue}{\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333}} + \sqrt[3]{{\left(1 + x\right)}^{2}}\right)} \]

   9. pow1/3 38.2%

      \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \color{blue}{{\left({\left(1 + x\right)}^{2}\right)}^{0.3333333333333333}}\right)} \]

   10. unpow2 38.2%

       \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + {\color{blue}{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right)}}^{0.3333333333333333}\right)} \]

   11. pow-prod-down 46.2%

       \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \color{blue}{{\left(1 + x\right)}^{0.3333333333333333} \cdot {\left(1 + x\right)}^{0.3333333333333333}}\right)} \]

   12. +-commutative 46.2%

       \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + {\color{blue}{\left(x + 1\right)}}^{0.3333333333333333} \cdot {\left(1 + x\right)}^{0.3333333333333333}\right)} \]

   13. pow1/3 46.3%

       \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \color{blue}{\sqrt[3]{x + 1}} \cdot {\left(1 + x\right)}^{0.3333333333333333}\right)} \]

   14. +-commutative 46.3%

       \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \sqrt[3]{x + 1} \cdot {\color{blue}{\left(x + 1\right)}}^{0.3333333333333333}\right)} \]

   15. pow1/3 46.5%

       \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \sqrt[3]{x + 1} \cdot \color{blue}{\sqrt[3]{x + 1}}\right)} \]

7. Applied egg-rr 99.2%

   \[\leadsto \frac{1}{\color{blue}{0 + \mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}} \]
8. Step-by-step derivation

   1. +-lft-identity 99.2%

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

9. Simplified 99.2%

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

10. Final simplification 99.2%

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

Alternative 2: 88.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + t_1}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot \left(1 + x\right)} + \sqrt[3]{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (* (cbrt x) (+ t_0 (cbrt x)))))
   (if (<= x -1.35e+154)
     (/ 1.0 (+ 1.0 t_1))
     (if (<= x 1.22e+154)
       (/ 1.0 (+ (pow t_0 2.0) (+ (cbrt (* x (+ 1.0 x))) (cbrt (* x x)))))
       (/ 1.0 (+ t_1 (exp (* 0.6666666666666666 (log1p x)))))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = cbrt(x) * (t_0 + cbrt(x));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / (1.0 + t_1);
	} else if (x <= 1.22e+154) {
		tmp = 1.0 / (pow(t_0, 2.0) + (cbrt((x * (1.0 + x))) + cbrt((x * x))));
	} else {
		tmp = 1.0 / (t_1 + exp((0.6666666666666666 * log1p(x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double t_1 = Math.cbrt(x) * (t_0 + Math.cbrt(x));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / (1.0 + t_1);
	} else if (x <= 1.22e+154) {
		tmp = 1.0 / (Math.pow(t_0, 2.0) + (Math.cbrt((x * (1.0 + x))) + Math.cbrt((x * x))));
	} else {
		tmp = 1.0 / (t_1 + Math.exp((0.6666666666666666 * Math.log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = Float64(cbrt(x) * Float64(t_0 + cbrt(x)))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(1.0 / Float64(1.0 + t_1));
	elseif (x <= 1.22e+154)
		tmp = Float64(1.0 / Float64((t_0 ^ 2.0) + Float64(cbrt(Float64(x * Float64(1.0 + x))) + cbrt(Float64(x * x)))));
	else
		tmp = Float64(1.0 / Float64(t_1 + exp(Float64(0.6666666666666666 * log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], N[(1.0 / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22e+154], N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 4.6%

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

      1. flip3-- 4.6%

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

      2. div-inv 4.6%

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

      3. rem-cube-cbrt 3.9%

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

      4. rem-cube-cbrt 4.6%

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

      5. cbrt-unprod 4.6%

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

      6. pow2 4.6%

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

      7. distribute-rgt-out 4.6%

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

      8. +-commutative 4.6%

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

   3. Applied egg-rr 4.6%

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

      1. associate-*r/ 4.6%

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

      2. *-rgt-identity 4.6%

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

      3. +-commutative 4.6%

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

      4. associate--l+ 4.6%

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

      5. +-inverses 4.6%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 4.6%

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

      7. +-commutative 4.6%

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

      8. fma-def 4.6%

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

      9. +-commutative 4.6%

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

      10. +-commutative 4.6%

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

   5. Simplified 4.6%

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

      1. fma-udef 4.6%

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

      2. pow1/3 4.6%

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

      3. unpow2 4.6%

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

      4. pow-prod-down 0.0%

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

      5. +-commutative 0.0%

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

      6. pow1/3 0.0%

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

      7. +-commutative 0.0%

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

      8. pow1/3 98.4%

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

      9. +-commutative 98.4%

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

      10. pow2 98.4%

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

      11. +-commutative 98.4%

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

      12. pow1/3 0.0%

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

      13. *-commutative 0.0%

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

      14. pow1/3 98.4%

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

   7. Applied egg-rr 98.4%

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

   8. Taylor expanded in x around 0 19.9%

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

   if -1.35000000000000003e154 < x < 1.22e154

   1. Initial program 71.4%

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

      1. flip3-- 71.4%

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

      2. div-inv 71.4%

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

      3. rem-cube-cbrt 72.1%

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

      4. rem-cube-cbrt 73.0%

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

      5. cbrt-unprod 73.0%

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

      6. pow2 73.0%

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

      7. distribute-rgt-out 72.9%

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

      8. +-commutative 72.9%

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

   3. Applied egg-rr 72.9%

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

      1. associate-*r/ 72.9%

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

      2. *-rgt-identity 72.9%

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

      3. +-commutative 72.9%

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

      4. associate--l+ 99.5%

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

      5. +-inverses 99.5%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

      8. fma-def 99.5%

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

      9. +-commutative 99.5%

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

      10. +-commutative 99.5%

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

   5. Simplified 99.5%

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

      1. fma-udef 99.5%

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

      2. pow1/3 98.2%

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

      3. unpow2 98.2%

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

      4. pow-prod-down 82.8%

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

      5. +-commutative 82.8%

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

      6. pow1/3 83.0%

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

      7. +-commutative 83.0%

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

      8. pow1/3 99.5%

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

      9. +-commutative 99.5%

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

      10. pow2 99.5%

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

      11. +-commutative 99.5%

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

      12. pow1/3 49.3%

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

      13. *-commutative 49.3%

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

      14. pow1/3 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. distribute-rgt-in 99.4%

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

      3. +-commutative 99.4%

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

      4. cbrt-unprod 99.5%

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

      5. +-commutative 99.5%

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

      6. cbrt-prod 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 1.22e154 < x

   1. Initial program 4.8%

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

      1. flip3-- 4.8%

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

      2. div-inv 4.8%

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

      3. rem-cube-cbrt 2.9%

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

      4. rem-cube-cbrt 4.8%

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

      5. cbrt-unprod 4.8%

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

      6. pow2 4.8%

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

      7. distribute-rgt-out 4.8%

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

      8. +-commutative 4.8%

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

   3. Applied egg-rr 4.8%

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

      1. associate-*r/ 4.8%

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

      2. *-rgt-identity 4.8%

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

      3. +-commutative 4.8%

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

      4. associate--l+ 4.8%

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

      5. +-inverses 4.8%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 4.8%

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

      7. +-commutative 4.8%

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

      8. fma-def 4.8%

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

      9. +-commutative 4.8%

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

      10. +-commutative 4.8%

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

   5. Simplified 4.8%

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

      1. pow1/3 4.8%

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

      2. pow-pow 91.6%

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

      3. metadata-eval 91.6%

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

   7. Applied egg-rr 91.6%

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

      1. fma-udef 91.6%

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

      2. +-commutative 91.6%

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

      3. add-exp-log 92.4%

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

      4. +-commutative 92.4%

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

      5. log-pow 91.7%

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

      6. +-commutative 91.7%

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

      7. log1p-udef 91.7%

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

   9. Applied egg-rr 91.7%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(\sqrt[3]{x \cdot \left(1 + x\right)} + \sqrt[3]{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \]

Alternative 3: 60.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(t_0 + -1\right)\right) - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (/ 1.0 (+ 1.0 (* (cbrt x) (+ t_0 (cbrt x)))))
     (- (+ 1.0 (+ t_0 -1.0)) (cbrt x)))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 1.0 / (1.0 + (cbrt(x) * (t_0 + cbrt(x))));
	} else {
		tmp = (1.0 + (t_0 + -1.0)) - cbrt(x);
	}
	return tmp;
}

Java

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

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))));
	else
		tmp = Float64(Float64(1.0 + Float64(t_0 + -1.0)) - cbrt(x));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 0.0

   1. Initial program 4.2%

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

      1. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 4.7%

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

      4. rem-cube-cbrt 5.0%

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

      5. cbrt-unprod 5.0%

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

      6. pow2 5.0%

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

      7. distribute-rgt-out 5.0%

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

      8. +-commutative 5.0%

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

   3. Applied egg-rr 5.0%

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

      1. associate-*r/ 5.0%

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

      2. *-rgt-identity 5.0%

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

      3. +-commutative 5.0%

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

      4. associate--l+ 50.5%

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

      5. +-inverses 50.5%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 50.5%

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

      7. +-commutative 50.5%

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

      8. fma-def 50.5%

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

      9. +-commutative 50.5%

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

      10. +-commutative 50.5%

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

   5. Simplified 50.5%

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

      1. fma-udef 50.5%

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

      2. pow1/3 48.3%

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

      3. unpow2 48.3%

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

      4. pow-prod-down 43.6%

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

      5. +-commutative 43.6%

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

      6. pow1/3 44.4%

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

      7. +-commutative 44.4%

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

      8. pow1/3 98.4%

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

      9. +-commutative 98.4%

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

      10. pow2 98.4%

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

      11. +-commutative 98.4%

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

      12. pow1/3 43.6%

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

      13. *-commutative 43.6%

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

      14. pow1/3 98.4%

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

   7. Applied egg-rr 98.4%

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

   8. Taylor expanded in x around 0 20.0%

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

   if 0.0 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 98.3%

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

      1. expm1-log1p-u 95.3%

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

      2. expm1-udef 95.3%

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

      3. log1p-udef 95.3%

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

      4. add-exp-log 98.3%

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

   3. Applied egg-rr 98.3%

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

      1. associate--l+ 98.4%

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

      2. +-commutative 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 63.1%

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

Alternative 4: 88.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 + \sqrt[3]{x}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot t_1}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot \left(1 + x\right)} + \sqrt[3]{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_1, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (+ t_0 (cbrt x))))
   (if (<= x -1.35e+154)
     (/ 1.0 (+ 1.0 (* (cbrt x) t_1)))
     (if (<= x 1.35e+154)
       (/ 1.0 (+ (pow t_0 2.0) (+ (cbrt (* x (+ 1.0 x))) (cbrt (* x x)))))
       (/ 1.0 (fma (cbrt x) t_1 (pow (+ 1.0 x) 0.6666666666666666)))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = t_0 + cbrt(x);
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / (1.0 + (cbrt(x) * t_1));
	} else if (x <= 1.35e+154) {
		tmp = 1.0 / (pow(t_0, 2.0) + (cbrt((x * (1.0 + x))) + cbrt((x * x))));
	} else {
		tmp = 1.0 / fma(cbrt(x), t_1, pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = Float64(t_0 + cbrt(x))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * t_1)));
	elseif (x <= 1.35e+154)
		tmp = Float64(1.0 / Float64((t_0 ^ 2.0) + Float64(cbrt(Float64(x * Float64(1.0 + x))) + cbrt(Float64(x * x)))));
	else
		tmp = Float64(1.0 / fma(cbrt(x), t_1, (Float64(1.0 + x) ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+154], N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$1 + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 4.6%

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

      1. flip3-- 4.6%

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

      2. div-inv 4.6%

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

      3. rem-cube-cbrt 3.9%

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

      4. rem-cube-cbrt 4.6%

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

      5. cbrt-unprod 4.6%

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

      6. pow2 4.6%

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

      7. distribute-rgt-out 4.6%

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

      8. +-commutative 4.6%

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

   3. Applied egg-rr 4.6%

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

      1. associate-*r/ 4.6%

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

      2. *-rgt-identity 4.6%

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

      3. +-commutative 4.6%

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

      4. associate--l+ 4.6%

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

      5. +-inverses 4.6%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 4.6%

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

      7. +-commutative 4.6%

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

      8. fma-def 4.6%

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

      9. +-commutative 4.6%

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

      10. +-commutative 4.6%

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

   5. Simplified 4.6%

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

      1. fma-udef 4.6%

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

      2. pow1/3 4.6%

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

      3. unpow2 4.6%

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

      4. pow-prod-down 0.0%

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

      5. +-commutative 0.0%

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

      6. pow1/3 0.0%

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

      7. +-commutative 0.0%

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

      8. pow1/3 98.4%

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

      9. +-commutative 98.4%

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

      10. pow2 98.4%

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

      11. +-commutative 98.4%

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

      12. pow1/3 0.0%

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

      13. *-commutative 0.0%

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

      14. pow1/3 98.4%

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

   7. Applied egg-rr 98.4%

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

   8. Taylor expanded in x around 0 19.9%

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

   if -1.35000000000000003e154 < x < 1.35000000000000003e154

   1. Initial program 71.4%

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

      1. flip3-- 71.4%

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

      2. div-inv 71.4%

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

      3. rem-cube-cbrt 72.1%

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

      4. rem-cube-cbrt 73.0%

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

      5. cbrt-unprod 73.0%

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

      6. pow2 73.0%

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

      7. distribute-rgt-out 72.9%

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

      8. +-commutative 72.9%

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

   3. Applied egg-rr 72.9%

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

      1. associate-*r/ 72.9%

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

      2. *-rgt-identity 72.9%

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

      3. +-commutative 72.9%

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

      4. associate--l+ 99.5%

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

      5. +-inverses 99.5%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

      8. fma-def 99.5%

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

      9. +-commutative 99.5%

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

      10. +-commutative 99.5%

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

   5. Simplified 99.5%

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

      1. fma-udef 99.5%

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

      2. pow1/3 98.2%

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

      3. unpow2 98.2%

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

      4. pow-prod-down 82.8%

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

      5. +-commutative 82.8%

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

      6. pow1/3 83.0%

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

      7. +-commutative 83.0%

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

      8. pow1/3 99.5%

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

      9. +-commutative 99.5%

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

      10. pow2 99.5%

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

      11. +-commutative 99.5%

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

      12. pow1/3 49.3%

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

      13. *-commutative 49.3%

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

      14. pow1/3 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. distribute-rgt-in 99.4%

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

      3. +-commutative 99.4%

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

      4. cbrt-unprod 99.5%

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

      5. +-commutative 99.5%

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

      6. cbrt-prod 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

      1. flip3-- 4.8%

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

      2. div-inv 4.8%

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

      3. rem-cube-cbrt 2.9%

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

      4. rem-cube-cbrt 4.8%

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

      5. cbrt-unprod 4.8%

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

      6. pow2 4.8%

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

      7. distribute-rgt-out 4.8%

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

      8. +-commutative 4.8%

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

   3. Applied egg-rr 4.8%

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

      1. associate-*r/ 4.8%

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

      2. *-rgt-identity 4.8%

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

      3. +-commutative 4.8%

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

      4. associate--l+ 4.8%

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

      5. +-inverses 4.8%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 4.8%

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

      7. +-commutative 4.8%

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

      8. fma-def 4.8%

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

      9. +-commutative 4.8%

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

      10. +-commutative 4.8%

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

   5. Simplified 4.8%

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

      1. pow1/3 4.8%

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

      2. pow-pow 91.6%

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

      3. metadata-eval 91.6%

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

   7. Applied egg-rr 91.6%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(\sqrt[3]{x \cdot \left(1 + x\right)} + \sqrt[3]{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]

Alternative 5: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. flip3-- 56.0%

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

   2. div-inv 56.0%

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

   3. rem-cube-cbrt 56.3%

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

   4. rem-cube-cbrt 57.2%

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

   5. cbrt-unprod 57.2%

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

   6. pow2 57.2%

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

   7. distribute-rgt-out 57.2%

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

   8. +-commutative 57.2%

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

3. Applied egg-rr 57.2%

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

   1. associate-*r/ 57.2%

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

   2. *-rgt-identity 57.2%

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

   3. +-commutative 57.2%

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

   4. associate--l+ 77.7%

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

   5. +-inverses 77.7%

      \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   6. metadata-eval 77.7%

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

   7. +-commutative 77.7%

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

   8. fma-def 77.7%

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

   9. +-commutative 77.7%

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

   10. +-commutative 77.7%

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

5. Simplified 77.7%

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

   1. fma-udef 77.7%

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

   2. pow1/3 76.7%

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

   3. unpow2 76.7%

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

   4. pow-prod-down 72.3%

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

   5. +-commutative 72.3%

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

   6. pow1/3 72.6%

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

   7. +-commutative 72.6%

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

   8. pow1/3 99.2%

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

   9. +-commutative 99.2%

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

   10. pow2 99.2%

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

   11. +-commutative 99.2%

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

   12. pow1/3 46.5%

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

   13. *-commutative 46.5%

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

   14. pow1/3 99.2%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 6: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot \left(1 + x\right)} + \sqrt[3]{x \cdot x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= x -1.35e+154)
     (/ 1.0 (+ 1.0 (* (cbrt x) (+ t_0 (cbrt x)))))
     (/ 1.0 (+ (pow t_0 2.0) (+ (cbrt (* x (+ 1.0 x))) (cbrt (* x x))))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / (1.0 + (cbrt(x) * (t_0 + cbrt(x))));
	} else {
		tmp = 1.0 / (pow(t_0, 2.0) + (cbrt((x * (1.0 + x))) + cbrt((x * x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (t_0 + Math.cbrt(x))));
	} else {
		tmp = 1.0 / (Math.pow(t_0, 2.0) + (Math.cbrt((x * (1.0 + x))) + Math.cbrt((x * x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))));
	else
		tmp = Float64(1.0 / Float64((t_0 ^ 2.0) + Float64(cbrt(Float64(x * Float64(1.0 + x))) + cbrt(Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, -1.35e+154], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 4.6%

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

      1. flip3-- 4.6%

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

      2. div-inv 4.6%

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

      3. rem-cube-cbrt 3.9%

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

      4. rem-cube-cbrt 4.6%

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

      5. cbrt-unprod 4.6%

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

      6. pow2 4.6%

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

      7. distribute-rgt-out 4.6%

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

      8. +-commutative 4.6%

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

   3. Applied egg-rr 4.6%

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

      1. associate-*r/ 4.6%

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

      2. *-rgt-identity 4.6%

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

      3. +-commutative 4.6%

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

      4. associate--l+ 4.6%

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

      5. +-inverses 4.6%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 4.6%

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

      7. +-commutative 4.6%

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

      8. fma-def 4.6%

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

      9. +-commutative 4.6%

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

      10. +-commutative 4.6%

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

   5. Simplified 4.6%

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

      1. fma-udef 4.6%

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

      2. pow1/3 4.6%

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

      3. unpow2 4.6%

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

      4. pow-prod-down 0.0%

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

      5. +-commutative 0.0%

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

      6. pow1/3 0.0%

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

      7. +-commutative 0.0%

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

      8. pow1/3 98.4%

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

      9. +-commutative 98.4%

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

      10. pow2 98.4%

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

      11. +-commutative 98.4%

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

      12. pow1/3 0.0%

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

      13. *-commutative 0.0%

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

      14. pow1/3 98.4%

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

   7. Applied egg-rr 98.4%

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

   8. Taylor expanded in x around 0 19.9%

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

   if -1.35000000000000003e154 < x

   1. Initial program 64.2%

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

      1. flip3-- 64.2%

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

      2. div-inv 64.2%

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

      3. rem-cube-cbrt 64.6%

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

      4. rem-cube-cbrt 65.6%

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

      5. cbrt-unprod 65.6%

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

      6. pow2 65.6%

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

      7. distribute-rgt-out 65.5%

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

      8. +-commutative 65.5%

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

   3. Applied egg-rr 65.5%

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

      1. associate-*r/ 65.5%

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

      2. *-rgt-identity 65.5%

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

      3. +-commutative 65.5%

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

      4. associate--l+ 89.2%

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

      5. +-inverses 89.2%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 89.2%

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

      7. +-commutative 89.2%

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

      8. fma-def 89.2%

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

      9. +-commutative 89.2%

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

      10. +-commutative 89.2%

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

   5. Simplified 89.2%

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

      1. fma-udef 89.2%

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

      2. pow1/3 88.1%

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

      3. unpow2 88.1%

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

      4. pow-prod-down 83.7%

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

      5. +-commutative 83.7%

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

      6. pow1/3 84.1%

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

      7. +-commutative 84.1%

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

      8. pow1/3 99.3%

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

      9. +-commutative 99.3%

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

      10. pow2 99.3%

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

      11. +-commutative 99.3%

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

      12. pow1/3 53.9%

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

      13. *-commutative 53.9%

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

      14. pow1/3 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. distribute-rgt-in 99.3%

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

      3. +-commutative 99.3%

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

      4. cbrt-unprod 89.2%

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

      5. +-commutative 89.2%

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

      6. cbrt-prod 89.3%

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

   9. Applied egg-rr 89.3%

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

4. Final simplification 79.8%

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

Alternative 7: 56.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(\sqrt[3]{1 + x} + -1\right)\right) - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;e^{0.6666666666666666 \cdot \left(-\log x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.95e+15)
   (- (+ 1.0 (+ (cbrt (+ 1.0 x)) -1.0)) (cbrt x))
   (exp (* 0.6666666666666666 (- (log x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.95e+15) {
		tmp = (1.0 + (cbrt((1.0 + x)) + -1.0)) - cbrt(x);
	} else {
		tmp = exp((0.6666666666666666 * -log(x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.95e+15) {
		tmp = (1.0 + (Math.cbrt((1.0 + x)) + -1.0)) - Math.cbrt(x);
	} else {
		tmp = Math.exp((0.6666666666666666 * -Math.log(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.95e+15)
		tmp = Float64(Float64(1.0 + Float64(cbrt(Float64(1.0 + x)) + -1.0)) - cbrt(x));
	else
		tmp = exp(Float64(0.6666666666666666 * Float64(-log(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.95e+15], N[(N[(1.0 + N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[Exp[N[(0.6666666666666666 * (-N[Log[x], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.95e15

   1. Initial program 69.9%

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

      1. expm1-log1p-u 66.5%

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

      2. expm1-udef 66.5%

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

      3. log1p-udef 66.5%

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

      4. add-exp-log 69.9%

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

   3. Applied egg-rr 69.9%

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

      1. associate--l+ 69.9%

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

      2. +-commutative 69.9%

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

   5. Simplified 69.9%

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

   if 1.95e15 < x

   1. Initial program 4.1%

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

      1. flip3-- 4.1%

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

      2. div-inv 4.1%

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

      3. rem-cube-cbrt 4.8%

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

      4. rem-cube-cbrt 5.9%

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

      5. cbrt-unprod 5.9%

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

      6. pow2 5.9%

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

      7. distribute-rgt-out 5.9%

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

      8. +-commutative 5.9%

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

   3. Applied egg-rr 5.9%

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

      1. associate-*r/ 5.9%

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

      2. *-rgt-identity 5.9%

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

      3. +-commutative 5.9%

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

      4. associate--l+ 57.0%

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

      5. +-inverses 57.0%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 57.0%

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

      7. +-commutative 57.0%

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

      8. fma-def 57.0%

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

      9. +-commutative 57.0%

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

      10. +-commutative 57.0%

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

   5. Simplified 57.0%

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

      1. add-log-exp 4.1%

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

      2. *-un-lft-identity 4.1%

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

      3. log-prod 4.1%

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

      4. metadata-eval 4.1%

         \[\leadsto \frac{1}{\color{blue}{0} + \log \left(e^{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\right)} \]

      5. add-log-exp 57.0%

         \[\leadsto \frac{1}{0 + \color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}} \]

      6. fma-udef 57.0%

         \[\leadsto \frac{1}{0 + \color{blue}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}} \]

      7. pow1/3 54.4%

         \[\leadsto \frac{1}{0 + \left(\color{blue}{{x}^{0.3333333333333333}} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}\right)} \]

      8. *-commutative 54.4%

         \[\leadsto \frac{1}{0 + \left(\color{blue}{\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333}} + \sqrt[3]{{\left(1 + x\right)}^{2}}\right)} \]

      9. pow1/3 53.5%

         \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \color{blue}{{\left({\left(1 + x\right)}^{2}\right)}^{0.3333333333333333}}\right)} \]

      10. unpow2 53.5%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + {\color{blue}{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right)}}^{0.3333333333333333}\right)} \]

      11. pow-prod-down 91.4%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \color{blue}{{\left(1 + x\right)}^{0.3333333333333333} \cdot {\left(1 + x\right)}^{0.3333333333333333}}\right)} \]

      12. +-commutative 91.4%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + {\color{blue}{\left(x + 1\right)}}^{0.3333333333333333} \cdot {\left(1 + x\right)}^{0.3333333333333333}\right)} \]

      13. pow1/3 92.0%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \color{blue}{\sqrt[3]{x + 1}} \cdot {\left(1 + x\right)}^{0.3333333333333333}\right)} \]

      14. +-commutative 92.0%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \sqrt[3]{x + 1} \cdot {\color{blue}{\left(x + 1\right)}}^{0.3333333333333333}\right)} \]

      15. pow1/3 92.9%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \sqrt[3]{x + 1} \cdot \color{blue}{\sqrt[3]{x + 1}}\right)} \]

   7. Applied egg-rr 98.5%

      \[\leadsto \frac{1}{\color{blue}{0 + \mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}} \]
   8. Step-by-step derivation

      1. +-lft-identity 98.5%

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

   9. Simplified 98.5%

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

   11. Applied egg-rr 91.8%

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

   12. Taylor expanded in x around inf 17.7%

       \[\leadsto \color{blue}{e^{-\log \left({\left(\frac{1}{x}\right)}^{-0.6666666666666666}\right)}} \]
   13. Step-by-step derivation

       1. log-pow 17.7%

          \[\leadsto e^{-\color{blue}{-0.6666666666666666 \cdot \log \left(\frac{1}{x}\right)}} \]

       2. distribute-lft-neg-in 17.7%

          \[\leadsto e^{\color{blue}{\left(--0.6666666666666666\right) \cdot \log \left(\frac{1}{x}\right)}} \]

       3. metadata-eval 17.7%

          \[\leadsto e^{\color{blue}{0.6666666666666666} \cdot \log \left(\frac{1}{x}\right)} \]

       4. log-rec 17.7%

          \[\leadsto e^{0.6666666666666666 \cdot \color{blue}{\left(-\log x\right)}} \]

   14. Simplified 17.7%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;\left(1 + \left(\sqrt[3]{1 + x} + -1\right)\right) - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;e^{0.6666666666666666 \cdot \left(-\log x\right)}\\ \end{array} \]

Alternative 8: 56.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;e^{0.6666666666666666 \cdot \left(-\log x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.95e+15)
   (- (cbrt (+ 1.0 x)) (cbrt x))
   (exp (* 0.6666666666666666 (- (log x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.95e+15) {
		tmp = cbrt((1.0 + x)) - cbrt(x);
	} else {
		tmp = exp((0.6666666666666666 * -log(x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.95e+15) {
		tmp = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	} else {
		tmp = Math.exp((0.6666666666666666 * -Math.log(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.95e+15)
		tmp = Float64(cbrt(Float64(1.0 + x)) - cbrt(x));
	else
		tmp = exp(Float64(0.6666666666666666 * Float64(-log(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.95e+15], N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[Exp[N[(0.6666666666666666 * (-N[Log[x], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.95e15

   1. Initial program 69.9%

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

   if 1.95e15 < x

   1. Initial program 4.1%

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

      1. flip3-- 4.1%

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

      2. div-inv 4.1%

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

      3. rem-cube-cbrt 4.8%

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

      4. rem-cube-cbrt 5.9%

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

      5. cbrt-unprod 5.9%

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

      6. pow2 5.9%

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

      7. distribute-rgt-out 5.9%

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

      8. +-commutative 5.9%

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

   3. Applied egg-rr 5.9%

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

      1. associate-*r/ 5.9%

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

      2. *-rgt-identity 5.9%

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

      3. +-commutative 5.9%

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

      4. associate--l+ 57.0%

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

      5. +-inverses 57.0%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 57.0%

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

      7. +-commutative 57.0%

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

      8. fma-def 57.0%

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

      9. +-commutative 57.0%

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

      10. +-commutative 57.0%

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

   5. Simplified 57.0%

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

      1. add-log-exp 4.1%

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

      2. *-un-lft-identity 4.1%

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

      3. log-prod 4.1%

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

      4. metadata-eval 4.1%

         \[\leadsto \frac{1}{\color{blue}{0} + \log \left(e^{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\right)} \]

      5. add-log-exp 57.0%

         \[\leadsto \frac{1}{0 + \color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}} \]

      6. fma-udef 57.0%

         \[\leadsto \frac{1}{0 + \color{blue}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}} \]

      7. pow1/3 54.4%

         \[\leadsto \frac{1}{0 + \left(\color{blue}{{x}^{0.3333333333333333}} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}\right)} \]

      8. *-commutative 54.4%

         \[\leadsto \frac{1}{0 + \left(\color{blue}{\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333}} + \sqrt[3]{{\left(1 + x\right)}^{2}}\right)} \]

      9. pow1/3 53.5%

         \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \color{blue}{{\left({\left(1 + x\right)}^{2}\right)}^{0.3333333333333333}}\right)} \]

      10. unpow2 53.5%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + {\color{blue}{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right)}}^{0.3333333333333333}\right)} \]

      11. pow-prod-down 91.4%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \color{blue}{{\left(1 + x\right)}^{0.3333333333333333} \cdot {\left(1 + x\right)}^{0.3333333333333333}}\right)} \]

      12. +-commutative 91.4%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + {\color{blue}{\left(x + 1\right)}}^{0.3333333333333333} \cdot {\left(1 + x\right)}^{0.3333333333333333}\right)} \]

      13. pow1/3 92.0%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \color{blue}{\sqrt[3]{x + 1}} \cdot {\left(1 + x\right)}^{0.3333333333333333}\right)} \]

      14. +-commutative 92.0%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \sqrt[3]{x + 1} \cdot {\color{blue}{\left(x + 1\right)}}^{0.3333333333333333}\right)} \]

      15. pow1/3 92.9%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \sqrt[3]{x + 1} \cdot \color{blue}{\sqrt[3]{x + 1}}\right)} \]

   7. Applied egg-rr 98.5%

      \[\leadsto \frac{1}{\color{blue}{0 + \mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}} \]
   8. Step-by-step derivation

      1. +-lft-identity 98.5%

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

   9. Simplified 98.5%

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

   11. Applied egg-rr 91.8%

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

   12. Taylor expanded in x around inf 17.7%

       \[\leadsto \color{blue}{e^{-\log \left({\left(\frac{1}{x}\right)}^{-0.6666666666666666}\right)}} \]
   13. Step-by-step derivation

       1. log-pow 17.7%

          \[\leadsto e^{-\color{blue}{-0.6666666666666666 \cdot \log \left(\frac{1}{x}\right)}} \]

       2. distribute-lft-neg-in 17.7%

          \[\leadsto e^{\color{blue}{\left(--0.6666666666666666\right) \cdot \log \left(\frac{1}{x}\right)}} \]

       3. metadata-eval 17.7%

          \[\leadsto e^{\color{blue}{0.6666666666666666} \cdot \log \left(\frac{1}{x}\right)} \]

       4. log-rec 17.7%

          \[\leadsto e^{0.6666666666666666 \cdot \color{blue}{\left(-\log x\right)}} \]

   14. Simplified 17.7%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;e^{0.6666666666666666 \cdot \left(-\log x\right)}\\ \end{array} \]

Alternative 9: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;1 - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;e^{0.6666666666666666 \cdot \left(-\log x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.8) (- 1.0 (cbrt x)) (exp (* 0.6666666666666666 (- (log x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.8) {
		tmp = 1.0 - cbrt(x);
	} else {
		tmp = exp((0.6666666666666666 * -log(x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.8) {
		tmp = 1.0 - Math.cbrt(x);
	} else {
		tmp = Math.exp((0.6666666666666666 * -Math.log(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.8)
		tmp = Float64(1.0 - cbrt(x));
	else
		tmp = exp(Float64(0.6666666666666666 * Float64(-log(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.8], N[(1.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[Exp[N[(0.6666666666666666 * (-N[Log[x], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 69.7%

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

      1. add-cube-cbrt 69.7%

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

      2. pow3 69.7%

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

   3. Applied egg-rr 69.7%

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

   4. Taylor expanded in x around 0 33.2%

      \[\leadsto \color{blue}{1 - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. metadata-eval 33.2%

         \[\leadsto 1 - {\color{blue}{1}}^{0.1111111111111111} \cdot {x}^{0.3333333333333333} \]

      2. pow-base-1 33.2%

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

      3. unpow1/3 66.8%

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

      4. *-lft-identity 66.8%

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

   6. Simplified 66.8%

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

   if 0.80000000000000004 < x

   1. Initial program 8.3%

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

      1. flip3-- 8.2%

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

      2. div-inv 8.2%

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

      3. rem-cube-cbrt 8.9%

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

      4. rem-cube-cbrt 10.8%

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

      5. cbrt-unprod 10.8%

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

      6. pow2 10.8%

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

      7. distribute-rgt-out 10.8%

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

      8. +-commutative 10.8%

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

   3. Applied egg-rr 10.8%

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

      1. associate-*r/ 10.8%

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

      2. *-rgt-identity 10.8%

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

      3. +-commutative 10.8%

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

      4. associate--l+ 59.2%

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

      5. +-inverses 59.2%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 59.2%

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

      7. +-commutative 59.2%

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

      8. fma-def 59.2%

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

      9. +-commutative 59.2%

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

      10. +-commutative 59.2%

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

   5. Simplified 59.2%

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

      1. add-log-exp 7.4%

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

      2. *-un-lft-identity 7.4%

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

      3. log-prod 7.4%

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

      4. metadata-eval 7.4%

         \[\leadsto \frac{1}{\color{blue}{0} + \log \left(e^{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\right)} \]

      5. add-log-exp 59.2%

         \[\leadsto \frac{1}{0 + \color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}} \]

      6. fma-udef 59.2%

         \[\leadsto \frac{1}{0 + \color{blue}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}} \]

      7. pow1/3 56.7%

         \[\leadsto \frac{1}{0 + \left(\color{blue}{{x}^{0.3333333333333333}} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}\right)} \]

      8. *-commutative 56.7%

         \[\leadsto \frac{1}{0 + \left(\color{blue}{\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333}} + \sqrt[3]{{\left(1 + x\right)}^{2}}\right)} \]

      9. pow1/3 55.9%

         \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \color{blue}{{\left({\left(1 + x\right)}^{2}\right)}^{0.3333333333333333}}\right)} \]

      10. unpow2 55.9%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + {\color{blue}{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right)}}^{0.3333333333333333}\right)} \]

      11. pow-prod-down 91.7%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \color{blue}{{\left(1 + x\right)}^{0.3333333333333333} \cdot {\left(1 + x\right)}^{0.3333333333333333}}\right)} \]

      12. +-commutative 91.7%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + {\color{blue}{\left(x + 1\right)}}^{0.3333333333333333} \cdot {\left(1 + x\right)}^{0.3333333333333333}\right)} \]

      13. pow1/3 92.3%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \color{blue}{\sqrt[3]{x + 1}} \cdot {\left(1 + x\right)}^{0.3333333333333333}\right)} \]

      14. +-commutative 92.3%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \sqrt[3]{x + 1} \cdot {\color{blue}{\left(x + 1\right)}}^{0.3333333333333333}\right)} \]

      15. pow1/3 93.2%

          \[\leadsto \frac{1}{0 + \left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot {x}^{0.3333333333333333} + \sqrt[3]{x + 1} \cdot \color{blue}{\sqrt[3]{x + 1}}\right)} \]

   7. Applied egg-rr 98.5%

      \[\leadsto \frac{1}{\color{blue}{0 + \mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}} \]
   8. Step-by-step derivation

      1. +-lft-identity 98.5%

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

   9. Simplified 98.5%

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

   11. Applied egg-rr 92.1%

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

   12. Taylor expanded in x around inf 17.7%

       \[\leadsto \color{blue}{e^{-\log \left({\left(\frac{1}{x}\right)}^{-0.6666666666666666}\right)}} \]
   13. Step-by-step derivation

       1. log-pow 17.7%

          \[\leadsto e^{-\color{blue}{-0.6666666666666666 \cdot \log \left(\frac{1}{x}\right)}} \]

       2. distribute-lft-neg-in 17.7%

          \[\leadsto e^{\color{blue}{\left(--0.6666666666666666\right) \cdot \log \left(\frac{1}{x}\right)}} \]

       3. metadata-eval 17.7%

          \[\leadsto e^{\color{blue}{0.6666666666666666} \cdot \log \left(\frac{1}{x}\right)} \]

       4. log-rec 17.7%

          \[\leadsto e^{0.6666666666666666 \cdot \color{blue}{\left(-\log x\right)}} \]

   14. Simplified 17.7%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;1 - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;e^{0.6666666666666666 \cdot \left(-\log x\right)}\\ \end{array} \]

Alternative 10: 50.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. add-cube-cbrt 55.9%

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

   2. pow3 55.9%

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

3. Applied egg-rr 55.9%

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

4. Taylor expanded in x around 0 26.8%

   \[\leadsto \color{blue}{\left(1 + 0.3333333333333333 \cdot x\right) - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. associate--l+ 26.8%

      \[\leadsto \color{blue}{1 + \left(0.3333333333333333 \cdot x - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}\right)} \]

   2. *-commutative 26.8%

      \[\leadsto 1 + \left(\color{blue}{x \cdot 0.3333333333333333} - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}\right) \]

   3. metadata-eval 26.8%

      \[\leadsto 1 + \left(x \cdot 0.3333333333333333 - {\color{blue}{1}}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}\right) \]

   4. pow-base-1 26.8%

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

   5. unpow1/3 52.4%

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

   6. *-lft-identity 52.4%

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

6. Simplified 52.4%

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

7. Final simplification 52.4%

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

Alternative 11: 50.2% 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 56.0%

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

   1. add-cube-cbrt 55.9%

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

   2. pow3 55.9%

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

3. Applied egg-rr 55.9%

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

4. Taylor expanded in x around 0 26.2%

   \[\leadsto \color{blue}{1 - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. metadata-eval 26.2%

      \[\leadsto 1 - {\color{blue}{1}}^{0.1111111111111111} \cdot {x}^{0.3333333333333333} \]

   2. pow-base-1 26.2%

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

   3. unpow1/3 52.3%

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

   4. *-lft-identity 52.3%

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

6. Simplified 52.3%

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

7. Final simplification 52.3%

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

Alternative 12: 3.6% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 56.0%

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

2. Taylor expanded in x around inf 3.6%

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

3. Final simplification 3.6%

   \[\leadsto 0 \]

Alternative 13: 49.3% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 56.0%

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

2. Taylor expanded in x around 0 51.7%

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

3. Final simplification 51.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023262 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))