2cbrt (problem 3.3.4)

Percentage Accurate: 53.3% → 99.2%

Time: 17.8s

Alternatives: 11

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

   1. add-cube-cbrt 52.7%

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

   2. pow3 52.7%

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

3. Applied egg-rr 52.7%

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

   1. rem-cube-cbrt 52.8%

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

   2. sub-neg 52.8%

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

   3. flip3-+ 52.8%

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

   4. pow3 52.7%

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

   5. add-cube-cbrt 52.7%

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

   6. +-commutative 52.7%

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

   7. pow2 52.7%

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

   8. +-commutative 52.7%

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

   9. +-commutative 52.7%

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

5. Applied egg-rr 52.7%

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

   1. cube-neg 52.7%

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

   2. rem-cube-cbrt 53.2%

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

   3. associate-+r+ 99.2%

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

   4. sub-neg 99.2%

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

   5. +-inverses 99.2%

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

   6. metadata-eval 99.2%

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

   7. *-commutative 99.2%

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

   8. cancel-sign-sub 99.2%

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

   9. sqr-neg 99.2%

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

   10. distribute-lft-in 99.2%

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

   11. +-commutative 99.2%

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

7. Simplified 99.2%

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

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 2: 88.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + t_0}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_0 + \sqrt[3]{{\left(1 + x\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (cbrt x) (+ (cbrt (+ 1.0 x)) (cbrt x)))))
   (if (<= x -1.35e+154)
     (/ 1.0 (+ 1.0 t_0))
     (if (<= x 1.32e+154)
       (/ 1.0 (+ t_0 (cbrt (pow (+ 1.0 x) 2.0))))
       (/ 1.0 (+ t_0 (exp (* 0.6666666666666666 (log1p x)))))))))

C

double code(double x) {
	double t_0 = cbrt(x) * (cbrt((1.0 + x)) + cbrt(x));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / (1.0 + t_0);
	} else if (x <= 1.32e+154) {
		tmp = 1.0 / (t_0 + cbrt(pow((1.0 + x), 2.0)));
	} else {
		tmp = 1.0 / (t_0 + exp((0.6666666666666666 * log1p(x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(x) * (Math.cbrt((1.0 + x)) + Math.cbrt(x));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / (1.0 + t_0);
	} else if (x <= 1.32e+154) {
		tmp = 1.0 / (t_0 + Math.cbrt(Math.pow((1.0 + x), 2.0)));
	} else {
		tmp = 1.0 / (t_0 + Math.exp((0.6666666666666666 * Math.log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(x) * Float64(cbrt(Float64(1.0 + x)) + cbrt(x)))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(1.0 / Float64(1.0 + t_0));
	elseif (x <= 1.32e+154)
		tmp = Float64(1.0 / Float64(t_0 + cbrt((Float64(1.0 + x) ^ 2.0))));
	else
		tmp = Float64(1.0 / Float64(t_0 + exp(Float64(0.6666666666666666 * log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e+154], N[(1.0 / N[(t$95$0 + N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 + 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 4.7%

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

      1. add-cube-cbrt 3.6%

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

      2. pow3 3.6%

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

   3. Applied egg-rr 3.6%

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

      1. rem-cube-cbrt 4.7%

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

      2. sub-neg 4.7%

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

      3. flip3-+ 4.7%

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

      4. pow3 4.4%

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

      5. add-cube-cbrt 3.6%

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

      6. +-commutative 3.6%

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

      7. pow2 3.6%

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

      8. +-commutative 3.6%

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

      9. +-commutative 3.6%

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

   5. Applied egg-rr 3.6%

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

      1. cube-neg 3.6%

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

      2. rem-cube-cbrt 4.7%

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

      3. associate-+r+ 98.3%

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

      4. sub-neg 98.3%

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

      5. +-inverses 98.3%

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

      6. metadata-eval 98.3%

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

      7. *-commutative 98.3%

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

      8. cancel-sign-sub 98.3%

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

      9. sqr-neg 98.3%

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

      10. distribute-lft-in 98.3%

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

      11. +-commutative 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x around 0 20.0%

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

   if -1.35000000000000003e154 < x < 1.31999999999999998e154

   1. Initial program 69.2%

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

      1. add-cube-cbrt 69.5%

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

      2. pow3 69.5%

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

   3. Applied egg-rr 69.5%

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

      1. rem-cube-cbrt 69.2%

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

      2. sub-neg 69.2%

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

      3. flip3-+ 69.2%

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

      4. pow3 69.2%

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

      5. add-cube-cbrt 69.5%

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

      6. +-commutative 69.5%

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

      7. pow2 69.5%

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

      8. +-commutative 69.5%

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

      9. +-commutative 69.5%

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

   5. Applied egg-rr 69.5%

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

      1. cube-neg 69.5%

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

      2. rem-cube-cbrt 69.7%

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

      3. associate-+r+ 99.4%

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

      4. sub-neg 99.4%

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

      5. +-inverses 99.4%

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

      6. metadata-eval 99.4%

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

      7. *-commutative 99.4%

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

      8. cancel-sign-sub 99.4%

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

      9. sqr-neg 99.4%

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

      10. distribute-lft-in 99.4%

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

      11. +-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. unpow2 99.4%

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

      2. cbrt-unprod 99.6%

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

      3. pow2 99.6%

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

      4. +-commutative 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 1.31999999999999998e154 < x

   1. Initial program 4.7%

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

      1. add-cube-cbrt 2.7%

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

      2. pow3 2.9%

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

   3. Applied egg-rr 2.9%

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

      1. rem-cube-cbrt 4.7%

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

      2. sub-neg 4.7%

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

      3. flip3-+ 4.7%

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

      4. pow3 4.1%

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

      5. add-cube-cbrt 2.9%

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

      6. +-commutative 2.9%

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

      7. pow2 2.9%

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

      8. +-commutative 2.9%

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

      9. +-commutative 2.9%

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

   5. Applied egg-rr 2.9%

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

      1. cube-neg 2.9%

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

      2. rem-cube-cbrt 4.7%

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

      3. associate-+r+ 98.4%

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

      4. sub-neg 98.4%

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

      5. +-inverses 98.4%

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

      6. metadata-eval 98.4%

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

      7. *-commutative 98.4%

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

      8. cancel-sign-sub 98.4%

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

      9. sqr-neg 98.4%

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

      10. distribute-lft-in 98.4%

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

      11. +-commutative 98.4%

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

   7. Simplified 98.4%

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

      1. pow1/3 91.6%

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

      2. pow-pow 91.6%

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

      3. metadata-eval 91.6%

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

      4. add-exp-log 91.8%

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

      5. log-pow 91.6%

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

      6. log1p-udef 91.6%

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

   9. Applied egg-rr 91.6%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \]

Alternative 3: 67.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0 + {x}^{0.3333333333333333}, \sqrt[3]{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x)))
        (t_1 (/ 1.0 (+ 1.0 (* (cbrt x) (+ t_0 (cbrt x)))))))
   (if (<= x 1.0)
     t_1
     (if (<= x 1.32e+154)
       (/ 1.0 (fma (cbrt x) (+ t_0 (pow x 0.3333333333333333)) (cbrt (* x x))))
       t_1))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = 1.0 / (1.0 + (cbrt(x) * (t_0 + cbrt(x))));
	double tmp;
	if (x <= 1.0) {
		tmp = t_1;
	} else if (x <= 1.32e+154) {
		tmp = 1.0 / fma(cbrt(x), (t_0 + pow(x, 0.3333333333333333)), cbrt((x * x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))))
	tmp = 0.0
	if (x <= 1.0)
		tmp = t_1;
	elseif (x <= 1.32e+154)
		tmp = Float64(1.0 / fma(cbrt(x), Float64(t_0 + (x ^ 0.3333333333333333)), cbrt(Float64(x * x))));
	else
		tmp = t_1;
	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[(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]}, If[LessEqual[x, 1.0], t$95$1, If[LessEqual[x, 1.32e+154], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 0.3333333333333333], $MachinePrecision]), $MachinePrecision] + N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1 or 1.31999999999999998e154 < x

   1. Initial program 59.4%

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

      1. add-cube-cbrt 59.2%

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

      2. pow3 59.2%

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

   3. Applied egg-rr 59.2%

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

      1. rem-cube-cbrt 59.4%

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

      2. sub-neg 59.4%

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

      3. flip3-+ 59.3%

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

      4. pow3 59.2%

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

      5. add-cube-cbrt 59.2%

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

      6. +-commutative 59.2%

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

      7. pow2 59.2%

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

      8. +-commutative 59.2%

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

      9. +-commutative 59.2%

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

   5. Applied egg-rr 59.2%

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

      1. cube-neg 59.2%

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

      2. rem-cube-cbrt 59.4%

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

      3. associate-+r+ 99.2%

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

      4. sub-neg 99.2%

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

      5. +-inverses 99.2%

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

      6. metadata-eval 99.2%

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

      7. *-commutative 99.2%

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

      8. cancel-sign-sub 99.2%

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

      9. sqr-neg 99.2%

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

      10. distribute-lft-in 99.2%

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

      11. +-commutative 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around 0 64.7%

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

   if 1 < x < 1.31999999999999998e154

   1. Initial program 7.0%

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

      1. flip3-- 7.0%

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

      2. div-inv 7.0%

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

      3. rem-cube-cbrt 7.3%

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

      4. rem-cube-cbrt 9.5%

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

      5. cbrt-unprod 9.6%

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

      6. pow2 9.6%

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

      7. distribute-rgt-out 9.6%

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

      8. +-commutative 9.6%

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

   3. Applied egg-rr 9.6%

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

      1. associate-*r/ 9.6%

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

      2. *-rgt-identity 9.6%

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

      3. +-commutative 9.6%

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

      4. associate--l+ 98.9%

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

      5. +-inverses 98.9%

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

      6. metadata-eval 98.9%

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

      7. +-commutative 98.9%

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

      8. fma-def 98.9%

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

      9. +-commutative 98.9%

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

      10. +-commutative 98.9%

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

   5. Simplified 98.9%

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

      1. pow1/3 95.8%

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

   7. Applied egg-rr 95.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \color{blue}{{x}^{0.3333333333333333}}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)} \]
   8. Step-by-step derivation

      1. pow1/3 93.4%

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

      2. pow-pow 93.4%

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

      3. metadata-eval 93.4%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + {x}^{0.3333333333333333}, {\left(1 + x\right)}^{\color{blue}{0.6666666666666666}}\right)} \]

   9. Applied egg-rr 93.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + {x}^{0.3333333333333333}, \color{blue}{{\left(1 + x\right)}^{0.6666666666666666}}\right)} \]

   10. Taylor expanded in x around inf 91.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + {x}^{0.3333333333333333}, \color{blue}{{\left({x}^{2}\right)}^{0.3333333333333333}}\right)} \]
   11. Step-by-step derivation

       1. unpow1/3 93.3%

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

       2. unpow2 93.3%

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

   12. Simplified 93.3%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + {x}^{0.3333333333333333}, \sqrt[3]{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\\ \end{array} \]

Alternative 4: 77.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (cbrt x) (+ (cbrt (+ 1.0 x)) (cbrt x)))))
   (if (<= x -1.0)
     (/ 1.0 (+ 1.0 t_0))
     (/ 1.0 (+ t_0 (exp (* 0.6666666666666666 (log1p x))))))))

C

double code(double x) {
	double t_0 = cbrt(x) * (cbrt((1.0 + x)) + cbrt(x));
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (1.0 + t_0);
	} else {
		tmp = 1.0 / (t_0 + exp((0.6666666666666666 * log1p(x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(x) * (Math.cbrt((1.0 + x)) + Math.cbrt(x));
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (1.0 + t_0);
	} else {
		tmp = 1.0 / (t_0 + Math.exp((0.6666666666666666 * Math.log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(x) * Float64(cbrt(Float64(1.0 + x)) + cbrt(x)))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 / Float64(1.0 + t_0));
	else
		tmp = Float64(1.0 / Float64(t_0 + exp(Float64(0.6666666666666666 * log1p(x)))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 4.1%

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

      1. add-cube-cbrt 4.4%

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

      2. pow3 4.4%

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

   3. Applied egg-rr 4.4%

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

      1. rem-cube-cbrt 4.1%

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

      2. sub-neg 4.1%

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

      3. flip3-+ 4.1%

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

      4. pow3 4.0%

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

      5. add-cube-cbrt 4.4%

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

      6. +-commutative 4.4%

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

      7. pow2 4.4%

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

      8. +-commutative 4.4%

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

      9. +-commutative 4.4%

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

   5. Applied egg-rr 4.4%

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

      1. cube-neg 4.4%

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

      2. rem-cube-cbrt 4.4%

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

      3. associate-+r+ 98.4%

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

      4. sub-neg 98.4%

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

      5. +-inverses 98.4%

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

      6. metadata-eval 98.4%

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

      7. *-commutative 98.4%

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

      8. cancel-sign-sub 98.4%

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

      9. sqr-neg 98.4%

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

      10. distribute-lft-in 98.4%

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

      11. +-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around 0 20.0%

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

   if -1 < x

   1. Initial program 69.1%

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

      1. add-cube-cbrt 68.8%

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

      2. pow3 68.8%

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

   3. Applied egg-rr 68.8%

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

      1. rem-cube-cbrt 69.1%

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

      2. sub-neg 69.1%

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

      3. flip3-+ 69.0%

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

      4. pow3 69.0%

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

      5. add-cube-cbrt 68.8%

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

      6. +-commutative 68.8%

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

      7. pow2 68.8%

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

      8. +-commutative 68.8%

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

      9. +-commutative 68.8%

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

   5. Applied egg-rr 68.8%

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

      1. cube-neg 68.8%

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

      2. rem-cube-cbrt 69.4%

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

      3. associate-+r+ 99.4%

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

      4. sub-neg 99.4%

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

      5. +-inverses 99.4%

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

      6. metadata-eval 99.4%

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

      7. *-commutative 99.4%

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

      8. cancel-sign-sub 99.4%

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

      9. sqr-neg 99.4%

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

      10. distribute-lft-in 99.4%

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

      11. +-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. pow1/3 97.6%

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

      2. pow-pow 97.6%

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

      3. metadata-eval 97.6%

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

      4. add-exp-log 97.7%

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

      5. log-pow 97.6%

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

      6. log1p-udef 97.6%

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

   9. Applied egg-rr 97.6%

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

4. Final simplification 78.2%

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

Alternative 5: 56.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{if}\;t_0 \leq 10^{-10}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (cbrt (+ 1.0 x)) (cbrt x))))
   (if (<= t_0 1e-10) (cbrt (/ 1.0 (* x x))) t_0)))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x)) - cbrt(x);
	double tmp;
	if (t_0 <= 1e-10) {
		tmp = cbrt((1.0 / (x * x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	double tmp;
	if (t_0 <= 1e-10) {
		tmp = Math.cbrt((1.0 / (x * x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(Float64(1.0 + x)) - cbrt(x))
	tmp = 0.0
	if (t_0 <= 1e-10)
		tmp = cbrt(Float64(1.0 / Float64(x * x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-10], N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 1.00000000000000004e-10

   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 4.0%

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

      4. rem-cube-cbrt 5.0%

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

      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+ 50.3%

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

      5. +-inverses 50.3%

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

      6. metadata-eval 50.3%

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

      7. +-commutative 50.3%

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

      8. fma-def 50.3%

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

      9. +-commutative 50.3%

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

      10. +-commutative 50.3%

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

   5. Simplified 50.3%

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

      1. add-sqr-sqrt 50.2%

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

      2. sqrt-unprod 50.3%

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

      3. inv-pow 50.3%

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

      4. inv-pow 50.3%

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

      5. pow-prod-up 50.3%

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

   7. Applied egg-rr 36.0%

      \[\leadsto \color{blue}{\sqrt{{\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{-2}}} \]

   8. Taylor expanded in x around inf 11.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. unpow1/3 11.0%

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

      2. unpow2 11.0%

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

   10. Simplified 11.0%

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

   if 1.00000000000000004e-10 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.9%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

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

Alternative 6: 59.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

   1. add-cube-cbrt 52.7%

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

   2. pow3 52.7%

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

3. Applied egg-rr 52.7%

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

   1. rem-cube-cbrt 52.8%

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

   2. sub-neg 52.8%

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

   3. flip3-+ 52.8%

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

   4. pow3 52.7%

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

   5. add-cube-cbrt 52.7%

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

   6. +-commutative 52.7%

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

   7. pow2 52.7%

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

   8. +-commutative 52.7%

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

   9. +-commutative 52.7%

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

5. Applied egg-rr 52.7%

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

   1. cube-neg 52.7%

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

   2. rem-cube-cbrt 53.2%

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

   3. associate-+r+ 99.2%

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

   4. sub-neg 99.2%

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

   5. +-inverses 99.2%

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

   6. metadata-eval 99.2%

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

   7. *-commutative 99.2%

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

   8. cancel-sign-sub 99.2%

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

   9. sqr-neg 99.2%

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

   10. distribute-lft-in 99.2%

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

   11. +-commutative 99.2%

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

7. Simplified 99.2%

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

8. Taylor expanded in x around 0 59.1%

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

9. Final simplification 59.1%

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

Alternative 7: 55.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.95 \lor \neg \left(x \leq 2.7\right):\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot 0.3333333333333333 - \sqrt[3]{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.95) (not (<= x 2.7)))
   (cbrt (/ 1.0 (* x x)))
   (+ 1.0 (- (* x 0.3333333333333333) (cbrt x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.95) || !(x <= 2.7)) {
		tmp = cbrt((1.0 / (x * x)));
	} else {
		tmp = 1.0 + ((x * 0.3333333333333333) - cbrt(x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.95) || !(x <= 2.7)) {
		tmp = Math.cbrt((1.0 / (x * x)));
	} else {
		tmp = 1.0 + ((x * 0.3333333333333333) - Math.cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.95) || !(x <= 2.7))
		tmp = cbrt(Float64(1.0 / Float64(x * x)));
	else
		tmp = Float64(1.0 + Float64(Float64(x * 0.3333333333333333) - cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.95], N[Not[LessEqual[x, 2.7]], $MachinePrecision]], N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(1.0 + N[(N[(x * 0.3333333333333333), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.94999999999999996 or 2.7000000000000002 < x

   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 4.0%

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

      4. rem-cube-cbrt 5.0%

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

      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+ 50.3%

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

      5. +-inverses 50.3%

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

      6. metadata-eval 50.3%

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

      7. +-commutative 50.3%

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

      8. fma-def 50.3%

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

      9. +-commutative 50.3%

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

      10. +-commutative 50.3%

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

   5. Simplified 50.3%

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

      1. add-sqr-sqrt 50.2%

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

      2. sqrt-unprod 50.3%

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

      3. inv-pow 50.3%

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

      4. inv-pow 50.3%

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

      5. pow-prod-up 50.3%

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

   7. Applied egg-rr 36.0%

      \[\leadsto \color{blue}{\sqrt{{\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{-2}}} \]

   8. Taylor expanded in x around inf 11.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. unpow1/3 11.0%

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

      2. unpow2 11.0%

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

   10. Simplified 11.0%

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

   if -0.94999999999999996 < x < 2.7000000000000002

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 45.9%

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

      1. associate--l+ 45.9%

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

      2. *-commutative 45.9%

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

      3. metadata-eval 45.9%

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

      4. pow-base-1 45.9%

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

      5. unpow1/3 98.7%

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

      6. *-lft-identity 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.95 \lor \neg \left(x \leq 2.7\right):\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot 0.3333333333333333 - \sqrt[3]{x}\right)\\ \end{array} \]

Alternative 8: 53.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;1 - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.8) (- 1.0 (cbrt x)) (cbrt (/ 1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.8) {
		tmp = 1.0 - cbrt(x);
	} else {
		tmp = cbrt((1.0 / (x * 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.cbrt((1.0 / (x * x)));
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 68.2%

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

      1. add-cube-cbrt 68.3%

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

      2. pow3 68.2%

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

   3. Applied egg-rr 68.2%

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

   4. Taylor expanded in x around 0 30.4%

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

      1. metadata-eval 30.4%

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

      2. pow-base-1 30.4%

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

      3. unpow1/3 67.1%

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

      4. *-lft-identity 67.1%

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

   6. Simplified 67.1%

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

   if 0.80000000000000004 < x

   1. Initial program 5.9%

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

      1. flip3-- 5.9%

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

      2. div-inv 5.9%

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

      3. rem-cube-cbrt 5.2%

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

      4. rem-cube-cbrt 7.2%

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

      5. cbrt-unprod 7.2%

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

      6. pow2 7.2%

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

      7. distribute-rgt-out 7.2%

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

      8. +-commutative 7.2%

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

   3. Applied egg-rr 7.2%

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

      1. associate-*r/ 7.2%

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

      2. *-rgt-identity 7.2%

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

      3. +-commutative 7.2%

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

      4. associate--l+ 52.6%

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

      5. +-inverses 52.6%

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

      6. metadata-eval 52.6%

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

      7. +-commutative 52.6%

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

      8. fma-def 52.6%

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

      9. +-commutative 52.6%

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

      10. +-commutative 52.6%

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

   5. Simplified 52.6%

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

      1. add-sqr-sqrt 52.3%

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

      2. sqrt-unprod 52.6%

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

      3. inv-pow 52.6%

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

      4. inv-pow 52.6%

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

      5. pow-prod-up 52.6%

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

   7. Applied egg-rr 73.5%

      \[\leadsto \color{blue}{\sqrt{{\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{-2}}} \]

   8. Taylor expanded in x around inf 11.3%

      \[\leadsto \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. unpow1/3 11.3%

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

      2. unpow2 11.3%

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

   10. Simplified 11.3%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;1 - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}}\\ \end{array} \]

Alternative 9: 50.9% 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 52.8%

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

   1. add-cube-cbrt 52.7%

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

   2. pow3 52.7%

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

3. Applied egg-rr 52.7%

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

4. Taylor expanded in x around 0 23.4%

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

   1. metadata-eval 23.4%

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

   2. pow-base-1 23.4%

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

   3. unpow1/3 51.1%

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

   4. *-lft-identity 51.1%

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

6. Simplified 51.1%

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

7. Final simplification 51.1%

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

Alternative 10: 3.7% 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 52.8%

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

2. Taylor expanded in x around inf 3.6%

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

3. Final simplification 3.6%

   \[\leadsto 0 \]

Alternative 11: 50.1% 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 52.8%

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

2. Taylor expanded in x around 0 50.0%

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

3. Final simplification 50.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))