2cbrt (problem 3.3.4)

Percentage Accurate: 54.1% → 99.2%

Time: 14.8s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.5%

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

   1. pow1/3 25.6%

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

3. Applied egg-rr 25.6%

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

   1. pow1/3 52.5%

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

   2. flip3-- 52.5%

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

   3. div-inv 52.5%

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

   4. add-sqr-sqrt 25.3%

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

   5. cbrt-unprod 25.1%

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

   6. rem-cube-cbrt 25.0%

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

   7. cbrt-unprod 25.0%

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

   8. add-sqr-sqrt 52.1%

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

   9. rem-cube-cbrt 52.8%

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

   10. pow2 52.8%

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

   11. distribute-rgt-out 52.8%

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

5. Applied egg-rr 52.8%

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

   1. associate-*r/ 52.8%

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

   2. *-rgt-identity 52.8%

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

   3. +-commutative 52.8%

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

   4. associate--l+ 99.3%

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

   5. +-commutative 99.3%

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

   6. fma-def 99.3%

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

   7. +-commutative 99.3%

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

   8. +-commutative 99.3%

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t_0, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (* 0.3333333333333333 (pow (cbrt x) -2.0))
     (/
      (+ 1.0 (- x x))
      (fma (cbrt x) (+ (cbrt x) t_0) (cbrt (pow (+ 1.0 x) 2.0)))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 0.3333333333333333 * pow(cbrt(x), -2.0);
	} else {
		tmp = (1.0 + (x - x)) / fma(cbrt(x), (cbrt(x) + t_0), cbrt(pow((1.0 + x), 2.0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(0.3333333333333333 * (cbrt(x) ^ -2.0));
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / fma(cbrt(x), Float64(cbrt(x) + t_0), cbrt((Float64(1.0 + x) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] + N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

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

      1. add-cube-cbrt 3.6%

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

      2. pow3 3.6%

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

   3. Applied egg-rr 3.6%

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

   4. Taylor expanded in x around inf 1.0%

      \[\leadsto \color{blue}{\left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \left(0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} + {x}^{0.3333333333333333}\right)\right)} - \sqrt[3]{x} \]
   5. Step-by-step derivation

      1. unpow1/3 4.2%

         \[\leadsto \left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \left(0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} + \color{blue}{\sqrt[3]{x}}\right)\right) - \sqrt[3]{x} \]

      2. associate-+r+ 4.2%

         \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \sqrt[3]{x}\right)} - \sqrt[3]{x} \]

      3. +-commutative 4.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right)} - \sqrt[3]{x} \]

      4. metadata-eval 4.2%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left({\color{blue}{1}}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      5. pow-base-1 4.2%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left(\color{blue}{1} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      6. *-lft-identity 4.2%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      7. distribute-rgt-out 4.2%

         \[\leadsto \left(\sqrt[3]{x} + \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} \cdot \left(0.1111111111111111 + 0.2222222222222222\right)}\right) - \sqrt[3]{x} \]

      8. unpow1/3 4.2%

         \[\leadsto \left(\sqrt[3]{x} + \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}}} \cdot \left(0.1111111111111111 + 0.2222222222222222\right)\right) - \sqrt[3]{x} \]

      9. metadata-eval 4.2%

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

   6. Simplified 4.2%

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

      1. associate--l+ 4.2%

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

      2. flip-+ 4.2%

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

   8. Applied egg-rr 4.2%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} - \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right) \cdot \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right)}{\sqrt[3]{x} - \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right)}} \]
   9. Step-by-step derivation

      1. unpow2 4.2%

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

      2. difference-of-squares 4.2%

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

      3. +-commutative 4.2%

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

      4. associate-+l- 98.5%

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

      5. +-inverses 98.5%

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

      6. --rgt-identity 98.5%

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

      7. associate--r- 98.5%

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

      8. +-commutative 98.5%

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

      9. associate--r- 98.5%

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

      10. +-commutative 98.5%

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

   10. Simplified 98.5%

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

       1. expm1-log1p-u 98.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)\right)}{\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)}\right)\right)} \]

       2. expm1-udef 4.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)\right)}{\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)}\right)} - 1} \]

   12. Applied egg-rr 4.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}}{1}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 98.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}}{1}\right)\right)} \]

       2. expm1-log1p 98.5%

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

       3. /-rgt-identity 98.5%

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

   14. Simplified 98.5%

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

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

   1. Initial program 99.3%

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

      1. pow1/3 47.7%

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

   3. Applied egg-rr 47.7%

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

      1. pow1/3 99.3%

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

      2. flip3-- 99.3%

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

      3. div-inv 99.3%

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

      4. add-sqr-sqrt 47.6%

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

      5. cbrt-unprod 47.6%

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

      6. rem-cube-cbrt 47.6%

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

      7. cbrt-unprod 47.6%

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

      8. add-sqr-sqrt 99.2%

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

      9. rem-cube-cbrt 99.8%

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

      10. pow2 99.8%

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

      11. distribute-rgt-out 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. +-commutative 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. cbrt-unprod 99.9%

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

      3. pow2 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\\ \end{array} \]

Alternative 3: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (* 0.3333333333333333 (pow (cbrt x) -2.0))
     (/ (- (+ 1.0 x) x) (+ (pow t_0 2.0) (* (cbrt x) (+ (cbrt x) t_0)))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 0.3333333333333333 * pow(cbrt(x), -2.0);
	} else {
		tmp = ((1.0 + x) - x) / (pow(t_0, 2.0) + (cbrt(x) * (cbrt(x) + t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double tmp;
	if ((t_0 - Math.cbrt(x)) <= 0.0) {
		tmp = 0.3333333333333333 * Math.pow(Math.cbrt(x), -2.0);
	} else {
		tmp = ((1.0 + x) - x) / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (Math.cbrt(x) + t_0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(0.3333333333333333 * (cbrt(x) ^ -2.0));
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64((t_0 ^ 2.0) + Float64(cbrt(x) * Float64(cbrt(x) + t_0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

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

      1. add-cube-cbrt 3.6%

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

      2. pow3 3.6%

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

   3. Applied egg-rr 3.6%

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

   4. Taylor expanded in x around inf 1.0%

      \[\leadsto \color{blue}{\left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \left(0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} + {x}^{0.3333333333333333}\right)\right)} - \sqrt[3]{x} \]
   5. Step-by-step derivation

      1. unpow1/3 4.2%

         \[\leadsto \left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \left(0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} + \color{blue}{\sqrt[3]{x}}\right)\right) - \sqrt[3]{x} \]

      2. associate-+r+ 4.2%

         \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \sqrt[3]{x}\right)} - \sqrt[3]{x} \]

      3. +-commutative 4.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right)} - \sqrt[3]{x} \]

      4. metadata-eval 4.2%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left({\color{blue}{1}}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      5. pow-base-1 4.2%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left(\color{blue}{1} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      6. *-lft-identity 4.2%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      7. distribute-rgt-out 4.2%

         \[\leadsto \left(\sqrt[3]{x} + \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} \cdot \left(0.1111111111111111 + 0.2222222222222222\right)}\right) - \sqrt[3]{x} \]

      8. unpow1/3 4.2%

         \[\leadsto \left(\sqrt[3]{x} + \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}}} \cdot \left(0.1111111111111111 + 0.2222222222222222\right)\right) - \sqrt[3]{x} \]

      9. metadata-eval 4.2%

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

   6. Simplified 4.2%

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

      1. associate--l+ 4.2%

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

      2. flip-+ 4.2%

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

   8. Applied egg-rr 4.2%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} - \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right) \cdot \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right)}{\sqrt[3]{x} - \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right)}} \]
   9. Step-by-step derivation

      1. unpow2 4.2%

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

      2. difference-of-squares 4.2%

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

      3. +-commutative 4.2%

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

      4. associate-+l- 98.5%

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

      5. +-inverses 98.5%

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

      6. --rgt-identity 98.5%

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

      7. associate--r- 98.5%

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

      8. +-commutative 98.5%

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

      9. associate--r- 98.5%

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

      10. +-commutative 98.5%

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

   10. Simplified 98.5%

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

       1. expm1-log1p-u 98.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)\right)}{\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)}\right)\right)} \]

       2. expm1-udef 4.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)\right)}{\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)}\right)} - 1} \]

   12. Applied egg-rr 4.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}}{1}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 98.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}}{1}\right)\right)} \]

       2. expm1-log1p 98.5%

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

       3. /-rgt-identity 98.5%

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

   14. Simplified 98.5%

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

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

   1. Initial program 99.3%

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

      1. pow1/3 47.7%

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

   3. Applied egg-rr 47.7%

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

      1. pow1/3 99.3%

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

      2. add-sqr-sqrt 47.7%

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

      3. cbrt-unprod 47.7%

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

      4. flip3-- 47.6%

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

      5. rem-cube-cbrt 47.6%

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

      6. cbrt-unprod 47.6%

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

      7. add-sqr-sqrt 47.6%

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

      8. rem-cube-cbrt 47.6%

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

      9. pow2 47.6%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \end{array} \]

Alternative 4: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;e^{\log t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (cbrt (+ 1.0 x)) (cbrt x))))
   (if (<= t_0 2e-6)
     (* 0.3333333333333333 (pow (cbrt x) -2.0))
     (exp (log t_0)))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x)) - cbrt(x);
	double tmp;
	if (t_0 <= 2e-6) {
		tmp = 0.3333333333333333 * pow(cbrt(x), -2.0);
	} else {
		tmp = exp(log(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 <= 2e-6) {
		tmp = 0.3333333333333333 * Math.pow(Math.cbrt(x), -2.0);
	} else {
		tmp = Math.exp(Math.log(t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(Float64(1.0 + x)) - cbrt(x))
	tmp = 0.0
	if (t_0 <= 2e-6)
		tmp = Float64(0.3333333333333333 * (cbrt(x) ^ -2.0));
	else
		tmp = exp(log(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, 2e-6], N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[t$95$0], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 1.99999999999999991e-6

   1. Initial program 4.7%

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

      1. add-cube-cbrt 4.0%

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

      2. pow3 4.0%

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

   3. Applied egg-rr 4.0%

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

   4. Taylor expanded in x around inf 1.0%

      \[\leadsto \color{blue}{\left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \left(0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} + {x}^{0.3333333333333333}\right)\right)} - \sqrt[3]{x} \]
   5. Step-by-step derivation

      1. unpow1/3 4.7%

         \[\leadsto \left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \left(0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} + \color{blue}{\sqrt[3]{x}}\right)\right) - \sqrt[3]{x} \]

      2. associate-+r+ 4.7%

         \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \sqrt[3]{x}\right)} - \sqrt[3]{x} \]

      3. +-commutative 4.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right)} - \sqrt[3]{x} \]

      4. metadata-eval 4.7%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left({\color{blue}{1}}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      5. pow-base-1 4.7%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left(\color{blue}{1} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      6. *-lft-identity 4.7%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      7. distribute-rgt-out 4.7%

         \[\leadsto \left(\sqrt[3]{x} + \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} \cdot \left(0.1111111111111111 + 0.2222222222222222\right)}\right) - \sqrt[3]{x} \]

      8. unpow1/3 4.7%

         \[\leadsto \left(\sqrt[3]{x} + \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}}} \cdot \left(0.1111111111111111 + 0.2222222222222222\right)\right) - \sqrt[3]{x} \]

      9. metadata-eval 4.7%

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

   6. Simplified 4.7%

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

      1. associate--l+ 4.7%

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

      2. flip-+ 4.7%

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

   8. Applied egg-rr 4.7%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} - \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right) \cdot \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right)}{\sqrt[3]{x} - \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right)}} \]
   9. Step-by-step derivation

      1. unpow2 4.7%

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

      2. difference-of-squares 4.7%

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

      3. +-commutative 4.7%

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

      4. associate-+l- 98.2%

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

      5. +-inverses 98.2%

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

      6. --rgt-identity 98.2%

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

      7. associate--r- 98.2%

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

      8. +-commutative 98.2%

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

      9. associate--r- 98.2%

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

      10. +-commutative 98.2%

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

   10. Simplified 98.2%

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

       1. expm1-log1p-u 98.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)\right)}{\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)}\right)\right)} \]

       2. expm1-udef 5.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)\right)}{\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)}\right)} - 1} \]

   12. Applied egg-rr 5.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}}{1}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 98.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}}{1}\right)\right)} \]

       2. expm1-log1p 98.2%

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

       3. /-rgt-identity 98.2%

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

   14. Simplified 98.2%

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

   if 1.99999999999999991e-6 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.7%

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

      1. add-exp-log 99.7%

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

   3. Applied egg-rr 99.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)}\\ \end{array} \]

Alternative 5: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (cbrt (+ 1.0 x)) (cbrt x))))
   (if (<= t_0 2e-6) (* 0.3333333333333333 (pow (cbrt x) -2.0)) t_0)))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x)) - cbrt(x);
	double tmp;
	if (t_0 <= 2e-6) {
		tmp = 0.3333333333333333 * pow(cbrt(x), -2.0);
	} 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 <= 2e-6) {
		tmp = 0.3333333333333333 * Math.pow(Math.cbrt(x), -2.0);
	} 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 <= 2e-6)
		tmp = Float64(0.3333333333333333 * (cbrt(x) ^ -2.0));
	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, 2e-6], N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 1.99999999999999991e-6

   1. Initial program 4.7%

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

      1. add-cube-cbrt 4.0%

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

      2. pow3 4.0%

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

   3. Applied egg-rr 4.0%

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

   4. Taylor expanded in x around inf 1.0%

      \[\leadsto \color{blue}{\left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \left(0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} + {x}^{0.3333333333333333}\right)\right)} - \sqrt[3]{x} \]
   5. Step-by-step derivation

      1. unpow1/3 4.7%

         \[\leadsto \left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \left(0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} + \color{blue}{\sqrt[3]{x}}\right)\right) - \sqrt[3]{x} \]

      2. associate-+r+ 4.7%

         \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \sqrt[3]{x}\right)} - \sqrt[3]{x} \]

      3. +-commutative 4.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right)} - \sqrt[3]{x} \]

      4. metadata-eval 4.7%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left({\color{blue}{1}}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      5. pow-base-1 4.7%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left(\color{blue}{1} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      6. *-lft-identity 4.7%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      7. distribute-rgt-out 4.7%

         \[\leadsto \left(\sqrt[3]{x} + \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} \cdot \left(0.1111111111111111 + 0.2222222222222222\right)}\right) - \sqrt[3]{x} \]

      8. unpow1/3 4.7%

         \[\leadsto \left(\sqrt[3]{x} + \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}}} \cdot \left(0.1111111111111111 + 0.2222222222222222\right)\right) - \sqrt[3]{x} \]

      9. metadata-eval 4.7%

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

   6. Simplified 4.7%

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

      1. associate--l+ 4.7%

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

      2. flip-+ 4.7%

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

   8. Applied egg-rr 4.7%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} - \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right) \cdot \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right)}{\sqrt[3]{x} - \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right)}} \]
   9. Step-by-step derivation

      1. unpow2 4.7%

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

      2. difference-of-squares 4.7%

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

      3. +-commutative 4.7%

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

      4. associate-+l- 98.2%

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

      5. +-inverses 98.2%

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

      6. --rgt-identity 98.2%

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

      7. associate--r- 98.2%

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

      8. +-commutative 98.2%

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

      9. associate--r- 98.2%

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

      10. +-commutative 98.2%

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

   10. Simplified 98.2%

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

       1. expm1-log1p-u 98.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)\right)}{\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)}\right)\right)} \]

       2. expm1-udef 5.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)\right)}{\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)}\right)} - 1} \]

   12. Applied egg-rr 5.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}}{1}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 98.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}}{1}\right)\right)} \]

       2. expm1-log1p 98.2%

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

       3. /-rgt-identity 98.2%

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

   14. Simplified 98.2%

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

   if 1.99999999999999991e-6 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array} \]

Alternative 6: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 0.25\right):\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot -0.6666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 0.25)))
   (* 0.3333333333333333 (pow (cbrt x) -2.0))
   (+ 1.0 (* x -0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 0.25)) {
		tmp = 0.3333333333333333 * pow(cbrt(x), -2.0);
	} else {
		tmp = 1.0 + (x * -0.6666666666666666);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 0.25)) {
		tmp = 0.3333333333333333 * Math.pow(Math.cbrt(x), -2.0);
	} else {
		tmp = 1.0 + (x * -0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 0.25))
		tmp = Float64(0.3333333333333333 * (cbrt(x) ^ -2.0));
	else
		tmp = Float64(1.0 + Float64(x * -0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 0.25]], $MachinePrecision]], N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 0.25 < x

   1. Initial program 5.9%

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

      1. add-cube-cbrt 5.2%

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

      2. pow3 5.2%

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

   3. Applied egg-rr 5.2%

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

   4. Taylor expanded in x around inf 1.0%

      \[\leadsto \color{blue}{\left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \left(0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} + {x}^{0.3333333333333333}\right)\right)} - \sqrt[3]{x} \]
   5. Step-by-step derivation

      1. unpow1/3 5.2%

         \[\leadsto \left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \left(0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} + \color{blue}{\sqrt[3]{x}}\right)\right) - \sqrt[3]{x} \]

      2. associate-+r+ 5.2%

         \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + \sqrt[3]{x}\right)} - \sqrt[3]{x} \]

      3. +-commutative 5.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left({\left({1}^{4}\right)}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right)} - \sqrt[3]{x} \]

      4. metadata-eval 5.2%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left({\color{blue}{1}}^{0.1111111111111111} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      5. pow-base-1 5.2%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \left(\color{blue}{1} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right) + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      6. *-lft-identity 5.2%

         \[\leadsto \left(\sqrt[3]{x} + \left(0.1111111111111111 \cdot \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} + 0.2222222222222222 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)\right) - \sqrt[3]{x} \]

      7. distribute-rgt-out 5.2%

         \[\leadsto \left(\sqrt[3]{x} + \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} \cdot \left(0.1111111111111111 + 0.2222222222222222\right)}\right) - \sqrt[3]{x} \]

      8. unpow1/3 5.2%

         \[\leadsto \left(\sqrt[3]{x} + \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}}} \cdot \left(0.1111111111111111 + 0.2222222222222222\right)\right) - \sqrt[3]{x} \]

      9. metadata-eval 5.2%

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

   6. Simplified 5.2%

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

      1. associate--l+ 5.2%

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

      2. flip-+ 5.2%

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

   8. Applied egg-rr 5.2%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} - \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right) \cdot \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right)}{\sqrt[3]{x} - \left(\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} - \sqrt[3]{x}\right)}} \]
   9. Step-by-step derivation

      1. unpow2 5.2%

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

      2. difference-of-squares 5.2%

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

      3. +-commutative 5.2%

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

      4. associate-+l- 97.3%

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

      5. +-inverses 97.3%

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

      6. --rgt-identity 97.3%

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

      7. associate--r- 97.3%

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

      8. +-commutative 97.3%

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

      9. associate--r- 97.3%

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

      10. +-commutative 97.3%

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

   10. Simplified 97.3%

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

       1. expm1-log1p-u 97.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)\right)}{\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)}\right)\right)} \]

       2. expm1-udef 5.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)\right)}{\sqrt[3]{x} + \left(\sqrt[3]{x} - \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\right)}\right)} - 1} \]

   12. Applied egg-rr 5.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}}{1}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 97.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}}{1}\right)\right)} \]

       2. expm1-log1p 97.3%

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

       3. /-rgt-identity 97.3%

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

   14. Simplified 97.3%

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

   if -1.05000000000000004 < x < 0.25

   1. Initial program 100.0%

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

      1. pow1/3 48.8%

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

   3. Applied egg-rr 48.8%

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

      1. pow1/3 100.0%

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

      2. flip3-- 99.9%

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

      3. div-inv 99.9%

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

      4. add-sqr-sqrt 48.7%

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

      5. cbrt-unprod 48.7%

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

      6. rem-cube-cbrt 48.7%

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

      7. cbrt-unprod 48.7%

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

      8. add-sqr-sqrt 99.9%

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

      9. rem-cube-cbrt 99.9%

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

      10. pow2 99.9%

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

      11. distribute-rgt-out 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-def 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 95.8%

      \[\leadsto \color{blue}{1 + -0.6666666666666666 \cdot x} \]
   9. Step-by-step derivation

      1. *-commutative 95.8%

         \[\leadsto 1 + \color{blue}{x \cdot -0.6666666666666666} \]

   10. Simplified 95.8%

       \[\leadsto \color{blue}{1 + x \cdot -0.6666666666666666} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 0.25\right):\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot -0.6666666666666666\\ \end{array} \]

Alternative 7: 3.6% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 52.5%

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

2. Taylor expanded in x around inf 3.7%

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

3. Final simplification 3.7%

   \[\leadsto 0 \]

Alternative 8: 50.6% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 52.5%

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

2. Taylor expanded in x around 0 50.6%

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

3. Final simplification 50.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023332 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))