2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 93.1%

Time: 2.7s

Alternatives: 6

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 93.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x - -1}\\ \frac{1}{\mathsf{fma}\left({x}^{0.3333333333333333}, \sqrt[3]{x} + t\_0, {t\_0}^{2}\right)} \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double t_0 = cbrt((x - -1.0));
	return 1.0 / fma(pow(x, 0.3333333333333333), (cbrt(x) + t_0), pow(t_0, 2.0));
}

Julia

function code(x)
	t_0 = cbrt(Float64(x - -1.0))
	return Float64(1.0 / fma((x ^ 0.3333333333333333), Float64(cbrt(x) + t_0), (t_0 ^ 2.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[(N[Power[x, 0.3333333333333333], $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 6.9%

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

   1. lift--.f64 N/A

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

   2. flip3-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. rem-cube-cbrt N/A

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

   6. lift-cbrt.f64 N/A

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

   7. rem-cube-cbrt N/A

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

   8. lower--.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. metadata-eval N/A

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

   11. fp-cancel-sign-sub-inv N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

3. Applied rewrites 9.0%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lift-pow.f64 8.9

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

5. Applied rewrites 8.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 93.1%

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

Alternative 2: 58.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+161}:\\ \;\;\;\;\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\sqrt[3]{x} + 1, \sqrt[3]{x}, 1\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.4e+161)
   (* (cbrt (pow x -2.0)) 0.3333333333333333)
   (pow (fma (+ (cbrt x) 1.0) (cbrt x) 1.0) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 6.4e+161) {
		tmp = cbrt(pow(x, -2.0)) * 0.3333333333333333;
	} else {
		tmp = pow(fma((cbrt(x) + 1.0), cbrt(x), 1.0), -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.4e+161)
		tmp = Float64(cbrt((x ^ -2.0)) * 0.3333333333333333);
	else
		tmp = fma(Float64(cbrt(x) + 1.0), cbrt(x), 1.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6.4e+161], N[(N[Power[N[Power[x, -2.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[Power[N[(N[(N[Power[x, 1/3], $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.40000000000000004e161

   1. Initial program 8.8%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-cbrt.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval N/A

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

      19. *-rgt-identity N/A

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

      20. lower--.f64 8.4

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

   3. Applied rewrites 8.4%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3} \]

      6. lower-pow.f64 93.3

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   6. Applied rewrites 93.3%

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

   if 6.40000000000000004e161 < x

   1. Initial program 4.8%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. rem-cube-cbrt N/A

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

      6. lift-cbrt.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. lower--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

   3. Applied rewrites 4.8%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-pow.f64 4.8

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in x around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. lift-cbrt.f64 17.7

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

   8. Applied rewrites 17.7%

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

Alternative 3: 52.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip3-+ N/A

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

   4. cbrt-div N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. fp-cancel-sign-sub-inv N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower--.f64 N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-cbrt.f64 N/A

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

   17. lower-fma.f64 N/A

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

   18. metadata-eval N/A

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

   19. *-rgt-identity N/A

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

   20. lower--.f64 4.5

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

3. Applied rewrites 4.5%

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

4. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. metadata-eval N/A

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot \frac{1}{3} \]

   6. lower-pow.f64 52.4

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

6. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
7. Add Preprocessing

Alternative 4: 5.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	return ((x - -1.0) - x) / (cbrt((x * x)) * 2.0);
}

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. lift--.f64 N/A

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

   2. flip3-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. rem-cube-cbrt N/A

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

   6. lift-cbrt.f64 N/A

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

   7. rem-cube-cbrt N/A

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

   8. lower--.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. metadata-eval N/A

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

   11. fp-cancel-sign-sub-inv N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

3. Applied rewrites 9.0%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lift-pow.f64 8.9

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

5. Applied rewrites 8.9%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow2 N/A

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

   5. lower-*.f64 5.0

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

8. Applied rewrites 5.0%

   \[\leadsto \frac{\left(x - -1\right) - x}{\color{blue}{\sqrt[3]{x \cdot x} \cdot 2}} \]
9. Add Preprocessing

Alternative 5: 4.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333, x, 1\right) - \sqrt[3]{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (fma 0.3333333333333333 x 1.0) (cbrt x)))

C

double code(double x) {
	return fma(0.3333333333333333, x, 1.0) - cbrt(x);
}

Julia

function code(x)
	return Float64(fma(0.3333333333333333, x, 1.0) - cbrt(x))
end

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. rem-cube-cbrt N/A

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

   2. lift-cbrt.f64 N/A

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

   3. unpow3 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. metadata-eval N/A

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

   7. 1-exp N/A

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

   8. 1-exp N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. metadata-eval N/A

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

   13. fp-cancel-sign-sub-inv N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower--.f64 N/A

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

   18. metadata-eval N/A

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

   19. 1-exp N/A

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

   20. metadata-eval N/A

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

   21. 1-exp N/A

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

   22. metadata-eval 7.0

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

   23. lift-+.f64 N/A

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

   24. metadata-eval N/A

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

   25. fp-cancel-sign-sub-inv N/A

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

3. Applied rewrites 7.0%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. cbrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 7.0

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

5. Applied rewrites 7.0%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. pow1/3 N/A

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

   8. log-pow N/A

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

   9. lower-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. sub-negate1 N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

   14. lower-log1p.f64 4.9

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

7. Applied rewrites 4.9%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 4.3

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

10. Applied rewrites 4.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, x, 1\right)} - \sqrt[3]{x} \]
11. Add Preprocessing

Alternative 6: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. rem-cube-cbrt N/A

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

   2. lift-cbrt.f64 N/A

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

   3. unpow3 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. metadata-eval N/A

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

   7. 1-exp N/A

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

   8. 1-exp N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. metadata-eval N/A

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

   13. fp-cancel-sign-sub-inv N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower--.f64 N/A

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

   18. metadata-eval N/A

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

   19. 1-exp N/A

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

   20. metadata-eval N/A

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

   21. 1-exp N/A

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

   22. metadata-eval 7.0

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

   23. lift-+.f64 N/A

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

   24. metadata-eval N/A

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

   25. fp-cancel-sign-sub-inv N/A

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

3. Applied rewrites 7.0%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. cbrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 7.0

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

5. Applied rewrites 7.0%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. pow1/3 N/A

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

   8. log-pow N/A

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

   9. lower-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. sub-negate1 N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

   14. lower-log1p.f64 4.9

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

7. Applied rewrites 4.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 1.8%

    \[\leadsto \color{blue}{1} - \sqrt[3]{x} \]
11. Add Preprocessing

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \frac{1}{\left(t\_0 \cdot t\_0 + \sqrt[3]{x} \cdot t\_0\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2025107 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (cbrt (+ x 1)) (cbrt (+ x 1))) (* (cbrt x) (cbrt (+ x 1))) (* (cbrt x) (cbrt x)))))

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