2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.6%

Time: 4.0s

Alternatives: 12

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{x \cdot x} + \frac{1}{x}\\ t_1 := \mathsf{fma}\left(\sqrt[3]{t\_0}, -1, \mathsf{fma}\left(\frac{\sqrt[3]{\frac{1}{x} + 1}}{\sqrt[3]{x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)\\ -\frac{\mathsf{fma}\left(\left({t\_1}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {t\_0}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{t\_1}\right)}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* x x)) (/ 1.0 x)))
        (t_1
         (fma
          (cbrt t_0)
          -1.0
          (fma
           (/ (cbrt (+ (/ 1.0 x) 1.0)) (cbrt x))
           -1.0
           (* (/ 1.0 (cbrt x)) -1.0)))))
   (-
    (/
     (fma
      (*
       (* (pow t_1 -2.0) (/ 1.0 (* (* x x) x)))
       (pow t_0 -0.6666666666666666))
      0.3333333333333333
      (/ 1.0 t_1))
     x))))

C

double code(double x) {
	double t_0 = (2.0 / (x * x)) + (1.0 / x);
	double t_1 = fma(cbrt(t_0), -1.0, fma((cbrt(((1.0 / x) + 1.0)) / cbrt(x)), -1.0, ((1.0 / cbrt(x)) * -1.0)));
	return -(fma(((pow(t_1, -2.0) * (1.0 / ((x * x) * x))) * pow(t_0, -0.6666666666666666)), 0.3333333333333333, (1.0 / t_1)) / x);
}

Julia

function code(x)
	t_0 = Float64(Float64(2.0 / Float64(x * x)) + Float64(1.0 / x))
	t_1 = fma(cbrt(t_0), -1.0, fma(Float64(cbrt(Float64(Float64(1.0 / x) + 1.0)) / cbrt(x)), -1.0, Float64(Float64(1.0 / cbrt(x)) * -1.0)))
	return Float64(-Float64(fma(Float64(Float64((t_1 ^ -2.0) * Float64(1.0 / Float64(Float64(x * x) * x))) * (t_0 ^ -0.6666666666666666)), 0.3333333333333333, Float64(1.0 / t_1)) / x))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[t$95$0, 1/3], $MachinePrecision] * -1.0 + N[(N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] + 1.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, (-N[(N[(N[(N[(N[Power[t$95$1, -2.0], $MachinePrecision] * N[(1.0 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, -0.6666666666666666], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision])]]

Derivation

1. Initial program 7.0%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. flip3-- N/A

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

   6. lower-/.f64 N/A

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

   7. rem-cube-cbrt N/A

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

   8. rem-cube-cbrt N/A

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

   9. lower--.f64 N/A

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

   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. lower--.f64 N/A

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

   15. lower-+.f64 N/A

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

3. Applied rewrites 9.1%

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

4. Taylor expanded in x around -inf

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

6. Applied rewrites 94.0%

   \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)}\right)}{x}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. inv-pow N/A

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

   5. pow1/3 N/A

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

   6. cbrt-div N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cbrt.f64 94.0

      \[\leadsto -\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)}\right)}{x} \]

8. Applied rewrites 94.0%

   \[\leadsto -\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)}\right)}{x} \]
9. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. inv-pow N/A

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

   5. pow1/3 N/A

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

   6. cbrt-div N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cbrt.f64 98.6

      \[\leadsto -\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)}\right)}{x} \]

10. Applied rewrites 98.6%

    \[\leadsto -\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)}\right)}{x} \]
11. Step-by-step derivation

    1. lift-cbrt.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-/r* N/A

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

    7. div-add N/A

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

    8. +-commutative N/A

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

    9. cbrt-div N/A

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

    10. lower-/.f64 N/A

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

    11. lower-cbrt.f64 N/A

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

    12. lift-/.f64 N/A

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

    13. lift-+.f64 N/A

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

    14. lift-cbrt.f64 98.6

        \[\leadsto -\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\frac{\sqrt[3]{\frac{1}{x} + 1}}{\sqrt[3]{x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)}\right)}{x} \]

12. Applied rewrites 98.6%

    \[\leadsto -\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\frac{\sqrt[3]{\frac{1}{x} + 1}}{\sqrt[3]{x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)}\right)}{x} \]
13. Step-by-step derivation

    1. lift-cbrt.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-/r* N/A

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

    7. div-add N/A

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

    8. +-commutative N/A

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

    9. cbrt-div N/A

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

    10. lower-/.f64 N/A

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

    11. lower-cbrt.f64 N/A

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

    12. lift-/.f64 N/A

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

    13. lift-+.f64 N/A

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

    14. lift-cbrt.f64 98.6

        \[\leadsto -\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\frac{\sqrt[3]{\frac{1}{x} + 1}}{\sqrt[3]{x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\frac{\sqrt[3]{\frac{1}{x} + 1}}{\sqrt[3]{x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)}\right)}{x} \]

14. Applied rewrites 98.6%

    \[\leadsto -\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\frac{\sqrt[3]{\frac{1}{x} + 1}}{\sqrt[3]{x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\frac{\sqrt[3]{\frac{1}{x} + 1}}{\sqrt[3]{x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)}\right)}{x} \]
15. Add Preprocessing

Alternative 2: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{x \cdot x} + \frac{1}{x}\\ -\frac{\mathsf{fma}\left(\left({\left(\left(\left(-{x}^{0.6666666666666666}\right) + \left(-\sqrt[3]{\mathsf{fma}\left(x, x, x + x\right)}\right)\right) + \left(-\sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right)\right)}^{-2} \cdot \frac{1}{x}\right) \cdot {t\_0}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{t\_0}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)}\right)}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* x x)) (/ 1.0 x))))
   (-
    (/
     (fma
      (*
       (*
        (pow
         (+
          (+ (- (pow x 0.6666666666666666)) (- (cbrt (fma x x (+ x x)))))
          (- (cbrt (fma x x x))))
         -2.0)
        (/ 1.0 x))
       (pow t_0 -0.6666666666666666))
      0.3333333333333333
      (/
       1.0
       (fma
        (cbrt t_0)
        -1.0
        (fma
         (cbrt (+ (/ 1.0 x) (/ 1.0 (* x x))))
         -1.0
         (* (/ 1.0 (cbrt x)) -1.0)))))
     x))))

C

double code(double x) {
	double t_0 = (2.0 / (x * x)) + (1.0 / x);
	return -(fma(((pow(((-pow(x, 0.6666666666666666) + -cbrt(fma(x, x, (x + x)))) + -cbrt(fma(x, x, x))), -2.0) * (1.0 / x)) * pow(t_0, -0.6666666666666666)), 0.3333333333333333, (1.0 / fma(cbrt(t_0), -1.0, fma(cbrt(((1.0 / x) + (1.0 / (x * x)))), -1.0, ((1.0 / cbrt(x)) * -1.0))))) / x);
}

Julia

function code(x)
	t_0 = Float64(Float64(2.0 / Float64(x * x)) + Float64(1.0 / x))
	return Float64(-Float64(fma(Float64(Float64((Float64(Float64(Float64(-(x ^ 0.6666666666666666)) + Float64(-cbrt(fma(x, x, Float64(x + x))))) + Float64(-cbrt(fma(x, x, x)))) ^ -2.0) * Float64(1.0 / x)) * (t_0 ^ -0.6666666666666666)), 0.3333333333333333, Float64(1.0 / fma(cbrt(t_0), -1.0, fma(cbrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x * x)))), -1.0, Float64(Float64(1.0 / cbrt(x)) * -1.0))))) / x))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, (-N[(N[(N[(N[(N[Power[N[(N[((-N[Power[x, 0.6666666666666666], $MachinePrecision]) + (-N[Power[N[(x * x + N[(x + x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision])), $MachinePrecision] + (-N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision])), $MachinePrecision], -2.0], $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, -0.6666666666666666], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(1.0 / N[(N[Power[t$95$0, 1/3], $MachinePrecision] * -1.0 + N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * -1.0 + N[(N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision])]

Derivation

1. Initial program 7.0%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. flip3-- N/A

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

   6. lower-/.f64 N/A

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

   7. rem-cube-cbrt N/A

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

   8. rem-cube-cbrt N/A

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

   9. lower--.f64 N/A

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

   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. lower--.f64 N/A

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

   15. lower-+.f64 N/A

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

3. Applied rewrites 9.1%

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

4. Taylor expanded in x around -inf

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

6. Applied rewrites 94.0%

   \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)}\right)}{x}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. inv-pow N/A

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

   5. pow1/3 N/A

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

   6. cbrt-div N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cbrt.f64 94.0

      \[\leadsto -\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)}\right)}{x} \]

8. Applied rewrites 94.0%

   \[\leadsto -\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)}\right)}{x} \]
9. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. inv-pow N/A

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

   5. pow1/3 N/A

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

   6. cbrt-div N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cbrt.f64 98.6

      \[\leadsto -\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)}\right)}{x} \]

10. Applied rewrites 98.6%

    \[\leadsto -\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)}\right)}{x} \]

11. Taylor expanded in x around 0

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

13. Applied rewrites 98.6%

    \[\leadsto -\frac{\mathsf{fma}\left(\left({\left(\left(\left(-{x}^{0.6666666666666666}\right) + \left(-\sqrt[3]{\mathsf{fma}\left(x, x, x + x\right)}\right)\right) + \left(-\sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right)\right)}^{-2} \cdot \frac{1}{x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, \frac{1}{\sqrt[3]{x}} \cdot -1\right)\right)}\right)}{x} \]
14. Add Preprocessing

Alternative 3: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{1}{\left(\left(-\sqrt[3]{\frac{1}{x}}\right) + \left(-\sqrt[3]{\frac{\frac{1}{x} + 1}{x}}\right)\right) + \left(-{x}^{-0.3333333333333333}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4e+154)
   (fma
    (pow x -1.6666666666666667)
    -0.1111111111111111
    (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333))
   (-
    (/
     (/
      1.0
      (+
       (+ (- (cbrt (/ 1.0 x))) (- (cbrt (/ (+ (/ 1.0 x) 1.0) x))))
       (- (pow x -0.3333333333333333))))
     x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4e+154) {
		tmp = fma(pow(x, -1.6666666666666667), -0.1111111111111111, (cbrt((1.0 / (x * x))) * 0.3333333333333333));
	} else {
		tmp = -((1.0 / ((-cbrt((1.0 / x)) + -cbrt((((1.0 / x) + 1.0) / x))) + -pow(x, -0.3333333333333333))) / x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4e+154)
		tmp = fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333));
	else
		tmp = Float64(-Float64(Float64(1.0 / Float64(Float64(Float64(-cbrt(Float64(1.0 / x))) + Float64(-cbrt(Float64(Float64(Float64(1.0 / x) + 1.0) / x)))) + Float64(-(x ^ -0.3333333333333333)))) / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.4e+154], N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], (-N[(N[(1.0 / N[(N[((-N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]) + (-N[Power[N[(N[(N[(1.0 / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision])), $MachinePrecision] + (-N[Power[x, -0.3333333333333333], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e154

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

         \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}} \]

      4. lower-fma.f64 N/A

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

      5. pow1/3 N/A

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

      6. pow-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cbrt.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 44.6

          \[\leadsto \frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 44.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      3. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left({\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{fma}\left({\left({x}^{\left(\mathsf{neg}\left(5\right)\right)}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{fma}\left({x}^{\left(\left(\mathsf{neg}\left(5\right)\right) \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\left(\left(\mathsf{neg}\left(5\right)\right) \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\left(-5 \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      9. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}\right) \]

      10. pow-flip N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {\left({x}^{\left(\mathsf{neg}\left(2\right)\right)}\right)}^{\frac{1}{3}}\right) \]

      11. pow-pow N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{3}\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\left(-2 \cdot \frac{1}{3}\right)}\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\frac{-2}{3}}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\frac{-2}{3}}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

      16. lift-*.f64 89.7

          \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

   7. Applied rewrites 89.7%

      \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, \color{blue}{-0.1111111111111111}, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

      3. pow-pow N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {\left({x}^{\left(\mathsf{neg}\left(2\right)\right)}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      5. pow-flip N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      6. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}\right) \]

      7. lower-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3}\right) \]

      10. lift-*.f64 49.8

          \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\right) \]

   9. Applied rewrites 49.8%

      \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\right) \]

   if 1.4e154 < x

   1. Initial program 7.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. rem-cube-cbrt N/A

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

      9. lower--.f64 N/A

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

      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. lower--.f64 N/A

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

      15. lower-+.f64 N/A

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

   3. Applied rewrites 9.1%

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

   4. Taylor expanded in x around -inf

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

   6. Applied rewrites 94.0%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)}\right)}{x}} \]

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 93.5%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 92.6%

       \[\leadsto -\frac{\frac{1}{\left(\left(-\sqrt[3]{\frac{1}{x}}\right) + \left(-\sqrt[3]{\frac{\frac{1}{x} + 1}{x}}\right)\right) + \left(-{x}^{-0.3333333333333333}\right)}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. flip3-- N/A

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

   6. lower-/.f64 N/A

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

   7. rem-cube-cbrt N/A

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

   8. rem-cube-cbrt N/A

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

   9. lower--.f64 N/A

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

   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. lower--.f64 N/A

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

   15. lower-+.f64 N/A

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

3. Applied rewrites 9.1%

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

4. Taylor expanded in x around -inf

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

6. Applied rewrites 94.0%

   \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)}\right)}{x}} \]

7. Taylor expanded in x around -inf

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

9. Applied rewrites 93.5%

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

Alternative 5: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\left(\sqrt[3]{\frac{\frac{2}{x} + 1}{x}} + \sqrt[3]{\frac{\frac{1}{x} + 1}{x}}\right) + {x}^{-0.3333333333333333}\right) \cdot x} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x) {
	return 1.0 / (((Math.cbrt((((2.0 / x) + 1.0) / x)) + Math.cbrt((((1.0 / x) + 1.0) / x))) + Math.pow(x, -0.3333333333333333)) * x);
}

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(cbrt(Float64(Float64(Float64(2.0 / x) + 1.0) / x)) + cbrt(Float64(Float64(Float64(1.0 / x) + 1.0) / x))) + (x ^ -0.3333333333333333)) * x))
end

Wolfram

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

Derivation

1. Initial program 7.0%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. flip3-- N/A

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

   6. lower-/.f64 N/A

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

   7. rem-cube-cbrt N/A

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

   8. rem-cube-cbrt N/A

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

   9. lower--.f64 N/A

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

   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. lower--.f64 N/A

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

   15. lower-+.f64 N/A

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

3. Applied rewrites 9.1%

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

4. Taylor expanded in x around -inf

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

6. Applied rewrites 94.0%

   \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\left({\left(\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right) \cdot {\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666}, 0.3333333333333333, \frac{1}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}, -1, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}, -1, {x}^{-0.3333333333333333} \cdot -1\right)\right)}\right)}{x}} \]

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

9. Applied rewrites 93.5%

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

Alternative 6: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4e+154)
   (fma
    (pow x -1.6666666666666667)
    -0.1111111111111111
    (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333))
   (fma
    (pow x -1.6666666666666667)
    -0.1111111111111111
    (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4e+154) {
		tmp = fma(pow(x, -1.6666666666666667), -0.1111111111111111, (cbrt((1.0 / (x * x))) * 0.3333333333333333));
	} else {
		tmp = fma(pow(x, -1.6666666666666667), -0.1111111111111111, (exp((log(x) * -0.6666666666666666)) * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4e+154)
		tmp = fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333));
	else
		tmp = fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64(exp(Float64(log(x) * -0.6666666666666666)) * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.4e+154], N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Exp[N[(N[Log[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e154

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

         \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}} \]

      4. lower-fma.f64 N/A

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

      5. pow1/3 N/A

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

      6. pow-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cbrt.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 44.6

          \[\leadsto \frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 44.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      3. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left({\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{fma}\left({\left({x}^{\left(\mathsf{neg}\left(5\right)\right)}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{fma}\left({x}^{\left(\left(\mathsf{neg}\left(5\right)\right) \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\left(\left(\mathsf{neg}\left(5\right)\right) \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\left(-5 \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      9. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}\right) \]

      10. pow-flip N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {\left({x}^{\left(\mathsf{neg}\left(2\right)\right)}\right)}^{\frac{1}{3}}\right) \]

      11. pow-pow N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{3}\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\left(-2 \cdot \frac{1}{3}\right)}\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\frac{-2}{3}}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\frac{-2}{3}}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

      16. lift-*.f64 89.7

          \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

   7. Applied rewrites 89.7%

      \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, \color{blue}{-0.1111111111111111}, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right) \]

      3. pow-pow N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {\left({x}^{-2}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {\left({x}^{\left(\mathsf{neg}\left(2\right)\right)}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      5. pow-flip N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right) \]

      6. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}\right) \]

      7. lower-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3}\right) \]

      10. lift-*.f64 49.8

          \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\right) \]

   9. Applied rewrites 49.8%

      \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\right) \]

   if 1.4e154 < x

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

         \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}} \]

      4. lower-fma.f64 N/A

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

      5. pow1/3 N/A

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

      6. pow-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cbrt.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 44.6

          \[\leadsto \frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 44.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      3. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left({\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{fma}\left({\left({x}^{\left(\mathsf{neg}\left(5\right)\right)}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{fma}\left({x}^{\left(\left(\mathsf{neg}\left(5\right)\right) \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\left(\left(\mathsf{neg}\left(5\right)\right) \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\left(-5 \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

      9. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}\right) \]

      10. pow-flip N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {\left({x}^{\left(\mathsf{neg}\left(2\right)\right)}\right)}^{\frac{1}{3}}\right) \]

      11. pow-pow N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{3}\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\left(-2 \cdot \frac{1}{3}\right)}\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\frac{-2}{3}}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\frac{-2}{3}}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

      16. lift-*.f64 89.7

          \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

   7. Applied rewrites 89.7%

      \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, \color{blue}{-0.1111111111111111}, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3}\right) \]

      5. lower-log.f64 90.1

         \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\right) \]

   9. Applied rewrites 90.1%

      \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 92.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{\sqrt[3]{-x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4e+154)
   (* (/ -1.0 (cbrt (- (* x x)))) 0.3333333333333333)
   (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.4e+154) {
		tmp = (-1.0 / cbrt(-(x * x))) * 0.3333333333333333;
	} else {
		tmp = exp((log(x) * -0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4e+154) {
		tmp = (-1.0 / Math.cbrt(-(x * x))) * 0.3333333333333333;
	} else {
		tmp = Math.exp((Math.log(x) * -0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4e+154)
		tmp = Float64(Float64(-1.0 / cbrt(Float64(-Float64(x * x)))) * 0.3333333333333333);
	else
		tmp = Float64(exp(Float64(log(x) * -0.6666666666666666)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.4e+154], N[(N[(-1.0 / N[Power[(-N[(x * x), $MachinePrecision]), 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e154

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   3. 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. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 88.8

         \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. cbrt-div N/A

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

      10. metadata-eval N/A

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

      11. rem-cbrt-cube N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 48.9

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

   6. Applied rewrites 48.9%

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

   if 1.4e154 < x

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   3. 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. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 88.8

         \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 89.1

         \[\leadsto e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333 \]

   6. Applied rewrites 89.1%

      \[\leadsto e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4e+154)
   (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333)
   (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.4e+154) {
		tmp = cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = exp((log(x) * -0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4e+154) {
		tmp = Math.cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = Math.exp((Math.log(x) * -0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4e+154)
		tmp = Float64(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64(exp(Float64(log(x) * -0.6666666666666666)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.4e+154], N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e154

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   3. 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. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 88.8

         \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 48.8

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

   6. Applied rewrites 48.8%

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

   if 1.4e154 < x

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   3. 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. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 88.8

         \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 89.1

         \[\leadsto e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333 \]

   6. Applied rewrites 89.1%

      \[\leadsto e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 90.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (pow x -1.6666666666666667)
  -0.1111111111111111
  (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333)))

C

double code(double x) {
	return fma(pow(x, -1.6666666666666667), -0.1111111111111111, (exp((log(x) * -0.6666666666666666)) * 0.3333333333333333));
}

Julia

function code(x)
	return fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64(exp(Float64(log(x) * -0.6666666666666666)) * 0.3333333333333333))
end

Wolfram

code[x_] := N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Exp[N[(N[Log[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}} \]

   4. lower-fma.f64 N/A

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

   5. pow1/3 N/A

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

   6. pow-pow N/A

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

   7. lower-pow.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 44.6

       \[\leadsto \frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

4. Applied rewrites 44.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   3. pow1/3 N/A

      \[\leadsto \mathsf{fma}\left({\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   4. pow-flip N/A

      \[\leadsto \mathsf{fma}\left({\left({x}^{\left(\mathsf{neg}\left(5\right)\right)}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   5. pow-pow N/A

      \[\leadsto \mathsf{fma}\left({x}^{\left(\left(\mathsf{neg}\left(5\right)\right) \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{\left(\left(\mathsf{neg}\left(5\right)\right) \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\left(-5 \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   9. pow1/3 N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}\right) \]

   10. pow-flip N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {\left({x}^{\left(\mathsf{neg}\left(2\right)\right)}\right)}^{\frac{1}{3}}\right) \]

   11. pow-pow N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{3}\right)}\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\left(-2 \cdot \frac{1}{3}\right)}\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\frac{-2}{3}}\right) \]

   14. lift-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{1}{3} \cdot {x}^{\frac{-2}{3}}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

   16. lift-*.f64 89.7

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

7. Applied rewrites 89.7%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, \color{blue}{-0.1111111111111111}, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right) \]

   2. pow-to-exp N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3}\right) \]

   3. lower-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3}\right) \]

   5. lower-log.f64 90.1

      \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\right) \]

9. Applied rewrites 90.1%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\right) \]
10. Add Preprocessing

Alternative 10: 89.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333))

C

double code(double x) {
	return exp((log(x) * -0.6666666666666666)) * 0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = exp((log(x) * (-0.6666666666666666d0))) * 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.exp((Math.log(x) * -0.6666666666666666)) * 0.3333333333333333;
}

Python

def code(x):
	return math.exp((math.log(x) * -0.6666666666666666)) * 0.3333333333333333

Julia

function code(x)
	return Float64(exp(Float64(log(x) * -0.6666666666666666)) * 0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = exp((log(x) * -0.6666666666666666)) * 0.3333333333333333;
end

Wolfram

code[x_] := N[(N[Exp[N[(N[Log[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
3. 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. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval 88.8

      \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]

4. Applied rewrites 88.8%

   \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. pow-to-exp N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-log.f64 89.1

      \[\leadsto e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333 \]

6. Applied rewrites 89.1%

   \[\leadsto e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333 \]
7. Add Preprocessing

Alternative 11: 88.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow x -0.6666666666666666) 0.3333333333333333))

C

double code(double x) {
	return pow(x, -0.6666666666666666) * 0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (x ** (-0.6666666666666666d0)) * 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
}

Python

def code(x):
	return math.pow(x, -0.6666666666666666) * 0.3333333333333333

Julia

function code(x)
	return Float64((x ^ -0.6666666666666666) * 0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.6666666666666666) * 0.3333333333333333;
end

Wolfram

code[x_] := N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
3. 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. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval 88.8

      \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]

4. Applied rewrites 88.8%

   \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
5. Add Preprocessing

Alternative 12: 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 7.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 1.8%

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

Developer Target 1: 98.4% 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 2025134 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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