2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.5%

Time: 11.1s

Alternatives: 18

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: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt[3]{\left(x + 1\right) \cdot \left(x + 1\right)} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{x}} \cdot 0.3333333333333333}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+15)
   (/
    (+ x (- 1.0 x))
    (+ (cbrt (* (+ x 1.0) (+ x 1.0))) (+ (cbrt (* x x)) (cbrt (fma x x x)))))
   (/ (* (cbrt (/ 1.0 x)) 0.3333333333333333) (cbrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 2e+15) {
		tmp = (x + (1.0 - x)) / (cbrt(((x + 1.0) * (x + 1.0))) + (cbrt((x * x)) + cbrt(fma(x, x, x))));
	} else {
		tmp = (cbrt((1.0 / x)) * 0.3333333333333333) / cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+15)
		tmp = Float64(Float64(x + Float64(1.0 - x)) / Float64(cbrt(Float64(Float64(x + 1.0) * Float64(x + 1.0))) + Float64(cbrt(Float64(x * x)) + cbrt(fma(x, x, x)))));
	else
		tmp = Float64(Float64(cbrt(Float64(1.0 / x)) * 0.3333333333333333) / cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+15], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e15

   1. Initial program 52.3%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. clear-num N/A

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

      10. flip3-+ N/A

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

      11. lift-+.f64 N/A

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

      12. lower-/.f64 52.2

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

   4. Applied rewrites 52.2%

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

   6. Applied rewrites 97.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow2 N/A

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

      5. pow1/3 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-*.f64 98.3

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

   8. Applied rewrites 98.3%

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

      1. rem-cbrt-cube N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 99.3

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

   10. Applied rewrites 99.3%

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

   if 2e15 < x

   1. Initial program 4.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 50.4

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

   5. Applied rewrites 50.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.4%

      \[\leadsto \frac{\frac{1}{\sqrt[3]{x}} \cdot 0.3333333333333333}{\color{blue}{\sqrt[3]{x}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.5%

      \[\leadsto \frac{\sqrt[3]{\frac{1}{x}} \cdot 0.3333333333333333}{\sqrt[3]{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{x}} \cdot 0.3333333333333333}{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\mathsf{fma}\left({x}^{-0.3333333333333333}, x, \sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + {\left(x + 1\right)}^{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 0.0)
   (/ (* (cbrt (/ 1.0 x)) 0.3333333333333333) (cbrt x))
   (/
    (+ x (- 1.0 x))
    (fma
     (pow x -0.3333333333333333)
     x
     (+ (cbrt (fma x x x)) (pow (+ x 1.0) 0.6666666666666666))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((x + 1.0)) - cbrt(x)) <= 0.0) {
		tmp = (cbrt((1.0 / x)) * 0.3333333333333333) / cbrt(x);
	} else {
		tmp = (x + (1.0 - x)) / fma(pow(x, -0.3333333333333333), x, (cbrt(fma(x, x, x)) + pow((x + 1.0), 0.6666666666666666)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(x + 1.0)) - cbrt(x)) <= 0.0)
		tmp = Float64(Float64(cbrt(Float64(1.0 / x)) * 0.3333333333333333) / cbrt(x));
	else
		tmp = Float64(Float64(x + Float64(1.0 - x)) / fma((x ^ -0.3333333333333333), x, Float64(cbrt(fma(x, x, x)) + (Float64(x + 1.0) ^ 0.6666666666666666))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.3333333333333333], $MachinePrecision] * x + N[(N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 0.0

   1. Initial program 4.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 49.9

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

   5. Applied rewrites 49.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.4%

      \[\leadsto \frac{\frac{1}{\sqrt[3]{x}} \cdot 0.3333333333333333}{\color{blue}{\sqrt[3]{x}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.4%

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

   if 0.0 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 58.6%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. clear-num N/A

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

      10. flip3-+ N/A

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

      11. lift-+.f64 N/A

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

      12. lower-/.f64 57.9

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

   4. Applied rewrites 57.9%

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

   6. Applied rewrites 97.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow2 N/A

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

      5. pow1/3 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-*.f64 98.1

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

   8. Applied rewrites 98.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-cbrt.f64 N/A

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

      6. pow1/3 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. pow-sqr N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. pow-pow N/A

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

      17. inv-pow N/A

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

      18. lift-/.f64 N/A

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

      19. pow1/3 N/A

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

      20. lift-cbrt.f64 N/A

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

      21. lower-fma.f64 N/A

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

   10. Applied rewrites 98.7%

       \[\leadsto \frac{x + \left(1 - x\right)}{\color{blue}{\mathsf{fma}\left({x}^{-0.3333333333333333}, x, \sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + {\left(x + 1\right)}^{0.6666666666666666}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024226 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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