2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.2%

Time: 10.7s

Alternatives: 15

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: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{\frac{1}{x \cdot x}}, 0.06172839506172839, \mathsf{fma}\left(0.3333333333333333, \sqrt[3]{t\_0}, \mathsf{fma}\left(-0.0411522633744856, \sqrt[3]{\frac{1}{x \cdot t\_0}}, \sqrt[3]{x} \cdot -0.1111111111111111\right)\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x 1.15e+77)
     (/
      (fma
       (cbrt (/ 1.0 (* x x)))
       0.06172839506172839
       (fma
        0.3333333333333333
        (cbrt t_0)
        (fma
         -0.0411522633744856
         (cbrt (/ 1.0 (* x t_0)))
         (* (cbrt x) -0.1111111111111111))))
      (* x x))
     (* 0.3333333333333333 (/ (cbrt (/ 1.0 x)) (cbrt x))))))

C

double code(double x) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= 1.15e+77) {
		tmp = fma(cbrt((1.0 / (x * x))), 0.06172839506172839, fma(0.3333333333333333, cbrt(t_0), fma(-0.0411522633744856, cbrt((1.0 / (x * t_0))), (cbrt(x) * -0.1111111111111111)))) / (x * x);
	} else {
		tmp = 0.3333333333333333 * (cbrt((1.0 / x)) / cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= 1.15e+77)
		tmp = Float64(fma(cbrt(Float64(1.0 / Float64(x * x))), 0.06172839506172839, fma(0.3333333333333333, cbrt(t_0), fma(-0.0411522633744856, cbrt(Float64(1.0 / Float64(x * t_0))), Float64(cbrt(x) * -0.1111111111111111)))) / Float64(x * x));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(Float64(1.0 / x)) / cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.15e+77], N[(N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.06172839506172839 + N[(0.3333333333333333 * N[Power[t$95$0, 1/3], $MachinePrecision] + N[(-0.0411522633744856 * N[Power[N[(1.0 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.14999999999999997e77

   1. Initial program 12.7%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 92.5

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

   5. Applied rewrites 92.5%

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

   7. Applied rewrites 92.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{-10}{243} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right)}{{x}^{2}}} \]

   10. Applied rewrites 98.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{\frac{1}{x \cdot x}}, 0.06172839506172839, \mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}, \mathsf{fma}\left(-0.0411522633744856, \sqrt[3]{\frac{1}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x}}, \sqrt[3]{x} \cdot -0.1111111111111111\right)\right)\right)}{x \cdot x}} \]

   if 1.14999999999999997e77 < x

   1. Initial program 4.4%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 35.8

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

   5. Applied rewrites 35.8%

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{\frac{1}{x \cdot x}}, 0.06172839506172839, \mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}, \mathsf{fma}\left(-0.0411522633744856, \sqrt[3]{\frac{1}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}, \sqrt[3]{x} \cdot -0.1111111111111111\right)\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{{\left(x + 1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 5e-11)
   (* 0.3333333333333333 (/ (cbrt (/ 1.0 x)) (cbrt x)))
   (/
    (+ x (- 1.0 x))
    (+
     (pow (+ x 1.0) 0.6666666666666666)
     (+ (pow x 0.6666666666666666) (cbrt (fma x x x)))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((x + 1.0)) - cbrt(x)) <= 5e-11) {
		tmp = 0.3333333333333333 * (cbrt((1.0 / x)) / cbrt(x));
	} else {
		tmp = (x + (1.0 - x)) / (pow((x + 1.0), 0.6666666666666666) + (pow(x, 0.6666666666666666) + cbrt(fma(x, x, x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(x + 1.0)) - cbrt(x)) <= 5e-11)
		tmp = Float64(0.3333333333333333 * Float64(cbrt(Float64(1.0 / x)) / cbrt(x)));
	else
		tmp = Float64(Float64(x + Float64(1.0 - x)) / Float64((Float64(x + 1.0) ^ 0.6666666666666666) + Float64((x ^ 0.6666666666666666) + cbrt(fma(x, x, x)))));
	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], 5e-11], N[(0.3333333333333333 * N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision] + N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[Power[N[(x * x + x), $MachinePrecision], 1/3], $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)) < 5.00000000000000018e-11

   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 50.3

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

   5. Applied rewrites 50.3%

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.5%

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

   if 5.00000000000000018e-11 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 55.4%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 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. clear-num N/A

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

      6. flip3-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 55.4

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 97.1%

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

Alternative 3: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\left({\left(x + 1\right)}^{0.6666666666666666} + {x}^{0.6666666666666666}\right) + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e+14)
   (/
    (+ x (- 1.0 x))
    (+
     (+ (pow (+ x 1.0) 0.6666666666666666) (pow x 0.6666666666666666))
     (cbrt (fma x x x))))
   (* 0.3333333333333333 (/ (cbrt (/ 1.0 x)) (cbrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 4e+14) {
		tmp = (x + (1.0 - x)) / ((pow((x + 1.0), 0.6666666666666666) + pow(x, 0.6666666666666666)) + cbrt(fma(x, x, x)));
	} else {
		tmp = 0.3333333333333333 * (cbrt((1.0 / x)) / cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e+14)
		tmp = Float64(Float64(x + Float64(1.0 - x)) / Float64(Float64((Float64(x + 1.0) ^ 0.6666666666666666) + (x ^ 0.6666666666666666)) + cbrt(fma(x, x, x))));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(Float64(1.0 / x)) / cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4e+14], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision] + N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision] + N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4e14

   1. Initial program 55.4%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 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. clear-num N/A

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

      6. flip3-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 55.4

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 97.1%

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

   if 4e14 < 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 50.3

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

   5. Applied rewrites 50.3%

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\left({\left(x + 1\right)}^{0.6666666666666666} + {x}^{0.6666666666666666}\right) + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.3333333333333333 (/ (cbrt (/ 1.0 x)) (cbrt x))))

C

double code(double x) {
	return 0.3333333333333333 * (cbrt((1.0 / x)) / cbrt(x));
}

Java

public static double code(double x) {
	return 0.3333333333333333 * (Math.cbrt((1.0 / x)) / Math.cbrt(x));
}

Julia

function code(x)
	return Float64(0.3333333333333333 * Float64(cbrt(Float64(1.0 / x)) / cbrt(x)))
end

Wolfram

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

Derivation

1. Initial program 6.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 50.8

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

5. Applied rewrites 50.8%

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

7. Applied rewrites 96.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 96.8%

    \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{\color{blue}{x}}} \]
11. Add Preprocessing

Alternative 5: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{--1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \end{array} \]

FPCore

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

C

double code(double x) {
	return 0.3333333333333333 * (-(-1.0) / (cbrt(x) * cbrt(x)));
}

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 50.8

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

5. Applied rewrites 50.8%

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

7. Applied rewrites 96.8%

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

9. Applied rewrites 96.7%

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

10. Final simplification 96.7%

    \[\leadsto 0.3333333333333333 \cdot \frac{--1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \]
11. Add Preprocessing

Alternative 6: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 50.8

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

5. Applied rewrites 50.8%

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

7. Applied rewrites 96.7%

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

Alternative 7: 95.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{\sqrt[3]{x} \cdot \mathsf{fma}\left(x, 0.3333333333333333, -0.1111111111111111\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.6e+231)
   (/
    (/ 1.0 x)
    (/ x (* (cbrt x) (fma x 0.3333333333333333 -0.1111111111111111))))
   (/ 0.3333333333333333 (pow x 0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 3.6e+231) {
		tmp = (1.0 / x) / (x / (cbrt(x) * fma(x, 0.3333333333333333, -0.1111111111111111)));
	} else {
		tmp = 0.3333333333333333 / pow(x, 0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.6e+231)
		tmp = Float64(Float64(1.0 / x) / Float64(x / Float64(cbrt(x) * fma(x, 0.3333333333333333, -0.1111111111111111))));
	else
		tmp = Float64(0.3333333333333333 / (x ^ 0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.6e+231], N[(N[(1.0 / x), $MachinePrecision] / N[(x / N[(N[Power[x, 1/3], $MachinePrecision] * N[(x * 0.3333333333333333 + -0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5999999999999999e231

   1. Initial program 7.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cbrt.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 35.3

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

   5. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.9%

      \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x \cdot x} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.0%

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

   if 3.5999999999999999e231 < x

   1. Initial program 5.1%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 88.7%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{0.6666666666666666}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 95.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{x} \cdot \mathsf{fma}\left(x, 0.3333333333333333, -0.1111111111111111\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.6e+231)
   (/ (/ (* (cbrt x) (fma x 0.3333333333333333 -0.1111111111111111)) x) x)
   (/ 0.3333333333333333 (pow x 0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 3.6e+231) {
		tmp = ((cbrt(x) * fma(x, 0.3333333333333333, -0.1111111111111111)) / x) / x;
	} else {
		tmp = 0.3333333333333333 / pow(x, 0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.6e+231)
		tmp = Float64(Float64(Float64(cbrt(x) * fma(x, 0.3333333333333333, -0.1111111111111111)) / x) / x);
	else
		tmp = Float64(0.3333333333333333 / (x ^ 0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.6e+231], N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * N[(x * 0.3333333333333333 + -0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(0.3333333333333333 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5999999999999999e231

   1. Initial program 7.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cbrt.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 35.3

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

   5. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.9%

      \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x \cdot x} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.0%

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

   if 3.5999999999999999e231 < x

   1. Initial program 5.1%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 88.7%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{0.6666666666666666}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 93.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \mathsf{fma}\left(x, 0.3333333333333333, -0.1111111111111111\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ (* (cbrt x) (fma x 0.3333333333333333 -0.1111111111111111)) (* x x))
   (/ 0.3333333333333333 (pow x 0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (cbrt(x) * fma(x, 0.3333333333333333, -0.1111111111111111)) / (x * x);
	} else {
		tmp = 0.3333333333333333 / pow(x, 0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(cbrt(x) * fma(x, 0.3333333333333333, -0.1111111111111111)) / Float64(x * x));
	else
		tmp = Float64(0.3333333333333333 / (x ^ 0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(N[Power[x, 1/3], $MachinePrecision] * N[(x * 0.3333333333333333 + -0.1111111111111111), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cbrt.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 51.8

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

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x \cdot x} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 4.8

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

   5. Applied rewrites 4.8%

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

   7. Applied rewrites 89.1%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{0.6666666666666666}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 92.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \left(0.3333333333333333 \cdot \sqrt[3]{x}\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ (* x (* 0.3333333333333333 (cbrt x))) (* x x))
   (/ 0.3333333333333333 (pow x 0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (x * (0.3333333333333333 * cbrt(x))) / (x * x);
	} else {
		tmp = 0.3333333333333333 / pow(x, 0.6666666666666666);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (x * (0.3333333333333333 * Math.cbrt(x))) / (x * x);
	} else {
		tmp = 0.3333333333333333 / Math.pow(x, 0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(x * Float64(0.3333333333333333 * cbrt(x))) / Float64(x * x));
	else
		tmp = Float64(0.3333333333333333 / (x ^ 0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(x * N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cbrt.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 49.8%

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

   10. Applied rewrites 95.7%

       \[\leadsto \frac{\left(\sqrt[3]{x} \cdot 0.3333333333333333\right) \cdot x}{x \cdot x} \]

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 4.8

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

   5. Applied rewrites 4.8%

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

   7. Applied rewrites 89.1%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{0.6666666666666666}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \left(0.3333333333333333 \cdot \sqrt[3]{x}\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ (* 0.3333333333333333 (* x (cbrt x))) (* x x))
   (/ 0.3333333333333333 (pow x 0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (0.3333333333333333 * (x * cbrt(x))) / (x * x);
	} else {
		tmp = 0.3333333333333333 / pow(x, 0.6666666666666666);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (0.3333333333333333 * (x * Math.cbrt(x))) / (x * x);
	} else {
		tmp = 0.3333333333333333 / Math.pow(x, 0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(0.3333333333333333 * Float64(x * cbrt(x))) / Float64(x * x));
	else
		tmp = Float64(0.3333333333333333 / (x ^ 0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(0.3333333333333333 * N[(x * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cbrt.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 49.8%

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

   10. Applied rewrites 95.6%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot x\right)}{x \cdot x} \]

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 4.8

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

   5. Applied rewrites 4.8%

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

   7. Applied rewrites 89.1%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{0.6666666666666666}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333)
   (/ 0.3333333333333333 (pow x 0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 / pow(x, 0.6666666666666666);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = Math.cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 / Math.pow(x, 0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64(0.3333333333333333 / (x ^ 0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(0.3333333333333333 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.5%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 95.5

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

   5. Applied rewrites 95.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 4.8

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

   5. Applied rewrites 4.8%

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

   7. Applied rewrites 89.1%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{0.6666666666666666}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 88.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot {\left(\frac{1}{x}\right)}^{0.6666666666666666} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.3333333333333333 (pow (/ 1.0 x) 0.6666666666666666)))

C

double code(double x) {
	return 0.3333333333333333 * pow((1.0 / x), 0.6666666666666666);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3333333333333333d0 * ((1.0d0 / x) ** 0.6666666666666666d0)
end function

Java

public static double code(double x) {
	return 0.3333333333333333 * Math.pow((1.0 / x), 0.6666666666666666);
}

Python

def code(x):
	return 0.3333333333333333 * math.pow((1.0 / x), 0.6666666666666666)

Julia

function code(x)
	return Float64(0.3333333333333333 * (Float64(1.0 / x) ^ 0.6666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333 * ((1.0 / x) ^ 0.6666666666666666);
end

Wolfram

code[x_] := N[(0.3333333333333333 * N[Power[N[(1.0 / x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 50.8

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

5. Applied rewrites 50.8%

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

7. Applied rewrites 96.8%

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

9. Applied rewrites 89.1%

   \[\leadsto 0.3333333333333333 \cdot {\left(\frac{1}{x}\right)}^{\color{blue}{0.6666666666666666}} \]
10. Add Preprocessing

Alternative 14: 88.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 50.8

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

5. Applied rewrites 50.8%

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

7. Applied rewrites 89.1%

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

8. Final simplification 89.1%

   \[\leadsto 0.3333333333333333 \cdot {x}^{-0.6666666666666666} \]
9. Add Preprocessing

Alternative 15: 4.1% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 6.6%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. pow-to-exp N/A

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

   5. pow-exp N/A

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

   6. *-commutative N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. rem-log-exp N/A

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

   11. pow-to-exp N/A

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

   12. lift-cbrt.f64 N/A

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

   13. rem-cube-cbrt N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-log1p.f64 5.2

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

4. Applied rewrites 5.2%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{0} \]
6. Step-by-step derivation

7. Applied rewrites 4.1%

   \[\leadsto \color{blue}{0} \]
8. 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 2024222 
(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)))