2cbrt (problem 3.3.4)

Percentage Accurate: 6.7% → 98.4%

Time: 8.2s

Alternatives: 13

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: 6.7% 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: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t\_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\frac{-1}{x}} \cdot 0.3333333333333333}{\sqrt[3]{-x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, t\_0 + \sqrt[3]{x}, {t\_0}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 52.8

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

   5. Applied rewrites 52.8%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 98.5%

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

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

   1. Initial program 52.1%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 50.1

         \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

   4. Applied rewrites 50.1%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
   5. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{x + 1}} - {\left({x}^{\frac{1}{6}}\right)}^{2} \]

      2. pow1/3 N/A

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{\frac{1}{3}}} - {\left({x}^{\frac{1}{6}}\right)}^{2} \]

      3. lower-pow.f64 52.6

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{0.3333333333333333}} - {\left({x}^{0.16666666666666666}\right)}^{2} \]

      4. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + 1\right)}}^{\frac{1}{3}} - {\left({x}^{\frac{1}{6}}\right)}^{2} \]

      5. +-commutative N/A

         \[\leadsto {\color{blue}{\left(1 + x\right)}}^{\frac{1}{3}} - {\left({x}^{\frac{1}{6}}\right)}^{2} \]

      6. lower-+.f64 52.6

         \[\leadsto {\color{blue}{\left(1 + x\right)}}^{0.3333333333333333} - {\left({x}^{0.16666666666666666}\right)}^{2} \]

   6. Applied rewrites 52.6%

      \[\leadsto \color{blue}{{\left(1 + x\right)}^{0.3333333333333333}} - {\left({x}^{0.16666666666666666}\right)}^{2} \]

   8. Applied rewrites 98.8%

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

4. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{if}\;x \leq 3.9 \cdot 10^{+223}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt[3]{x} \cdot x, 0.3333333333333333, \mathsf{fma}\left(0.06172839506172839, t\_0, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (cbrt x) -2.0)))
   (if (<= x 3.9e+223)
     (/
      (/
       (fma
        (* (cbrt x) x)
        0.3333333333333333
        (fma 0.06172839506172839 t_0 (* -0.1111111111111111 (cbrt x))))
       x)
      x)
     (* (/ 1.0 (/ 1.0 t_0)) 0.3333333333333333))))

C

double code(double x) {
	double t_0 = pow(cbrt(x), -2.0);
	double tmp;
	if (x <= 3.9e+223) {
		tmp = (fma((cbrt(x) * x), 0.3333333333333333, fma(0.06172839506172839, t_0, (-0.1111111111111111 * cbrt(x)))) / x) / x;
	} else {
		tmp = (1.0 / (1.0 / t_0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(x) ^ -2.0
	tmp = 0.0
	if (x <= 3.9e+223)
		tmp = Float64(Float64(fma(Float64(cbrt(x) * x), 0.3333333333333333, fma(0.06172839506172839, t_0, Float64(-0.1111111111111111 * cbrt(x)))) / x) / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 / t_0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, If[LessEqual[x, 3.9e+223], N[(N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333 + N[(0.06172839506172839 * t$95$0 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.8999999999999999e223

   1. Initial program 7.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

   5. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.2%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\sqrt[3]{x} \cdot x, 0.3333333333333333, \mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x}}{\color{blue}{x}} \]

   if 3.8999999999999999e223 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 98.6%

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

Alternative 3: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+143}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{x} \cdot x, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{-x}} \cdot \sqrt[3]{\frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3e+143)
   (/
    (fma
     (* (cbrt x) x)
     0.3333333333333333
     (fma
      (cbrt (/ (/ 1.0 x) x))
      0.06172839506172839
      (* -0.1111111111111111 (cbrt x))))
    (* x x))
   (* (/ 0.3333333333333333 (cbrt (- x))) (cbrt (/ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 3e+143) {
		tmp = fma((cbrt(x) * x), 0.3333333333333333, fma(cbrt(((1.0 / x) / x)), 0.06172839506172839, (-0.1111111111111111 * cbrt(x)))) / (x * x);
	} else {
		tmp = (0.3333333333333333 / cbrt(-x)) * cbrt((-1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3e+143)
		tmp = Float64(fma(Float64(cbrt(x) * x), 0.3333333333333333, fma(cbrt(Float64(Float64(1.0 / x) / x)), 0.06172839506172839, Float64(-0.1111111111111111 * cbrt(x)))) / Float64(x * x));
	else
		tmp = Float64(Float64(0.3333333333333333 / cbrt(Float64(-x))) * cbrt(Float64(-1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3e+143], N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333 + N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision] * 0.06172839506172839 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[Power[(-x), 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(-1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.0000000000000001e143

   1. Initial program 9.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

   5. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.1%

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

   if 3.0000000000000001e143 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 12.2

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

   5. Applied rewrites 12.2%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 98.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+143}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{x} \cdot x, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{-x}} \cdot \sqrt[3]{\frac{-1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{+223}:\\ \;\;\;\;\frac{--1}{\frac{x}{\mathsf{fma}\left(\sqrt[3]{x} \cdot x, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{{\left(\sqrt[3]{x}\right)}^{-2}}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.9e+223)
   (/
    (- -1.0)
    (*
     (/
      x
      (fma (* (cbrt x) x) 0.3333333333333333 (* -0.1111111111111111 (cbrt x))))
     x))
   (* (/ 1.0 (/ 1.0 (pow (cbrt x) -2.0))) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 3.9e+223) {
		tmp = -(-1.0) / ((x / fma((cbrt(x) * x), 0.3333333333333333, (-0.1111111111111111 * cbrt(x)))) * x);
	} else {
		tmp = (1.0 / (1.0 / pow(cbrt(x), -2.0))) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.9e+223)
		tmp = Float64(Float64(-(-1.0)) / Float64(Float64(x / fma(Float64(cbrt(x) * x), 0.3333333333333333, Float64(-0.1111111111111111 * cbrt(x)))) * x));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 / (cbrt(x) ^ -2.0))) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.9e+223], N[((--1.0) / N[(N[(x / N[(N[(N[Power[x, 1/3], $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 / N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.8999999999999999e223

   1. Initial program 7.8%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 10.2

         \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

   4. Applied rewrites 10.2%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
   5. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{x + 1}} - {\left({x}^{\frac{1}{6}}\right)}^{2} \]

      2. pow1/3 N/A

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{\frac{1}{3}}} - {\left({x}^{\frac{1}{6}}\right)}^{2} \]

      3. lower-pow.f64 7.9

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{0.3333333333333333}} - {\left({x}^{0.16666666666666666}\right)}^{2} \]

      4. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + 1\right)}}^{\frac{1}{3}} - {\left({x}^{\frac{1}{6}}\right)}^{2} \]

      5. +-commutative N/A

         \[\leadsto {\color{blue}{\left(1 + x\right)}}^{\frac{1}{3}} - {\left({x}^{\frac{1}{6}}\right)}^{2} \]

      6. lower-+.f64 7.9

         \[\leadsto {\color{blue}{\left(1 + x\right)}}^{0.3333333333333333} - {\left({x}^{0.16666666666666666}\right)}^{2} \]

   6. Applied rewrites 7.9%

      \[\leadsto \color{blue}{{\left(1 + x\right)}^{0.3333333333333333}} - {\left({x}^{0.16666666666666666}\right)}^{2} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \]
   8. 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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 41.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{\color{blue}{x \cdot x}} \]

   9. Applied rewrites 41.5%

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

   11. Applied rewrites 98.1%

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

   if 3.8999999999999999e223 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 98.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{+223}:\\ \;\;\;\;\frac{--1}{\frac{x}{\mathsf{fma}\left(\sqrt[3]{x} \cdot x, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{{\left(\sqrt[3]{x}\right)}^{-2}}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{+223}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt[3]{x} \cdot x, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{{\left(\sqrt[3]{x}\right)}^{-2}}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.9e+223)
   (/
    (/
     (fma (* (cbrt x) x) 0.3333333333333333 (* -0.1111111111111111 (cbrt x)))
     x)
    x)
   (* (/ 1.0 (/ 1.0 (pow (cbrt x) -2.0))) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 3.9e+223) {
		tmp = (fma((cbrt(x) * x), 0.3333333333333333, (-0.1111111111111111 * cbrt(x))) / x) / x;
	} else {
		tmp = (1.0 / (1.0 / pow(cbrt(x), -2.0))) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.9e+223)
		tmp = Float64(Float64(fma(Float64(cbrt(x) * x), 0.3333333333333333, Float64(-0.1111111111111111 * cbrt(x))) / x) / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 / (cbrt(x) ^ -2.0))) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.9e+223], N[(N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 / N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.8999999999999999e223

   1. Initial program 7.8%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 10.2

         \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

   4. Applied rewrites 10.2%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
   5. Step-by-step derivation

      1. lift-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{x + 1}} - {\left({x}^{\frac{1}{6}}\right)}^{2} \]

      2. pow1/3 N/A

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{\frac{1}{3}}} - {\left({x}^{\frac{1}{6}}\right)}^{2} \]

      3. lower-pow.f64 7.9

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{0.3333333333333333}} - {\left({x}^{0.16666666666666666}\right)}^{2} \]

      4. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(x + 1\right)}}^{\frac{1}{3}} - {\left({x}^{\frac{1}{6}}\right)}^{2} \]

      5. +-commutative N/A

         \[\leadsto {\color{blue}{\left(1 + x\right)}}^{\frac{1}{3}} - {\left({x}^{\frac{1}{6}}\right)}^{2} \]

      6. lower-+.f64 7.9

         \[\leadsto {\color{blue}{\left(1 + x\right)}}^{0.3333333333333333} - {\left({x}^{0.16666666666666666}\right)}^{2} \]

   6. Applied rewrites 7.9%

      \[\leadsto \color{blue}{{\left(1 + x\right)}^{0.3333333333333333}} - {\left({x}^{0.16666666666666666}\right)}^{2} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \]
   8. 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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 41.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{\color{blue}{x \cdot x}} \]

   9. Applied rewrites 41.5%

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

   11. Applied rewrites 98.0%

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

   if 3.8999999999999999e223 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 98.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{+223}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt[3]{x} \cdot x, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{{\left(\sqrt[3]{x}\right)}^{-2}}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 53.6

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

5. Applied rewrites 53.6%

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

7. Applied rewrites 96.6%

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

9. Applied rewrites 96.6%

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

10. Final simplification 96.6%

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

Alternative 7: 96.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 53.6

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

5. Applied rewrites 53.6%

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

7. Applied rewrites 96.5%

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

9. Applied rewrites 96.6%

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

Alternative 8: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 53.6

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

5. Applied rewrites 53.6%

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

7. Applied rewrites 96.6%

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

Alternative 9: 92.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.9%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

   7. Applied rewrites 94.9%

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

   9. Applied rewrites 95.4%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 6.5

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

   5. Applied rewrites 6.5%

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

   7. Applied rewrites 89.0%

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

Alternative 10: 92.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.9%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

   7. Applied rewrites 94.9%

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

   9. Applied rewrites 95.3%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 6.5

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

   5. Applied rewrites 6.5%

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

   7. Applied rewrites 89.0%

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

Alternative 11: 92.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	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 = Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	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((x ^ -0.6666666666666666) * 0.3333333333333333);
	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[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.9%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

   7. Applied rewrites 95.1%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 6.5

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

   5. Applied rewrites 6.5%

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

   7. Applied rewrites 89.0%

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

Alternative 12: 89.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 53.6

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

5. Applied rewrites 53.6%

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

7. Applied rewrites 89.0%

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

Alternative 13: 4.2% 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 7.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 5.6

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

4. Applied rewrites 5.6%

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

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 2024242 
(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)))