2cbrt (problem 3.3.4)

Percentage Accurate: 6.8% → 98.1%

Time: 2.8s

Alternatives: 10

Speedup: 1.7×

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: 6.8% 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: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{{x}^{-1}}\\ t_1 := \sqrt[3]{x - -1}\\ \mathbf{if}\;x \leq 10^{+15}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{{t\_1}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (pow x -1.0))) (t_1 (cbrt (- x -1.0))))
   (if (<= x 1e+15)
     (/ (- (- x -1.0) x) (+ (pow t_1 2.0) (* (cbrt x) (+ (cbrt x) t_1))))
     (* (* t_0 t_0) 0.3333333333333333))))

C

double code(double x) {
	double t_0 = cbrt(pow(x, -1.0));
	double t_1 = cbrt((x - -1.0));
	double tmp;
	if (x <= 1e+15) {
		tmp = ((x - -1.0) - x) / (pow(t_1, 2.0) + (cbrt(x) * (cbrt(x) + t_1)));
	} else {
		tmp = (t_0 * t_0) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(Math.pow(x, -1.0));
	double t_1 = Math.cbrt((x - -1.0));
	double tmp;
	if (x <= 1e+15) {
		tmp = ((x - -1.0) - x) / (Math.pow(t_1, 2.0) + (Math.cbrt(x) * (Math.cbrt(x) + t_1)));
	} else {
		tmp = (t_0 * t_0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt((x ^ -1.0))
	t_1 = cbrt(Float64(x - -1.0))
	tmp = 0.0
	if (x <= 1e+15)
		tmp = Float64(Float64(Float64(x - -1.0) - x) / Float64((t_1 ^ 2.0) + Float64(cbrt(x) * Float64(cbrt(x) + t_1))));
	else
		tmp = Float64(Float64(t_0 * t_0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[x, -1.0], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 1e+15], N[(N[(N[(x - -1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e15

   1. Initial program 57.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. rem-cube-cbrt N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-+.f64 N/A

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

   3. Applied rewrites 98.6%

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

   if 1e15 < x

   1. Initial program 4.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 50.4

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

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 50.4

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

   6. Applied rewrites 50.4%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. inv-pow N/A

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

      6. inv-pow N/A

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

      7. cbrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. inv-pow N/A

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

      11. lift-pow.f64 N/A

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

      12. lower-cbrt.f64 N/A

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

      13. inv-pow N/A

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

      14. lift-pow.f64 98.1

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

   8. Applied rewrites 98.1%

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

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{{x}^{-1}}\\ \mathbf{if}\;x \leq 10^{+65}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (pow x -1.0))))
   (if (<= x 1e+65)
     (/
      (fma
       (cbrt (pow x -2.0))
       0.06172839506172839
       (fma
        (cbrt (* (* x x) (* x x)))
        0.3333333333333333
        (* -0.1111111111111111 (cbrt x))))
      (* x x))
     (* (* t_0 t_0) 0.3333333333333333))))

C

double code(double x) {
	double t_0 = cbrt(pow(x, -1.0));
	double tmp;
	if (x <= 1e+65) {
		tmp = fma(cbrt(pow(x, -2.0)), 0.06172839506172839, fma(cbrt(((x * x) * (x * x))), 0.3333333333333333, (-0.1111111111111111 * cbrt(x)))) / (x * x);
	} else {
		tmp = (t_0 * t_0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt((x ^ -1.0))
	tmp = 0.0
	if (x <= 1e+65)
		tmp = Float64(fma(cbrt((x ^ -2.0)), 0.06172839506172839, fma(cbrt(Float64(Float64(x * x) * Float64(x * x))), 0.3333333333333333, Float64(-0.1111111111111111 * cbrt(x)))) / Float64(x * x));
	else
		tmp = Float64(Float64(t_0 * t_0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[x, -1.0], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 1e+65], N[(N[(N[Power[N[Power[x, -2.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.06172839506172839 + N[(N[Power[N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999999e64

   1. Initial program 15.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      2. sqr-pow N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{2} \cdot {x}^{2}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{2} \cdot {x}^{2}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{\left(x \cdot x\right) \cdot {x}^{2}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{\left(x \cdot x\right) \cdot {x}^{2}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      8. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      9. lift-*.f64 96.6

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

   6. Applied rewrites 96.6%

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

   if 9.9999999999999999e64 < x

   1. Initial program 4.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 40.3

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

   4. Applied rewrites 40.3%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 40.3

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

   6. Applied rewrites 40.3%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. inv-pow N/A

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

      6. inv-pow N/A

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

      7. cbrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. inv-pow N/A

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

      11. lift-pow.f64 N/A

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

      12. lower-cbrt.f64 N/A

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

      13. inv-pow N/A

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

      14. lift-pow.f64 98.1

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

   8. Applied rewrites 98.1%

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

Alternative 3: 96.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{{x}^{-1}}\\ \left(t\_0 \cdot t\_0\right) \cdot 0.3333333333333333 \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (pow x -1.0)))) (* (* t_0 t_0) 0.3333333333333333)))

C

double code(double x) {
	double t_0 = cbrt(pow(x, -1.0));
	return (t_0 * t_0) * 0.3333333333333333;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(Math.pow(x, -1.0));
	return (t_0 * t_0) * 0.3333333333333333;
}

Julia

function code(x)
	t_0 = cbrt((x ^ -1.0))
	return Float64(Float64(t_0 * t_0) * 0.3333333333333333)
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 51.0

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

4. Applied rewrites 51.0%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. inv-pow N/A

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

   7. lower-*.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 51.0

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

6. Applied rewrites 51.0%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. inv-pow N/A

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

   6. inv-pow N/A

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

   7. cbrt-prod N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. inv-pow N/A

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

   11. lift-pow.f64 N/A

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

   12. lower-cbrt.f64 N/A

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

   13. inv-pow N/A

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

   14. lift-pow.f64 96.3

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

8. Applied rewrites 96.3%

   \[\leadsto \left(\sqrt[3]{{x}^{-1}} \cdot \sqrt[3]{{x}^{-1}}\right) \cdot 0.3333333333333333 \]
9. Add Preprocessing

Alternative 4: 93.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \frac{1}{\sqrt[3]{x \cdot x}} \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log x \cdot -2\right) \cdot 0.3333333333333333} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (fma
    (cbrt (pow x -5.0))
    -0.1111111111111111
    (* (/ 1.0 (cbrt (* x x))) 0.3333333333333333))
   (* (exp (* (* (log x) -2.0) 0.3333333333333333)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = fma(cbrt(pow(x, -5.0)), -0.1111111111111111, ((1.0 / cbrt((x * x))) * 0.3333333333333333));
	} else {
		tmp = exp(((log(x) * -2.0) * 0.3333333333333333)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = fma(cbrt((x ^ -5.0)), -0.1111111111111111, Float64(Float64(1.0 / cbrt(Float64(x * x))) * 0.3333333333333333));
	else
		tmp = Float64(exp(Float64(Float64(log(x) * -2.0) * 0.3333333333333333)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[Power[N[Power[x, -5.0], $MachinePrecision], 1/3], $MachinePrecision] * -0.1111111111111111 + N[(N[(1.0 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[x], $MachinePrecision] * -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $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. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-flip N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-cbrt.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 97.1

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]

   7. Applied rewrites 97.1%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right) \]
   8. Step-by-step derivation

      1. lift-cbrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 97.3

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \frac{1}{\sqrt[3]{x \cdot x}} \cdot 0.3333333333333333\right) \]

   9. Applied rewrites 97.3%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \frac{1}{\sqrt[3]{x \cdot x}} \cdot 0.3333333333333333\right) \]

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 7.6

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

   4. Applied rewrites 7.6%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 7.6

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

   6. Applied rewrites 7.6%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. exp-to-pow N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-up N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-exp.f64 7.6

          \[\leadsto e^{\log \left({x}^{-2}\right) \cdot 0.3333333333333333} \cdot 0.3333333333333333 \]

      13. lift-log.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. log-pow-rev N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-log.f64 89.5

          \[\leadsto e^{\left(\log x \cdot -2\right) \cdot 0.3333333333333333} \cdot 0.3333333333333333 \]

   8. Applied rewrites 89.5%

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

Alternative 5: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log x \cdot -2\right) \cdot 0.3333333333333333} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* (/ 1.0 (cbrt (* x x))) 0.3333333333333333)
   (* (exp (* (* (log x) -2.0) 0.3333333333333333)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (1.0 / cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = exp(((log(x) * -2.0) * 0.3333333333333333)) * 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.exp(((Math.log(x) * -2.0) * 0.3333333333333333)) * 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(exp(Float64(Float64(log(x) * -2.0) * 0.3333333333333333)) * 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[Exp[N[(N[(N[Log[x], $MachinePrecision] * -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $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. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 95.1

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

   4. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-cbrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 95.3

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

   6. Applied rewrites 95.3%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 7.6

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

   4. Applied rewrites 7.6%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 7.6

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

   6. Applied rewrites 7.6%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. exp-to-pow N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-up N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-exp.f64 7.6

          \[\leadsto e^{\log \left({x}^{-2}\right) \cdot 0.3333333333333333} \cdot 0.3333333333333333 \]

      13. lift-log.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. log-pow-rev N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-log.f64 89.5

          \[\leadsto e^{\left(\log x \cdot -2\right) \cdot 0.3333333333333333} \cdot 0.3333333333333333 \]

   8. Applied rewrites 89.5%

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

Alternative 6: 51.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 51.0

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

4. Applied rewrites 51.0%

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

Alternative 7: 49.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 51.0

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

4. Applied rewrites 51.0%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
5. Step-by-step derivation

   1. lift-cbrt.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-flip N/A

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

   5. cbrt-div N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 49.6

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

6. Applied rewrites 49.6%

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

Alternative 8: 49.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 51.0

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

4. Applied rewrites 51.0%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-flip N/A

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

   4. lower-/.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 49.5

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

6. Applied rewrites 49.5%

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

Alternative 9: 4.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lift-cbrt.f64 4.2

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

4. Applied rewrites 4.2%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\frac{1}{3}, x, -1 \cdot \sqrt[3]{x}\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, x, \mathsf{neg}\left(\sqrt[3]{x}\right)\right) \]

   2. lower-neg.f64 N/A

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

   3. lift-cbrt.f64 4.2

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

7. Applied rewrites 4.2%

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

Alternative 10: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 1.8%

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

Developer Target 1: 98.5% 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 2025101 
(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)))