2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 50.5%

Time: 8.2s

Alternatives: 6

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: 6.9% 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: 50.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (cbrt (/ 1.0 (pow x 5.0)))
  -0.1111111111111111
  (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333)))

C

double code(double x) {
	return fma(cbrt((1.0 / pow(x, 5.0))), -0.1111111111111111, (cbrt((1.0 / (x * x))) * 0.3333333333333333));
}

Julia

function code(x)
	return fma(cbrt(Float64(1.0 / (x ^ 5.0))), -0.1111111111111111, Float64(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333))
end

Wolfram

code[x_] := N[(N[Power[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.4%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cbrt.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 24.8

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

5. Simplified 24.8%

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

6. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 52.2

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

8. Simplified 52.2%

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

9. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-cbrt.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. lower-pow.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. lower-cbrt.f64 N/A

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

    9. lower-/.f64 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f64 52.2

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

11. Simplified 52.2%

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

Alternative 2: 50.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (cbrt (/ 1.0 (* x x)))
  0.3333333333333333
  (* (cbrt (/ 1.0 (pow x 5.0))) -0.1111111111111111)))

C

double code(double x) {
	return fma(cbrt((1.0 / (x * x))), 0.3333333333333333, (cbrt((1.0 / pow(x, 5.0))) * -0.1111111111111111));
}

Julia

function code(x)
	return fma(cbrt(Float64(1.0 / Float64(x * x))), 0.3333333333333333, Float64(cbrt(Float64(1.0 / (x ^ 5.0))) * -0.1111111111111111))
end

Wolfram

code[x_] := N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333 + N[(N[Power[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.4%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cbrt.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 24.8

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

5. Simplified 24.8%

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

6. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 52.2

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

8. Simplified 52.2%

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

9. Final simplification 52.2%

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

Alternative 3: 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 7.4%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-cbrt.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 51.1

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

5. Simplified 51.1%

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

6. Final simplification 51.1%

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

Alternative 4: 4.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-cbrt.f64 4.3

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

5. Simplified 4.3%

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

6. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-cbrt.f64 4.3

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

8. Simplified 4.3%

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

Alternative 5: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-cbrt.f64 1.8

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

5. Simplified 1.8%

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

Alternative 6: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-cbrt.f64 1.8

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

5. Simplified 1.8%

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

6. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)} \]

   3. lower-cbrt.f64 1.8

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

8. Simplified 1.8%

   \[\leadsto \color{blue}{-\sqrt[3]{x}} \]
9. 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 2024215 
(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)))