2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 99.4%

Time: 14.9s

Alternatives: 12

Speedup: 1.0×

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: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\\ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{1 + x}{t\_0 + 0 \cdot \sqrt[3]{1 + x}} + \frac{x}{{\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x} \cdot 0}, t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* 0.6666666666666666 (log1p x)))))
   (/
    1.0
    (fma
     (cbrt x)
     (+
      (/ (+ 1.0 x) (+ t_0 (* 0.0 (cbrt (+ 1.0 x)))))
      (/ x (+ (pow (cbrt x) 2.0) (* (cbrt x) 0.0))))
     t_0))))

C

double code(double x) {
	double t_0 = exp((0.6666666666666666 * log1p(x)));
	return 1.0 / fma(cbrt(x), (((1.0 + x) / (t_0 + (0.0 * cbrt((1.0 + x))))) + (x / (pow(cbrt(x), 2.0) + (cbrt(x) * 0.0)))), t_0);
}

Julia

function code(x)
	t_0 = exp(Float64(0.6666666666666666 * log1p(x)))
	return Float64(1.0 / fma(cbrt(x), Float64(Float64(Float64(1.0 + x) / Float64(t_0 + Float64(0.0 * cbrt(Float64(1.0 + x))))) + Float64(x / Float64((cbrt(x) ^ 2.0) + Float64(cbrt(x) * 0.0)))), t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[(N[(1.0 + x), $MachinePrecision] / N[(t$95$0 + N[(0.0 * N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.8%

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

   1. flip3-- 6.7%

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

   2. rem-cube-cbrt 5.9%

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

   3. rem-cube-cbrt 8.3%

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

   4. +-commutative 8.3%

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

   5. distribute-rgt-out 8.3%

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

   6. +-commutative 8.3%

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

   7. fma-define 8.3%

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

   8. add-exp-log 8.3%

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

   9. pow1/3 8.3%

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

   10. pow1/3 8.3%

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

   11. pow-prod-up 8.3%

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

4. Applied egg-rr 8.3%

   \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]

5. Taylor expanded in x around 0 93.0%

   \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]
6. Step-by-step derivation

   1. add0 93.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\left(\sqrt[3]{x + 1} + 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   2. flip3-+ 93.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} + {0}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)}} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   3. metadata-eval 93.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{{\left(\sqrt[3]{x + 1}\right)}^{3} + \color{blue}{0}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   4. add0 93.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   5. rem-cube-cbrt 93.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{\color{blue}{x + 1}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   6. add-exp-log 95.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{\color{blue}{e^{\log \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   7. pow1/3 96.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{\log \left(\color{blue}{{\left(x + 1\right)}^{0.3333333333333333}} \cdot \sqrt[3]{x + 1}\right)} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   8. pow1/3 95.9%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{\log \left({\left(x + 1\right)}^{0.3333333333333333} \cdot \color{blue}{{\left(x + 1\right)}^{0.3333333333333333}}\right)} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   9. pow-prod-up 95.9%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{\log \color{blue}{\left({\left(x + 1\right)}^{\left(0.3333333333333333 + 0.3333333333333333\right)}\right)}} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   10. log-pow 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{\color{blue}{\left(0.3333333333333333 + 0.3333333333333333\right) \cdot \log \left(x + 1\right)}} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   11. metadata-eval 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{\color{blue}{0.6666666666666666} \cdot \log \left(x + 1\right)} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   12. +-commutative 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \log \color{blue}{\left(1 + x\right)}} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   13. log1p-define 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   14. metadata-eval 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \left(\color{blue}{0} - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

7. Applied egg-rr 98.9%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\frac{x + 1}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \left(0 - \sqrt[3]{x + 1} \cdot 0\right)}} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]
8. Step-by-step derivation

   1. add0 98.9%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \left(0 - \sqrt[3]{x + 1} \cdot 0\right)} + \color{blue}{\left(\sqrt[3]{x} + 0\right)}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   2. flip3-+ 98.9%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \left(0 - \sqrt[3]{x + 1} \cdot 0\right)} + \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{3} + {0}^{3}}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \left(0 \cdot 0 - \sqrt[3]{x} \cdot 0\right)}}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   3. metadata-eval 98.9%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \left(0 - \sqrt[3]{x + 1} \cdot 0\right)} + \frac{{\left(\sqrt[3]{x}\right)}^{3} + \color{blue}{0}}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \left(0 \cdot 0 - \sqrt[3]{x} \cdot 0\right)}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   4. add0 98.9%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \left(0 - \sqrt[3]{x + 1} \cdot 0\right)} + \frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{3}}}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \left(0 \cdot 0 - \sqrt[3]{x} \cdot 0\right)}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   5. rem-cube-cbrt 99.5%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \left(0 - \sqrt[3]{x + 1} \cdot 0\right)} + \frac{\color{blue}{x}}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \left(0 \cdot 0 - \sqrt[3]{x} \cdot 0\right)}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   6. pow2 99.5%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \left(0 - \sqrt[3]{x + 1} \cdot 0\right)} + \frac{x}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} + \left(0 \cdot 0 - \sqrt[3]{x} \cdot 0\right)}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   7. metadata-eval 99.5%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \left(0 - \sqrt[3]{x + 1} \cdot 0\right)} + \frac{x}{{\left(\sqrt[3]{x}\right)}^{2} + \left(\color{blue}{0} - \sqrt[3]{x} \cdot 0\right)}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

9. Applied egg-rr 99.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \left(0 - \sqrt[3]{x + 1} \cdot 0\right)} + \color{blue}{\frac{x}{{\left(\sqrt[3]{x}\right)}^{2} + \left(0 - \sqrt[3]{x} \cdot 0\right)}}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

10. Final simplification 99.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{1 + x}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + 0 \cdot \sqrt[3]{1 + x}} + \frac{x}{{\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x} \cdot 0}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]
11. Add Preprocessing

Alternative 2: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\\ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \frac{1 + x}{t\_0 + 0 \cdot \sqrt[3]{1 + x}}, t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* 0.6666666666666666 (log1p x)))))
   (/
    1.0
    (fma
     (cbrt x)
     (+ (cbrt x) (/ (+ 1.0 x) (+ t_0 (* 0.0 (cbrt (+ 1.0 x))))))
     t_0))))

C

double code(double x) {
	double t_0 = exp((0.6666666666666666 * log1p(x)));
	return 1.0 / fma(cbrt(x), (cbrt(x) + ((1.0 + x) / (t_0 + (0.0 * cbrt((1.0 + x)))))), t_0);
}

Julia

function code(x)
	t_0 = exp(Float64(0.6666666666666666 * log1p(x)))
	return Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + Float64(Float64(1.0 + x) / Float64(t_0 + Float64(0.0 * cbrt(Float64(1.0 + x)))))), t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] / N[(t$95$0 + N[(0.0 * N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.8%

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

   1. flip3-- 6.7%

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

   2. rem-cube-cbrt 5.9%

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

   3. rem-cube-cbrt 8.3%

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

   4. +-commutative 8.3%

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

   5. distribute-rgt-out 8.3%

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

   6. +-commutative 8.3%

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

   7. fma-define 8.3%

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

   8. add-exp-log 8.3%

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

   9. pow1/3 8.3%

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

   10. pow1/3 8.3%

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

   11. pow-prod-up 8.3%

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

4. Applied egg-rr 8.3%

   \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]

5. Taylor expanded in x around 0 93.0%

   \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]
6. Step-by-step derivation

   1. add0 93.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\left(\sqrt[3]{x + 1} + 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   2. flip3-+ 93.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} + {0}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)}} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   3. metadata-eval 93.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{{\left(\sqrt[3]{x + 1}\right)}^{3} + \color{blue}{0}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   4. add0 93.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{3}}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   5. rem-cube-cbrt 93.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{\color{blue}{x + 1}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   6. add-exp-log 95.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{\color{blue}{e^{\log \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   7. pow1/3 96.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{\log \left(\color{blue}{{\left(x + 1\right)}^{0.3333333333333333}} \cdot \sqrt[3]{x + 1}\right)} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   8. pow1/3 95.9%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{\log \left({\left(x + 1\right)}^{0.3333333333333333} \cdot \color{blue}{{\left(x + 1\right)}^{0.3333333333333333}}\right)} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   9. pow-prod-up 95.9%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{\log \color{blue}{\left({\left(x + 1\right)}^{\left(0.3333333333333333 + 0.3333333333333333\right)}\right)}} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   10. log-pow 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{\color{blue}{\left(0.3333333333333333 + 0.3333333333333333\right) \cdot \log \left(x + 1\right)}} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   11. metadata-eval 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{\color{blue}{0.6666666666666666} \cdot \log \left(x + 1\right)} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   12. +-commutative 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \log \color{blue}{\left(1 + x\right)}} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   13. log1p-define 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} + \left(0 \cdot 0 - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   14. metadata-eval 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{x + 1}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \left(\color{blue}{0} - \sqrt[3]{x + 1} \cdot 0\right)} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

7. Applied egg-rr 98.9%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\frac{x + 1}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \left(0 - \sqrt[3]{x + 1} \cdot 0\right)}} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

8. Final simplification 98.9%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \frac{1 + x}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + 0 \cdot \sqrt[3]{1 + x}}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]
9. Add Preprocessing

Alternative 3: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x} + \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(1 + x\right)}^{2}} + \sqrt[3]{x} \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t\_0, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (cbrt x) (cbrt (+ 1.0 x)))))
   (if (<= x 5e+153)
     (/ 1.0 (+ (cbrt (pow (+ 1.0 x) 2.0)) (* (cbrt x) t_0)))
     (/ 1.0 (fma (cbrt x) t_0 (exp (* 0.6666666666666666 (log1p x))))))))

C

double code(double x) {
	double t_0 = cbrt(x) + cbrt((1.0 + x));
	double tmp;
	if (x <= 5e+153) {
		tmp = 1.0 / (cbrt(pow((1.0 + x), 2.0)) + (cbrt(x) * t_0));
	} else {
		tmp = 1.0 / fma(cbrt(x), t_0, exp((0.6666666666666666 * log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(x) + cbrt(Float64(1.0 + x)))
	tmp = 0.0
	if (x <= 5e+153)
		tmp = Float64(1.0 / Float64(cbrt((Float64(1.0 + x) ^ 2.0)) + Float64(cbrt(x) * t_0)));
	else
		tmp = Float64(1.0 / fma(cbrt(x), t_0, exp(Float64(0.6666666666666666 * log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e+153], N[(1.0 / N[(N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0 + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000018e153

   1. Initial program 8.9%

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

      1. expm1-log1p-u 9.0%

         \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)} \]

   4. Applied egg-rr 9.0%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)} \]

   6. Applied egg-rr 12.1%

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

   7. Taylor expanded in x around 0 98.8%

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

   if 5.00000000000000018e153 < x

   1. Initial program 4.8%

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

      1. flip3-- 4.8%

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

      2. rem-cube-cbrt 2.9%

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

      3. rem-cube-cbrt 4.8%

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

      4. +-commutative 4.8%

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

      5. distribute-rgt-out 4.8%

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

      6. +-commutative 4.8%

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

      7. fma-define 4.8%

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

      8. add-exp-log 4.8%

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

      9. pow1/3 4.8%

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

      10. pow1/3 4.8%

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

      11. pow-prod-up 4.8%

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

   4. Applied egg-rr 4.8%

      \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]

   5. Taylor expanded in x around 0 92.2%

      \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(1 + x\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t\_0, {t\_0}^{2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (/ 1.0 (fma (cbrt x) (+ (cbrt x) t_0) (pow t_0 2.0)))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	return 1.0 / fma(cbrt(x), (cbrt(x) + t_0), pow(t_0, 2.0));
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	return Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + t_0), (t_0 ^ 2.0)))
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. flip3-- 6.7%

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

   2. rem-cube-cbrt 5.9%

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

   3. rem-cube-cbrt 8.3%

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

   4. +-commutative 8.3%

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

   5. distribute-rgt-out 8.3%

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

   6. +-commutative 8.3%

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

   7. fma-define 8.3%

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

   8. add-exp-log 8.3%

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

   9. pow1/3 8.3%

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

   10. pow1/3 8.3%

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

   11. pow-prod-up 8.3%

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

4. Applied egg-rr 8.3%

   \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]

5. Taylor expanded in x around 0 93.0%

   \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]
6. Step-by-step derivation

   1. *-commutative 93.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}}\right)} \]

   2. log1p-undefine 93.0%

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

   3. +-commutative 93.0%

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

   4. exp-to-pow 92.9%

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

   5. metadata-eval 92.9%

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

   6. pow-prod-up 92.9%

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

   7. pow1/3 94.3%

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

   8. pow1/3 98.4%

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

   9. pow2 98.4%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
9. Add Preprocessing

Alternative 5: 95.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x} + \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(1 + x\right)}^{2}} + \sqrt[3]{x} \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t\_0, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (cbrt x) (cbrt (+ 1.0 x)))))
   (if (<= x 1.35e+154)
     (/ 1.0 (+ (cbrt (pow (+ 1.0 x) 2.0)) (* (cbrt x) t_0)))
     (/ 1.0 (fma (cbrt x) t_0 (pow (+ 1.0 x) 0.6666666666666666))))))

C

double code(double x) {
	double t_0 = cbrt(x) + cbrt((1.0 + x));
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (cbrt(pow((1.0 + x), 2.0)) + (cbrt(x) * t_0));
	} else {
		tmp = 1.0 / fma(cbrt(x), t_0, pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(x) + cbrt(Float64(1.0 + x)))
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(1.0 / Float64(cbrt((Float64(1.0 + x) ^ 2.0)) + Float64(cbrt(x) * t_0)));
	else
		tmp = Float64(1.0 / fma(cbrt(x), t_0, (Float64(1.0 + x) ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.35e+154], N[(1.0 / N[(N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0 + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.8%

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

      1. expm1-log1p-u 8.9%

         \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)} \]

   4. Applied egg-rr 8.9%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)} \]

   6. Applied egg-rr 12.0%

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

   7. Taylor expanded in x around 0 98.8%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

      1. flip3-- 4.8%

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

      2. rem-cube-cbrt 2.9%

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

      3. rem-cube-cbrt 4.8%

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

      4. +-commutative 4.8%

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

      5. distribute-rgt-out 4.8%

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

      6. +-commutative 4.8%

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

      7. fma-define 4.8%

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

      8. add-exp-log 4.8%

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

      9. pow1/3 4.8%

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

      10. pow1/3 4.8%

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

      11. pow-prod-up 4.8%

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

   4. Applied egg-rr 4.8%

      \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]

   5. Taylor expanded in x around 0 92.2%

      \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 92.2%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}}\right)} \]

      2. log1p-undefine 92.2%

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

      3. +-commutative 92.2%

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

      4. exp-to-pow 91.6%

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

   7. Applied egg-rr 91.6%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(1 + x\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(1 + x\right)}^{0.6666666666666666}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (fma
   (cbrt x)
   (+ (cbrt x) (cbrt (+ 1.0 x)))
   (pow (+ 1.0 x) 0.6666666666666666))))

C

double code(double x) {
	return 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((1.0 + x))), pow((1.0 + x), 0.6666666666666666));
}

Julia

function code(x)
	return Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(1.0 + x))), (Float64(1.0 + x) ^ 0.6666666666666666)))
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. flip3-- 6.7%

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

   2. rem-cube-cbrt 5.9%

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

   3. rem-cube-cbrt 8.3%

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

   4. +-commutative 8.3%

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

   5. distribute-rgt-out 8.3%

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

   6. +-commutative 8.3%

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

   7. fma-define 8.3%

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

   8. add-exp-log 8.3%

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

   9. pow1/3 8.3%

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

   10. pow1/3 8.3%

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

   11. pow-prod-up 8.3%

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

4. Applied egg-rr 8.3%

   \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]

5. Taylor expanded in x around 0 93.0%

   \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]
6. Step-by-step derivation

   1. *-commutative 93.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}}\right)} \]

   2. log1p-undefine 93.0%

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

   3. +-commutative 93.0%

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

   4. exp-to-pow 92.9%

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

7. Applied egg-rr 92.9%

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

8. Final simplification 92.9%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(1 + x\right)}^{0.6666666666666666}\right)} \]
9. Add Preprocessing

Alternative 7: 12.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x} + \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq 1.35 \cdot 10^{+16}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\sqrt[3]{x} \cdot t\_0 + {\left(1 + x\right)}^{0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t\_0, 1 + x \cdot 0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (cbrt x) (cbrt (+ 1.0 x)))))
   (if (<= x 1.35e+16)
     (/
      (- (+ 1.0 x) x)
      (+ (* (cbrt x) t_0) (pow (+ 1.0 x) 0.6666666666666666)))
     (/ 1.0 (fma (cbrt x) t_0 (+ 1.0 (* x 0.6666666666666666)))))))

C

double code(double x) {
	double t_0 = cbrt(x) + cbrt((1.0 + x));
	double tmp;
	if (x <= 1.35e+16) {
		tmp = ((1.0 + x) - x) / ((cbrt(x) * t_0) + pow((1.0 + x), 0.6666666666666666));
	} else {
		tmp = 1.0 / fma(cbrt(x), t_0, (1.0 + (x * 0.6666666666666666)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(x) + cbrt(Float64(1.0 + x)))
	tmp = 0.0
	if (x <= 1.35e+16)
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64(Float64(cbrt(x) * t_0) + (Float64(1.0 + x) ^ 0.6666666666666666)));
	else
		tmp = Float64(1.0 / fma(cbrt(x), t_0, Float64(1.0 + Float64(x * 0.6666666666666666))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.35e+16], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0 + N[(1.0 + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e16

   1. Initial program 62.8%

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

      1. expm1-log1p-u 60.5%

         \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)} \]

   4. Applied egg-rr 60.5%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)} \]

   6. Applied egg-rr 99.1%

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

      1. pow1/3 97.9%

         \[\leadsto \frac{\left(x + 1\right) - x}{\color{blue}{{\left({\left(x + 1\right)}^{2}\right)}^{0.3333333333333333}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      2. pow-pow 97.9%

         \[\leadsto \frac{\left(x + 1\right) - x}{\color{blue}{{\left(x + 1\right)}^{\left(2 \cdot 0.3333333333333333\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      3. metadata-eval 97.9%

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

   8. Applied egg-rr 97.9%

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

   if 1.35e16 < x

   1. Initial program 4.2%

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

      1. flip3-- 4.2%

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

      2. rem-cube-cbrt 3.3%

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

      3. rem-cube-cbrt 4.2%

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

      4. +-commutative 4.2%

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

      5. distribute-rgt-out 4.2%

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

      6. +-commutative 4.2%

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

      7. fma-define 4.2%

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

      8. add-exp-log 4.2%

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

      9. pow1/3 4.2%

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

      10. pow1/3 4.2%

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

      11. pow-prod-up 4.2%

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

   4. Applied egg-rr 4.2%

      \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]

   5. Taylor expanded in x around 0 92.8%

      \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   6. Taylor expanded in x around 0 7.4%

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

4. Final simplification 11.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+16}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + {\left(1 + x\right)}^{0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, 1 + x \cdot 0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 10.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;{\left(1 + x\right)}^{0.3333333333333333} - {x}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, 1 + x \cdot 0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4e+15)
   (- (pow (+ 1.0 x) 0.3333333333333333) (pow x 0.3333333333333333))
   (/
    1.0
    (fma
     (cbrt x)
     (+ (cbrt x) (cbrt (+ 1.0 x)))
     (+ 1.0 (* x 0.6666666666666666))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.4e+15) {
		tmp = pow((1.0 + x), 0.3333333333333333) - pow(x, 0.3333333333333333);
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((1.0 + x))), (1.0 + (x * 0.6666666666666666)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4e+15)
		tmp = Float64((Float64(1.0 + x) ^ 0.3333333333333333) - (x ^ 0.3333333333333333));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(1.0 + x))), Float64(1.0 + Float64(x * 0.6666666666666666))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.4e+15], N[(N[Power[N[(1.0 + x), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[x, 0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4e15

   1. Initial program 62.8%

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

      1. pow1/3 60.0%

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

   4. Applied egg-rr 60.0%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow1/3 64.1%

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

   6. Applied egg-rr 64.1%

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

   if 2.4e15 < x

   1. Initial program 4.2%

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

      1. flip3-- 4.2%

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

      2. rem-cube-cbrt 3.3%

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

      3. rem-cube-cbrt 4.2%

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

      4. +-commutative 4.2%

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

      5. distribute-rgt-out 4.2%

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

      6. +-commutative 4.2%

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

      7. fma-define 4.2%

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

      8. add-exp-log 4.2%

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

      9. pow1/3 4.2%

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

      10. pow1/3 4.2%

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

      11. pow-prod-up 4.2%

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

   4. Applied egg-rr 4.2%

      \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]

   5. Taylor expanded in x around 0 92.8%

      \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   6. Taylor expanded in x around 0 7.4%

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

4. Final simplification 9.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;{\left(1 + x\right)}^{0.3333333333333333} - {x}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, 1 + x \cdot 0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 7.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{1 + x} + \left(0 - {x}^{0.3333333333333333}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (cbrt (+ 1.0 x)) (- 0.0 (pow x 0.3333333333333333))))

C

double code(double x) {
	return cbrt((1.0 + x)) + (0.0 - pow(x, 0.3333333333333333));
}

Java

public static double code(double x) {
	return Math.cbrt((1.0 + x)) + (0.0 - Math.pow(x, 0.3333333333333333));
}

Julia

function code(x)
	return Float64(cbrt(Float64(1.0 + x)) + Float64(0.0 - (x ^ 0.3333333333333333)))
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. pow1/3 7.6%

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

4. Applied egg-rr 7.6%

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

5. Final simplification 7.6%

   \[\leadsto \sqrt[3]{1 + x} + \left(0 - {x}^{0.3333333333333333}\right) \]
6. Add Preprocessing

Alternative 10: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

3. Final simplification 6.8%

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

Alternative 11: 4.2% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 6.8%

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

3. Taylor expanded in x around inf 4.2%

   \[\leadsto \color{blue}{0} \]

4. Final simplification 4.2%

   \[\leadsto 0 \]
5. Add Preprocessing

Alternative 12: 6.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 6.8%

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

3. Taylor expanded in x around 0 6.1%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 6.1%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 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 2024046 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (+ (* (cbrt (+ x 1.0)) (cbrt (+ x 1.0))) (* (cbrt x) (cbrt (+ x 1.0)))) (* (cbrt x) (cbrt x))))

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