2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 99.0%

Time: 12.5s

Alternatives: 16

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: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{1}{{x}^{2}}}\\ \mathbf{if}\;x \leq 3 \cdot 10^{+28}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.1388888888888889 \cdot t\_0 + \left(t\_0 \cdot 0.027777777777777776 + \sqrt[3]{x} \cdot 0.3333333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (/ 1.0 (pow x 2.0)))))
   (if (<= x 3e+28)
     (/
      1.0
      (fma (cbrt x) (+ (cbrt x) (cbrt (+ 1.0 x))) (cbrt (pow (+ 1.0 x) 2.0))))
     (/
      (+
       (* -0.1388888888888889 t_0)
       (+ (* t_0 0.027777777777777776) (* (cbrt x) 0.3333333333333333)))
      x))))

C

double code(double x) {
	double t_0 = cbrt((1.0 / pow(x, 2.0)));
	double tmp;
	if (x <= 3e+28) {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((1.0 + x))), cbrt(pow((1.0 + x), 2.0)));
	} else {
		tmp = ((-0.1388888888888889 * t_0) + ((t_0 * 0.027777777777777776) + (cbrt(x) * 0.3333333333333333))) / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 / (x ^ 2.0)))
	tmp = 0.0
	if (x <= 3e+28)
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(1.0 + x))), cbrt((Float64(1.0 + x) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(-0.1388888888888889 * t_0) + Float64(Float64(t_0 * 0.027777777777777776) + Float64(cbrt(x) * 0.3333333333333333))) / x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 3e+28], 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[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.1388888888888889 * t$95$0), $MachinePrecision] + N[(N[(t$95$0 * 0.027777777777777776), $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.0000000000000001e28

   1. Initial program 40.0%

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

      1. flip3-- 41.0%

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

      2. div-inv 41.0%

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

      3. rem-cube-cbrt 42.3%

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

      4. rem-cube-cbrt 61.1%

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

      5. +-commutative 61.1%

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

      6. distribute-rgt-out 61.1%

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

      7. +-commutative 61.1%

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

      8. fma-define 61.1%

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

      9. add-exp-log 60.7%

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

   4. Applied egg-rr 60.8%

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

      1. associate-*r/ 60.8%

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

      2. *-rgt-identity 60.8%

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

      3. +-commutative 60.8%

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

      4. associate--l+ 97.4%

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

      5. +-inverses 97.4%

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

      6. metadata-eval 97.4%

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

      7. +-commutative 97.4%

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

      8. exp-prod 97.2%

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

   6. Simplified 97.2%

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

      1. add-cbrt-cube 96.8%

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

      2. pow3 96.7%

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

   8. Applied egg-rr 96.7%

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

      1. pow-exp 96.5%

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

      2. rem-cbrt-cube 97.4%

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

      3. *-commutative 97.4%

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

      4. log1p-undefine 97.4%

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

      5. +-commutative 97.4%

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

      6. exp-to-pow 97.4%

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

      7. metadata-eval 97.4%

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

      8. pow-prod-up 97.4%

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

      9. pow1/3 98.5%

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

      10. pow1/3 98.3%

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

      11. cbrt-unprod 99.0%

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

      12. pow2 99.0%

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

      13. +-commutative 99.0%

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

   10. Applied egg-rr 99.0%

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

   if 3.0000000000000001e28 < x

   1. Initial program 4.3%

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

      1. add-sqr-sqrt 3.7%

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

      2. add-sqr-sqrt 4.3%

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

      3. difference-of-squares 4.3%

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

      4. pow1/3 4.3%

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

      5. sqrt-pow1 4.3%

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

      6. metadata-eval 4.3%

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

      7. pow1/3 4.3%

         \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + \sqrt{\color{blue}{{x}^{0.3333333333333333}}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}} - \sqrt{\sqrt[3]{x}}\right) \]

      8. sqrt-pow1 4.3%

         \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + \color{blue}{{x}^{\left(\frac{0.3333333333333333}{2}\right)}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}} - \sqrt{\sqrt[3]{x}}\right) \]

      9. metadata-eval 4.3%

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

      10. pow1/3 1.8%

          \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left(\sqrt{\color{blue}{{\left(x + 1\right)}^{0.3333333333333333}}} - \sqrt{\sqrt[3]{x}}\right) \]

      11. sqrt-pow1 1.8%

          \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left(\color{blue}{{\left(x + 1\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} - \sqrt{\sqrt[3]{x}}\right) \]

      12. metadata-eval 1.8%

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

      13. pow1/3 4.2%

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

      14. sqrt-pow1 4.3%

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

      15. metadata-eval 4.3%

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

   4. Applied egg-rr 4.3%

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

   5. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{\frac{-0.1388888888888889 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(0.027777777777777776 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.3333333333333333 \cdot \sqrt[3]{x}\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+28}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.1388888888888889 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.027777777777777776 + \sqrt[3]{x} \cdot 0.3333333333333333\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt{1 + x}}\\ t_1 := \sqrt[3]{1 + x}\\ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t\_0 \cdot t\_0, t\_1 \cdot t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (sqrt (+ 1.0 x)))) (t_1 (cbrt (+ 1.0 x))))
   (/ 1.0 (fma (cbrt x) (+ (cbrt x) (* t_0 t_0)) (* t_1 t_1)))))

C

double code(double x) {
	double t_0 = cbrt(sqrt((1.0 + x)));
	double t_1 = cbrt((1.0 + x));
	return 1.0 / fma(cbrt(x), (cbrt(x) + (t_0 * t_0)), (t_1 * t_1));
}

Julia

function code(x)
	t_0 = cbrt(sqrt(Float64(1.0 + x)))
	t_1 = cbrt(Float64(1.0 + x))
	return Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + Float64(t_0 * t_0)), Float64(t_1 * t_1)))
end

Wolfram

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

Derivation

1. Initial program 8.9%

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

   1. flip3-- 9.0%

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

   2. div-inv 9.0%

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

   3. rem-cube-cbrt 8.3%

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

   4. rem-cube-cbrt 11.6%

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

   5. +-commutative 11.6%

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

   6. distribute-rgt-out 11.6%

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

   7. +-commutative 11.6%

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

   8. fma-define 11.6%

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

   9. add-exp-log 11.5%

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

4. Applied egg-rr 11.6%

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

   1. associate-*r/ 11.6%

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

   2. *-rgt-identity 11.6%

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

   3. +-commutative 11.6%

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

   4. associate--l+ 93.4%

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

   5. +-inverses 93.4%

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

   6. metadata-eval 93.4%

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

   7. +-commutative 93.4%

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

   8. exp-prod 92.5%

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

6. Simplified 92.5%

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

   1. add-cbrt-cube 92.2%

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

   2. pow3 92.4%

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

8. Applied egg-rr 92.4%

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

   1. rem-cbrt-cube 92.5%

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

   2. exp-prod 93.4%

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

   3. *-commutative 93.4%

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

   4. log1p-undefine 93.4%

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

   5. +-commutative 93.4%

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

   6. exp-to-pow 93.2%

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

   7. metadata-eval 93.2%

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

   8. pow-prod-up 93.2%

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

   9. +-commutative 93.2%

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

   10. pow1/3 94.6%

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

   11. +-commutative 94.6%

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

   12. pow1/3 98.3%

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

10. Applied egg-rr 98.3%

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

    1. pow1/3 94.6%

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

    2. +-commutative 94.6%

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

    3. add-sqr-sqrt 94.6%

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

    4. unpow-prod-down 94.6%

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

    5. +-commutative 94.6%

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

    6. +-commutative 94.6%

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

12. Applied egg-rr 94.6%

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

    1. unpow1/3 95.9%

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

    2. unpow1/3 98.5%

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

14. Simplified 98.5%

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

15. Final simplification 98.5%

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

Alternative 3: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{\frac{1}{{x}^{2}}}\\ \mathbf{if}\;t\_0 - \sqrt[3]{x} \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.1388888888888889 \cdot t\_1 + \left(t\_1 \cdot 0.027777777777777776 + \sqrt[3]{x} \cdot 0.3333333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (cbrt (/ 1.0 (pow x 2.0)))))
   (if (<= (- t_0 (cbrt x)) 3.5e-7)
     (/
      (+
       (* -0.1388888888888889 t_1)
       (+ (* t_1 0.027777777777777776) (* (cbrt x) 0.3333333333333333)))
      x)
     (/
      (- (+ 1.0 x) x)
      (+
       (exp (* 0.6666666666666666 (log1p x)))
       (* (cbrt x) (+ (cbrt x) t_0)))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = cbrt((1.0 / pow(x, 2.0)));
	double tmp;
	if ((t_0 - cbrt(x)) <= 3.5e-7) {
		tmp = ((-0.1388888888888889 * t_1) + ((t_1 * 0.027777777777777776) + (cbrt(x) * 0.3333333333333333))) / x;
	} else {
		tmp = ((1.0 + x) - x) / (exp((0.6666666666666666 * log1p(x))) + (cbrt(x) * (cbrt(x) + t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double t_1 = Math.cbrt((1.0 / Math.pow(x, 2.0)));
	double tmp;
	if ((t_0 - Math.cbrt(x)) <= 3.5e-7) {
		tmp = ((-0.1388888888888889 * t_1) + ((t_1 * 0.027777777777777776) + (Math.cbrt(x) * 0.3333333333333333))) / x;
	} else {
		tmp = ((1.0 + x) - x) / (Math.exp((0.6666666666666666 * Math.log1p(x))) + (Math.cbrt(x) * (Math.cbrt(x) + t_0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = cbrt(Float64(1.0 / (x ^ 2.0)))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 3.5e-7)
		tmp = Float64(Float64(Float64(-0.1388888888888889 * t_1) + Float64(Float64(t_1 * 0.027777777777777776) + Float64(cbrt(x) * 0.3333333333333333))) / x);
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64(exp(Float64(0.6666666666666666 * log1p(x))) + Float64(cbrt(x) * Float64(cbrt(x) + t_0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 3.5e-7], N[(N[(N[(-0.1388888888888889 * t$95$1), $MachinePrecision] + N[(N[(t$95$1 * 0.027777777777777776), $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 3.49999999999999984e-7

   1. Initial program 5.3%

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

      1. add-sqr-sqrt 4.8%

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

      2. add-sqr-sqrt 5.3%

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

      3. difference-of-squares 5.3%

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

      4. pow1/3 5.3%

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

      5. sqrt-pow1 5.3%

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

      6. metadata-eval 5.3%

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

      7. pow1/3 5.3%

         \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + \sqrt{\color{blue}{{x}^{0.3333333333333333}}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}} - \sqrt{\sqrt[3]{x}}\right) \]

      8. sqrt-pow1 5.3%

         \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + \color{blue}{{x}^{\left(\frac{0.3333333333333333}{2}\right)}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}} - \sqrt{\sqrt[3]{x}}\right) \]

      9. metadata-eval 5.3%

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

      10. pow1/3 2.8%

          \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left(\sqrt{\color{blue}{{\left(x + 1\right)}^{0.3333333333333333}}} - \sqrt{\sqrt[3]{x}}\right) \]

      11. sqrt-pow1 2.9%

          \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left(\color{blue}{{\left(x + 1\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} - \sqrt{\sqrt[3]{x}}\right) \]

      12. metadata-eval 2.9%

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

      13. pow1/3 5.2%

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

      14. sqrt-pow1 5.3%

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

      15. metadata-eval 5.3%

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

   4. Applied egg-rr 5.3%

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

   5. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{\frac{-0.1388888888888889 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(0.027777777777777776 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.3333333333333333 \cdot \sqrt[3]{x}\right)}{x}} \]

   if 3.49999999999999984e-7 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 76.3%

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

      1. sub-neg 76.3%

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

      2. flip3-+ 78.7%

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

      3. rem-cube-cbrt 81.3%

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

      4. neg-mul-1 81.3%

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

      5. metadata-eval 81.3%

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

      6. unpow-prod-down 81.3%

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

      7. metadata-eval 81.3%

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

      8. metadata-eval 81.3%

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

      9. rem-cube-cbrt 98.3%

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

   4. Applied egg-rr 98.6%

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

      1. sub-neg 98.6%

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

      2. distribute-rgt-neg-out 98.6%

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

      3. remove-double-neg 98.6%

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

      4. add-sqr-sqrt 98.8%

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

      5. sqrt-prod 98.6%

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

      6. sqr-neg 98.6%

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

      7. sqrt-prod 0.0%

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

      8. add-sqr-sqrt 17.7%

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

      9. distribute-rgt-in 17.7%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-prod 1.6%

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

      12. sqr-neg 1.6%

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

      13. sqrt-prod 1.6%

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

      14. add-sqr-sqrt 1.6%

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

      15. +-commutative 1.6%

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

      16. +-commutative 1.6%

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

   6. Applied egg-rr 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.1388888888888889 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.027777777777777776 + \sqrt[3]{x} \cdot 0.3333333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{\frac{1}{{x}^{2}}}\\ \mathbf{if}\;t\_0 - \sqrt[3]{x} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.1388888888888889 \cdot t\_1 + \left(t\_1 \cdot 0.027777777777777776 + \sqrt[3]{x} \cdot 0.3333333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{t\_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (cbrt (/ 1.0 (pow x 2.0)))))
   (if (<= (- t_0 (cbrt x)) 5e-7)
     (/
      (+
       (* -0.1388888888888889 t_1)
       (+ (* t_1 0.027777777777777776) (* (cbrt x) 0.3333333333333333)))
      x)
     (/ (- (+ 1.0 x) x) (+ (pow t_0 2.0) (* (cbrt x) (+ (cbrt x) t_0)))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = cbrt((1.0 / pow(x, 2.0)));
	double tmp;
	if ((t_0 - cbrt(x)) <= 5e-7) {
		tmp = ((-0.1388888888888889 * t_1) + ((t_1 * 0.027777777777777776) + (cbrt(x) * 0.3333333333333333))) / x;
	} else {
		tmp = ((1.0 + x) - x) / (pow(t_0, 2.0) + (cbrt(x) * (cbrt(x) + t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double t_1 = Math.cbrt((1.0 / Math.pow(x, 2.0)));
	double tmp;
	if ((t_0 - Math.cbrt(x)) <= 5e-7) {
		tmp = ((-0.1388888888888889 * t_1) + ((t_1 * 0.027777777777777776) + (Math.cbrt(x) * 0.3333333333333333))) / x;
	} else {
		tmp = ((1.0 + x) - x) / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (Math.cbrt(x) + t_0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = cbrt(Float64(1.0 / (x ^ 2.0)))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 5e-7)
		tmp = Float64(Float64(Float64(-0.1388888888888889 * t_1) + Float64(Float64(t_1 * 0.027777777777777776) + Float64(cbrt(x) * 0.3333333333333333))) / x);
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64((t_0 ^ 2.0) + Float64(cbrt(x) * Float64(cbrt(x) + t_0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(N[(-0.1388888888888889 * t$95$1), $MachinePrecision] + N[(N[(t$95$1 * 0.027777777777777776), $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 4.99999999999999977e-7

   1. Initial program 5.5%

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

      1. add-sqr-sqrt 5.0%

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

      2. add-sqr-sqrt 5.5%

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

      3. difference-of-squares 5.5%

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

      4. pow1/3 5.5%

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

      5. sqrt-pow1 5.5%

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

      6. metadata-eval 5.5%

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

      7. pow1/3 5.5%

         \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + \sqrt{\color{blue}{{x}^{0.3333333333333333}}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}} - \sqrt{\sqrt[3]{x}}\right) \]

      8. sqrt-pow1 5.5%

         \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + \color{blue}{{x}^{\left(\frac{0.3333333333333333}{2}\right)}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}} - \sqrt{\sqrt[3]{x}}\right) \]

      9. metadata-eval 5.5%

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

      10. pow1/3 3.0%

          \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left(\sqrt{\color{blue}{{\left(x + 1\right)}^{0.3333333333333333}}} - \sqrt{\sqrt[3]{x}}\right) \]

      11. sqrt-pow1 3.1%

          \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left(\color{blue}{{\left(x + 1\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} - \sqrt{\sqrt[3]{x}}\right) \]

      12. metadata-eval 3.1%

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

      13. pow1/3 5.4%

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

      14. sqrt-pow1 5.5%

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

      15. metadata-eval 5.5%

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

   4. Applied egg-rr 5.5%

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

   5. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{\frac{-0.1388888888888889 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(0.027777777777777776 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.3333333333333333 \cdot \sqrt[3]{x}\right)}{x}} \]

   if 4.99999999999999977e-7 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 78.2%

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

      1. add-exp-log 77.8%

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

      2. +-commutative 77.8%

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

      3. log1p-define 77.8%

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

   4. Applied egg-rr 77.8%

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

   6. Applied egg-rr 98.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.1388888888888889 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.027777777777777776 + \sqrt[3]{x} \cdot 0.3333333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 8.9%

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

   1. flip3-- 9.0%

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

   2. div-inv 9.0%

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

   3. rem-cube-cbrt 8.3%

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

   4. rem-cube-cbrt 11.6%

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

   5. +-commutative 11.6%

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

   6. distribute-rgt-out 11.6%

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

   7. +-commutative 11.6%

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

   8. fma-define 11.6%

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

   9. add-exp-log 11.5%

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

4. Applied egg-rr 11.6%

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

   1. associate-*r/ 11.6%

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

   2. *-rgt-identity 11.6%

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

   3. +-commutative 11.6%

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

   4. associate--l+ 93.4%

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

   5. +-inverses 93.4%

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

   6. metadata-eval 93.4%

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

   7. +-commutative 93.4%

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

   8. exp-prod 92.5%

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

6. Simplified 92.5%

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

   1. add-cbrt-cube 92.2%

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

   2. pow3 92.4%

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

8. Applied egg-rr 92.4%

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

   1. rem-cbrt-cube 92.5%

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

   2. exp-prod 93.4%

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

   3. *-commutative 93.4%

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

   4. log1p-undefine 93.4%

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

   5. +-commutative 93.4%

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

   6. exp-to-pow 93.2%

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

   7. metadata-eval 93.2%

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

   8. pow-prod-up 93.2%

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

   9. +-commutative 93.2%

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

   10. pow1/3 94.6%

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

   11. +-commutative 94.6%

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

   12. pow1/3 98.3%

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

10. Applied egg-rr 98.3%

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

    1. pow2 98.3%

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

12. Applied egg-rr 98.3%

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

13. Final simplification 98.3%

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

Alternative 6: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{1}{{x}^{2}}}\\ \mathbf{if}\;x \leq 20000000:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{\left(1 + x\right)}^{0.6666666666666666} - \left(\sqrt[3]{x} \cdot \left(-\sqrt[3]{x}\right) - \sqrt[3]{x} \cdot \sqrt[3]{1 + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.1388888888888889 \cdot t\_0 + \left(t\_0 \cdot 0.027777777777777776 + \sqrt[3]{x} \cdot 0.3333333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (/ 1.0 (pow x 2.0)))))
   (if (<= x 20000000.0)
     (/
      (- (+ 1.0 x) x)
      (-
       (pow (+ 1.0 x) 0.6666666666666666)
       (- (* (cbrt x) (- (cbrt x))) (* (cbrt x) (cbrt (+ 1.0 x))))))
     (/
      (+
       (* -0.1388888888888889 t_0)
       (+ (* t_0 0.027777777777777776) (* (cbrt x) 0.3333333333333333)))
      x))))

C

double code(double x) {
	double t_0 = cbrt((1.0 / pow(x, 2.0)));
	double tmp;
	if (x <= 20000000.0) {
		tmp = ((1.0 + x) - x) / (pow((1.0 + x), 0.6666666666666666) - ((cbrt(x) * -cbrt(x)) - (cbrt(x) * cbrt((1.0 + x)))));
	} else {
		tmp = ((-0.1388888888888889 * t_0) + ((t_0 * 0.027777777777777776) + (cbrt(x) * 0.3333333333333333))) / x;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 / Math.pow(x, 2.0)));
	double tmp;
	if (x <= 20000000.0) {
		tmp = ((1.0 + x) - x) / (Math.pow((1.0 + x), 0.6666666666666666) - ((Math.cbrt(x) * -Math.cbrt(x)) - (Math.cbrt(x) * Math.cbrt((1.0 + x)))));
	} else {
		tmp = ((-0.1388888888888889 * t_0) + ((t_0 * 0.027777777777777776) + (Math.cbrt(x) * 0.3333333333333333))) / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 / (x ^ 2.0)))
	tmp = 0.0
	if (x <= 20000000.0)
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64((Float64(1.0 + x) ^ 0.6666666666666666) - Float64(Float64(cbrt(x) * Float64(-cbrt(x))) - Float64(cbrt(x) * cbrt(Float64(1.0 + x))))));
	else
		tmp = Float64(Float64(Float64(-0.1388888888888889 * t_0) + Float64(Float64(t_0 * 0.027777777777777776) + Float64(cbrt(x) * 0.3333333333333333))) / x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 20000000.0], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision] - N[(N[(N[Power[x, 1/3], $MachinePrecision] * (-N[Power[x, 1/3], $MachinePrecision])), $MachinePrecision] - N[(N[Power[x, 1/3], $MachinePrecision] * N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.1388888888888889 * t$95$0), $MachinePrecision] + N[(N[(t$95$0 * 0.027777777777777776), $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e7

   1. Initial program 78.2%

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

      1. sub-neg 78.2%

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

      2. flip3-+ 81.0%

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

      3. rem-cube-cbrt 83.9%

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

      4. neg-mul-1 83.9%

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

      5. metadata-eval 83.9%

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

      6. unpow-prod-down 83.9%

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

      7. metadata-eval 83.9%

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

      8. metadata-eval 83.9%

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

      9. rem-cube-cbrt 98.3%

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

   4. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. exp-prod 98.9%

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

      3. log1p-undefine 98.9%

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

      4. add-exp-log 98.9%

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

   6. Applied egg-rr 98.9%

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

   if 2e7 < x

   1. Initial program 5.5%

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

      1. add-sqr-sqrt 5.0%

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

      2. add-sqr-sqrt 5.5%

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

      3. difference-of-squares 5.5%

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

      4. pow1/3 5.5%

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

      5. sqrt-pow1 5.5%

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

      6. metadata-eval 5.5%

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

      7. pow1/3 5.5%

         \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + \sqrt{\color{blue}{{x}^{0.3333333333333333}}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}} - \sqrt{\sqrt[3]{x}}\right) \]

      8. sqrt-pow1 5.5%

         \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + \color{blue}{{x}^{\left(\frac{0.3333333333333333}{2}\right)}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}} - \sqrt{\sqrt[3]{x}}\right) \]

      9. metadata-eval 5.5%

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

      10. pow1/3 3.0%

          \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left(\sqrt{\color{blue}{{\left(x + 1\right)}^{0.3333333333333333}}} - \sqrt{\sqrt[3]{x}}\right) \]

      11. sqrt-pow1 3.1%

          \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left(\color{blue}{{\left(x + 1\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} - \sqrt{\sqrt[3]{x}}\right) \]

      12. metadata-eval 3.1%

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

      13. pow1/3 5.4%

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

      14. sqrt-pow1 5.5%

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

      15. metadata-eval 5.5%

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

   4. Applied egg-rr 5.5%

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

   5. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{\frac{-0.1388888888888889 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(0.027777777777777776 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.3333333333333333 \cdot \sqrt[3]{x}\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 20000000:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{\left(1 + x\right)}^{0.6666666666666666} - \left(\sqrt[3]{x} \cdot \left(-\sqrt[3]{x}\right) - \sqrt[3]{x} \cdot \sqrt[3]{1 + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.1388888888888889 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.027777777777777776 + \sqrt[3]{x} \cdot 0.3333333333333333\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 8.9%

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

   1. flip3-- 9.0%

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

   2. div-inv 9.0%

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

   3. rem-cube-cbrt 8.3%

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

   4. rem-cube-cbrt 11.6%

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

   5. +-commutative 11.6%

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

   6. distribute-rgt-out 11.6%

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

   7. +-commutative 11.6%

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

   8. fma-define 11.6%

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

   9. add-exp-log 11.5%

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

4. Applied egg-rr 11.6%

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

   1. associate-*r/ 11.6%

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

   2. *-rgt-identity 11.6%

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

   3. +-commutative 11.6%

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

   4. associate--l+ 93.4%

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

   5. +-inverses 93.4%

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

   6. metadata-eval 93.4%

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

   7. +-commutative 93.4%

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

   8. exp-prod 92.5%

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

6. Simplified 92.5%

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

   1. add-cbrt-cube 92.2%

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

   2. pow3 92.4%

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

8. Applied egg-rr 92.4%

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

   1. rem-cbrt-cube 92.5%

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

   2. exp-prod 93.4%

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

   3. *-commutative 93.4%

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

   4. log1p-undefine 93.4%

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

   5. +-commutative 93.4%

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

   6. exp-to-pow 93.2%

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

   7. metadata-eval 93.2%

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

   8. pow-prod-up 93.2%

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

   9. +-commutative 93.2%

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

   10. pow1/3 94.6%

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

   11. +-commutative 94.6%

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

   12. pow1/3 98.3%

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

10. Applied egg-rr 98.3%

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

    1. pow2 98.3%

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

    2. pow1/3 93.2%

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

    3. pow-pow 93.2%

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

    4. metadata-eval 93.2%

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

12. Applied egg-rr 93.2%

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

13. Final simplification 93.2%

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

Alternative 8: 58.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+70)
   (/
    (+
     (* (cbrt x) -0.1111111111111111)
     (* 0.3333333333333333 (cbrt (pow x 4.0))))
    (pow x 2.0))
   (if (<= x 1.35e+154)
     (* (cbrt (/ 1.0 (pow x 2.0))) 0.3333333333333333)
     (/ 1.0 (fma (cbrt x) (+ (cbrt x) (cbrt (+ 1.0 x))) 1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 2e+70) {
		tmp = ((cbrt(x) * -0.1111111111111111) + (0.3333333333333333 * cbrt(pow(x, 4.0)))) / pow(x, 2.0);
	} else if (x <= 1.35e+154) {
		tmp = cbrt((1.0 / pow(x, 2.0))) * 0.3333333333333333;
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((1.0 + x))), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+70)
		tmp = Float64(Float64(Float64(cbrt(x) * -0.1111111111111111) + Float64(0.3333333333333333 * cbrt((x ^ 4.0)))) / (x ^ 2.0));
	elseif (x <= 1.35e+154)
		tmp = Float64(cbrt(Float64(1.0 / (x ^ 2.0))) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(1.0 + x))), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+70], N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision] + N[(0.3333333333333333 * N[Power[N[Power[x, 4.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+154], N[(N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $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] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.00000000000000015e70

   1. Initial program 21.7%

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

   3. Taylor expanded in x around inf 92.5%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \]

   if 2.00000000000000015e70 < x < 1.35000000000000003e154

   1. Initial program 3.8%

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

   3. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]

   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. div-inv 4.8%

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

      3. rem-cube-cbrt 3.1%

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

      4. rem-cube-cbrt 4.8%

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

      5. +-commutative 4.8%

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

      6. distribute-rgt-out 4.8%

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

      7. +-commutative 4.8%

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

      8. fma-define 4.8%

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

      9. add-exp-log 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. associate-*r/ 4.8%

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

      2. *-rgt-identity 4.8%

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

      3. +-commutative 4.8%

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

      4. associate--l+ 92.0%

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

      5. +-inverses 92.0%

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

      6. metadata-eval 92.0%

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

      7. +-commutative 92.0%

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

      8. exp-prod 90.9%

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

   6. Simplified 90.9%

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

      1. add-cbrt-cube 90.7%

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

      2. pow3 90.8%

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

   8. Applied egg-rr 90.8%

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

      1. rem-cbrt-cube 90.9%

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

      2. exp-prod 92.0%

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

      3. *-commutative 92.0%

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

      4. log1p-undefine 92.0%

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

      5. +-commutative 92.0%

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

      6. exp-to-pow 91.6%

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

      7. metadata-eval 91.6%

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

      8. pow-prod-up 91.6%

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

      9. +-commutative 91.6%

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

      10. pow1/3 93.1%

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

      11. +-commutative 93.1%

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

      12. pow1/3 98.2%

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

   10. Applied egg-rr 98.2%

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

   11. Taylor expanded in x around 0 20.0%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq 23000000:\\ \;\;\;\;\sqrt[3]{{t\_0}^{3}} - \sqrt[3]{x}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t\_0, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= x 23000000.0)
     (- (cbrt (pow t_0 3.0)) (cbrt x))
     (if (<= x 1.35e+154)
       (* (cbrt (/ 1.0 (pow x 2.0))) 0.3333333333333333)
       (/ 1.0 (fma (cbrt x) (+ (cbrt x) t_0) 1.0))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if (x <= 23000000.0) {
		tmp = cbrt(pow(t_0, 3.0)) - cbrt(x);
	} else if (x <= 1.35e+154) {
		tmp = cbrt((1.0 / pow(x, 2.0))) * 0.3333333333333333;
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + t_0), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (x <= 23000000.0)
		tmp = Float64(cbrt((t_0 ^ 3.0)) - cbrt(x));
	elseif (x <= 1.35e+154)
		tmp = Float64(cbrt(Float64(1.0 / (x ^ 2.0))) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + t_0), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 23000000.0], N[(N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+154], N[(N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.3e7

   1. Initial program 78.2%

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

      1. rem-cube-cbrt 79.2%

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

   4. Applied egg-rr 79.2%

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

   if 2.3e7 < x < 1.35000000000000003e154

   1. Initial program 6.3%

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

   3. Taylor expanded in x around inf 97.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]

   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. div-inv 4.8%

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

      3. rem-cube-cbrt 3.1%

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

      4. rem-cube-cbrt 4.8%

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

      5. +-commutative 4.8%

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

      6. distribute-rgt-out 4.8%

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

      7. +-commutative 4.8%

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

      8. fma-define 4.8%

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

      9. add-exp-log 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. associate-*r/ 4.8%

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

      2. *-rgt-identity 4.8%

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

      3. +-commutative 4.8%

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

      4. associate--l+ 92.0%

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

      5. +-inverses 92.0%

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

      6. metadata-eval 92.0%

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

      7. +-commutative 92.0%

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

      8. exp-prod 90.9%

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

   6. Simplified 90.9%

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

      1. add-cbrt-cube 90.7%

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

      2. pow3 90.8%

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

   8. Applied egg-rr 90.8%

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

      1. rem-cbrt-cube 90.9%

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

      2. exp-prod 92.0%

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

      3. *-commutative 92.0%

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

      4. log1p-undefine 92.0%

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

      5. +-commutative 92.0%

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

      6. exp-to-pow 91.6%

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

      7. metadata-eval 91.6%

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

      8. pow-prod-up 91.6%

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

      9. +-commutative 91.6%

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

      10. pow1/3 93.1%

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

      11. +-commutative 93.1%

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

      12. pow1/3 98.2%

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

   10. Applied egg-rr 98.2%

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

   11. Taylor expanded in x around 0 20.0%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 23000000:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{1 + x}\right)}^{3}} - \sqrt[3]{x}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 23000000:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{1 + x}\right)}^{3}} - \sqrt[3]{x}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 23000000.0)
   (- (cbrt (pow (cbrt (+ 1.0 x)) 3.0)) (cbrt x))
   (if (<= x 1.35e+154)
     (* (cbrt (/ 1.0 (pow x 2.0))) 0.3333333333333333)
     (/ 1.0 (+ 1.0 (* (cbrt x) (+ 1.0 (cbrt x))))))))

C

double code(double x) {
	double tmp;
	if (x <= 23000000.0) {
		tmp = cbrt(pow(cbrt((1.0 + x)), 3.0)) - cbrt(x);
	} else if (x <= 1.35e+154) {
		tmp = cbrt((1.0 / pow(x, 2.0))) * 0.3333333333333333;
	} else {
		tmp = 1.0 / (1.0 + (cbrt(x) * (1.0 + cbrt(x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 23000000.0) {
		tmp = Math.cbrt(Math.pow(Math.cbrt((1.0 + x)), 3.0)) - Math.cbrt(x);
	} else if (x <= 1.35e+154) {
		tmp = Math.cbrt((1.0 / Math.pow(x, 2.0))) * 0.3333333333333333;
	} else {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (1.0 + Math.cbrt(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 23000000.0)
		tmp = Float64(cbrt((cbrt(Float64(1.0 + x)) ^ 3.0)) - cbrt(x));
	elseif (x <= 1.35e+154)
		tmp = Float64(cbrt(Float64(1.0 / (x ^ 2.0))) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(1.0 + cbrt(x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 23000000.0], N[(N[Power[N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+154], N[(N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.3e7

   1. Initial program 78.2%

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

      1. rem-cube-cbrt 79.2%

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

   4. Applied egg-rr 79.2%

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

   if 2.3e7 < x < 1.35000000000000003e154

   1. Initial program 6.3%

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

   3. Taylor expanded in x around inf 97.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]

   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. div-inv 4.8%

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

      3. rem-cube-cbrt 3.1%

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

      4. rem-cube-cbrt 4.8%

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

      5. +-commutative 4.8%

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

      6. distribute-rgt-out 4.8%

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

      7. +-commutative 4.8%

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

      8. fma-define 4.8%

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

      9. add-exp-log 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. associate-*r/ 4.8%

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

      2. *-rgt-identity 4.8%

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

      3. +-commutative 4.8%

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

      4. associate--l+ 92.0%

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

      5. +-inverses 92.0%

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

      6. metadata-eval 92.0%

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

      7. +-commutative 92.0%

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

      8. exp-prod 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in x around 0 17.7%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 23000000:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{1 + x}\right)}^{3}} - \sqrt[3]{x}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 31000000:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{2}}{x + -1} + \frac{-1}{x + -1}} - \sqrt[3]{x}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 31000000.0)
   (- (cbrt (+ (/ (pow x 2.0) (+ x -1.0)) (/ -1.0 (+ x -1.0)))) (cbrt x))
   (if (<= x 1.35e+154)
     (* (cbrt (/ 1.0 (pow x 2.0))) 0.3333333333333333)
     (/ 1.0 (+ 1.0 (* (cbrt x) (+ 1.0 (cbrt x))))))))

C

double code(double x) {
	double tmp;
	if (x <= 31000000.0) {
		tmp = cbrt(((pow(x, 2.0) / (x + -1.0)) + (-1.0 / (x + -1.0)))) - cbrt(x);
	} else if (x <= 1.35e+154) {
		tmp = cbrt((1.0 / pow(x, 2.0))) * 0.3333333333333333;
	} else {
		tmp = 1.0 / (1.0 + (cbrt(x) * (1.0 + cbrt(x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 31000000.0) {
		tmp = Math.cbrt(((Math.pow(x, 2.0) / (x + -1.0)) + (-1.0 / (x + -1.0)))) - Math.cbrt(x);
	} else if (x <= 1.35e+154) {
		tmp = Math.cbrt((1.0 / Math.pow(x, 2.0))) * 0.3333333333333333;
	} else {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (1.0 + Math.cbrt(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 31000000.0)
		tmp = Float64(cbrt(Float64(Float64((x ^ 2.0) / Float64(x + -1.0)) + Float64(-1.0 / Float64(x + -1.0)))) - cbrt(x));
	elseif (x <= 1.35e+154)
		tmp = Float64(cbrt(Float64(1.0 / (x ^ 2.0))) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(1.0 + cbrt(x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 31000000.0], N[(N[Power[N[(N[(N[Power[x, 2.0], $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+154], N[(N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.1e7

   1. Initial program 78.2%

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

      1. flip-+ 78.4%

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

      2. metadata-eval 78.4%

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

      3. div-sub 78.6%

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

      4. pow2 78.6%

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

      5. sub-neg 78.6%

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

      6. metadata-eval 78.6%

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

      7. sub-neg 78.6%

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

      8. metadata-eval 78.6%

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

   4. Applied egg-rr 78.6%

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

   if 3.1e7 < x < 1.35000000000000003e154

   1. Initial program 6.3%

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

   3. Taylor expanded in x around inf 97.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]

   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. div-inv 4.8%

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

      3. rem-cube-cbrt 3.1%

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

      4. rem-cube-cbrt 4.8%

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

      5. +-commutative 4.8%

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

      6. distribute-rgt-out 4.8%

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

      7. +-commutative 4.8%

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

      8. fma-define 4.8%

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

      9. add-exp-log 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. associate-*r/ 4.8%

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

      2. *-rgt-identity 4.8%

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

      3. +-commutative 4.8%

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

      4. associate--l+ 92.0%

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

      5. +-inverses 92.0%

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

      6. metadata-eval 92.0%

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

      7. +-commutative 92.0%

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

      8. exp-prod 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in x around 0 17.7%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 31000000:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{2}}{x + -1} + \frac{-1}{x + -1}} - \sqrt[3]{x}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 25000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 25000000.0)
   (- (cbrt (+ 1.0 x)) (cbrt x))
   (if (<= x 1.35e+154)
     (* (cbrt (/ 1.0 (pow x 2.0))) 0.3333333333333333)
     (/ 1.0 (+ 1.0 (* (cbrt x) (+ 1.0 (cbrt x))))))))

C

double code(double x) {
	double tmp;
	if (x <= 25000000.0) {
		tmp = cbrt((1.0 + x)) - cbrt(x);
	} else if (x <= 1.35e+154) {
		tmp = cbrt((1.0 / pow(x, 2.0))) * 0.3333333333333333;
	} else {
		tmp = 1.0 / (1.0 + (cbrt(x) * (1.0 + cbrt(x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 25000000.0) {
		tmp = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	} else if (x <= 1.35e+154) {
		tmp = Math.cbrt((1.0 / Math.pow(x, 2.0))) * 0.3333333333333333;
	} else {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (1.0 + Math.cbrt(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 25000000.0)
		tmp = Float64(cbrt(Float64(1.0 + x)) - cbrt(x));
	elseif (x <= 1.35e+154)
		tmp = Float64(cbrt(Float64(1.0 / (x ^ 2.0))) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(1.0 + cbrt(x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 25000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+154], N[(N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.5e7

   1. Initial program 78.2%

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

   if 2.5e7 < x < 1.35000000000000003e154

   1. Initial program 6.3%

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

   3. Taylor expanded in x around inf 97.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]

   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. div-inv 4.8%

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

      3. rem-cube-cbrt 3.1%

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

      4. rem-cube-cbrt 4.8%

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

      5. +-commutative 4.8%

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

      6. distribute-rgt-out 4.8%

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

      7. +-commutative 4.8%

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

      8. fma-define 4.8%

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

      9. add-exp-log 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. associate-*r/ 4.8%

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

      2. *-rgt-identity 4.8%

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

      3. +-commutative 4.8%

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

      4. associate--l+ 92.0%

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

      5. +-inverses 92.0%

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

      6. metadata-eval 92.0%

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

      7. +-commutative 92.0%

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

      8. exp-prod 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in x around 0 17.7%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 25000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 25000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 25000000.0)
   (- (cbrt (+ 1.0 x)) (cbrt x))
   (* (cbrt (/ 1.0 (pow x 2.0))) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 25000000.0) {
		tmp = cbrt((1.0 + x)) - cbrt(x);
	} else {
		tmp = cbrt((1.0 / pow(x, 2.0))) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 25000000.0) {
		tmp = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	} else {
		tmp = Math.cbrt((1.0 / Math.pow(x, 2.0))) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 25000000.0)
		tmp = Float64(cbrt(Float64(1.0 + x)) - cbrt(x));
	else
		tmp = Float64(cbrt(Float64(1.0 / (x ^ 2.0))) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 25000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5e7

   1. Initial program 78.2%

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

   if 2.5e7 < x

   1. Initial program 5.5%

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

   3. Taylor expanded in x around inf 49.1%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 25000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{2}}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 25000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{0.037037037037037035}{{x}^{2}}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 25000000.0)
   (- (cbrt (+ 1.0 x)) (cbrt x))
   (pow (/ 0.037037037037037035 (pow x 2.0)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 25000000.0) {
		tmp = cbrt((1.0 + x)) - cbrt(x);
	} else {
		tmp = pow((0.037037037037037035 / pow(x, 2.0)), 0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 25000000.0) {
		tmp = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	} else {
		tmp = Math.pow((0.037037037037037035 / Math.pow(x, 2.0)), 0.3333333333333333);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 25000000.0)
		tmp = Float64(cbrt(Float64(1.0 + x)) - cbrt(x));
	else
		tmp = Float64(0.037037037037037035 / (x ^ 2.0)) ^ 0.3333333333333333;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 25000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[Power[N[(0.037037037037037035 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5e7

   1. Initial program 78.2%

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

   if 2.5e7 < x

   1. Initial program 5.5%

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

      1. add-cbrt-cube 5.5%

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

      2. pow1/3 5.5%

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

      3. pow3 5.5%

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

   4. Applied egg-rr 5.5%

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

   5. Taylor expanded in x around inf 45.9%

      \[\leadsto {\color{blue}{\left(\frac{0.037037037037037035}{{x}^{2}}\right)}}^{0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 25000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{0.037037037037037035}{{x}^{2}}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 7.0% 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 8.9%

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

3. Final simplification 8.9%

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

Alternative 16: 5.4% 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 cbrt(x)
end

Wolfram

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

Derivation

1. Initial program 8.9%

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

3. Taylor expanded in x around 0 1.8%

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

   1. sub-neg 1.8%

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

   2. rem-square-sqrt 0.0%

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

   3. fabs-sqr 0.0%

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

   4. rem-square-sqrt 5.5%

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

   5. fabs-neg 5.5%

      \[\leadsto 1 + \color{blue}{\left|\sqrt[3]{x}\right|} \]

   6. unpow1/3 5.5%

      \[\leadsto 1 + \left|\color{blue}{{x}^{0.3333333333333333}}\right| \]

   7. metadata-eval 5.5%

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

   8. pow-sqr 5.5%

      \[\leadsto 1 + \left|\color{blue}{{x}^{0.16666666666666666} \cdot {x}^{0.16666666666666666}}\right| \]

   9. fabs-sqr 5.5%

      \[\leadsto 1 + \color{blue}{{x}^{0.16666666666666666} \cdot {x}^{0.16666666666666666}} \]

   10. pow-sqr 5.5%

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

   11. metadata-eval 5.5%

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

   12. unpow1/3 5.5%

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

5. Simplified 5.5%

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

6. Taylor expanded in x around inf 5.5%

   \[\leadsto \color{blue}{\sqrt[3]{x}} \]
7. Add Preprocessing

Developer Target 1: 98.4% 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 2024137 
(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)))