2cbrt (problem 3.3.4)

Percentage Accurate: 7.2% → 98.8%

Time: 10.9s

Alternatives: 11

Speedup: 2.0×

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: 7.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (/ 1.0 (* x (* 3.0 (cbrt (/ 1.0 x)))))
     (/ (- (+ 1.0 x) x) (+ (* (cbrt x) (+ t_0 (cbrt x))) (pow t_0 2.0))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 1.0 / (x * (3.0 * cbrt((1.0 / x))));
	} else {
		tmp = ((1.0 + x) - x) / ((cbrt(x) * (t_0 + cbrt(x))) + pow(t_0, 2.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double tmp;
	if ((t_0 - Math.cbrt(x)) <= 0.0) {
		tmp = 1.0 / (x * (3.0 * Math.cbrt((1.0 / x))));
	} else {
		tmp = ((1.0 + x) - x) / ((Math.cbrt(x) * (t_0 + Math.cbrt(x))) + Math.pow(t_0, 2.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(1.0 / Float64(x * Float64(3.0 * cbrt(Float64(1.0 / x)))));
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64(Float64(cbrt(x) * Float64(t_0 + cbrt(x))) + (t_0 ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(x * N[(3.0 * N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

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

      1. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 3.6%

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

      4. rem-cube-cbrt 4.2%

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

      5. +-commutative 4.2%

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

      6. distribute-rgt-out 4.2%

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

      7. +-commutative 4.2%

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

      8. fma-define 4.2%

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

      9. add-exp-log 4.2%

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

   4. Applied egg-rr 4.2%

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

      1. associate-*r/ 4.2%

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

      2. *-rgt-identity 4.2%

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

      3. +-commutative 4.2%

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

      4. associate--l+ 92.9%

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

      5. +-inverses 92.9%

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

      6. metadata-eval 92.9%

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

      7. +-commutative 92.9%

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

      8. exp-prod 91.9%

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

   6. Simplified 91.9%

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

   7. Taylor expanded in x around inf 91.9%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{2 \cdot \sqrt[3]{x}}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 91.9%

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

   9. Simplified 91.9%

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

   10. Taylor expanded in x around inf 98.9%

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

       1. distribute-rgt1-in 98.9%

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

       2. metadata-eval 98.9%

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

   12. Simplified 98.9%

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

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

   1. Initial program 57.6%

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

      1. add-sqr-sqrt 56.9%

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

      2. pow2 56.9%

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

      3. pow1/3 54.9%

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

      4. sqrt-pow1 54.7%

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

      5. metadata-eval 54.7%

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

   4. Applied egg-rr 54.7%

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 98.9%

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

Alternative 2: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

   1. flip3-- 9.1%

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

   2. div-inv 9.1%

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

   3. rem-cube-cbrt 8.8%

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

   4. rem-cube-cbrt 12.3%

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

   5. +-commutative 12.3%

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

   6. distribute-rgt-out 12.4%

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

   7. +-commutative 12.4%

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

   8. fma-define 12.3%

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

   9. add-exp-log 12.3%

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

4. Applied egg-rr 12.2%

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

   1. associate-*r/ 12.2%

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

   2. *-rgt-identity 12.2%

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

   3. +-commutative 12.2%

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

   4. associate--l+ 93.3%

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

   5. +-inverses 93.3%

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

   6. metadata-eval 93.3%

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

   7. +-commutative 93.3%

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

   8. exp-prod 92.4%

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

6. Simplified 92.4%

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

   1. /-rgt-identity 92.4%

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

   2. add-exp-log 91.7%

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

   3. /-rgt-identity 91.7%

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

   4. +-commutative 91.7%

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

8. Applied egg-rr 91.7%

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

10. Applied egg-rr 98.6%

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

11. Final simplification 98.6%

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

Alternative 3: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 17000000:\\ \;\;\;\;\log \left(e^{-\sqrt[3]{x}} \cdot e^{\sqrt[3]{1 + x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{{x}^{2}}} + \sqrt[3]{\frac{1}{x}} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 17000000.0)
   (log (* (exp (- (cbrt x))) (exp (cbrt (+ 1.0 x)))))
   (/
    1.0
    (*
     x
     (+ (cbrt (+ (/ 1.0 x) (/ 2.0 (pow x 2.0)))) (* (cbrt (/ 1.0 x)) 2.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = log((exp(-cbrt(x)) * exp(cbrt((1.0 + x)))));
	} else {
		tmp = 1.0 / (x * (cbrt(((1.0 / x) + (2.0 / pow(x, 2.0)))) + (cbrt((1.0 / x)) * 2.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = Math.log((Math.exp(-Math.cbrt(x)) * Math.exp(Math.cbrt((1.0 + x)))));
	} else {
		tmp = 1.0 / (x * (Math.cbrt(((1.0 / x) + (2.0 / Math.pow(x, 2.0)))) + (Math.cbrt((1.0 / x)) * 2.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 17000000.0)
		tmp = log(Float64(exp(Float64(-cbrt(x))) * exp(cbrt(Float64(1.0 + x)))));
	else
		tmp = Float64(1.0 / Float64(x * Float64(cbrt(Float64(Float64(1.0 / x) + Float64(2.0 / (x ^ 2.0)))) + Float64(cbrt(Float64(1.0 / x)) * 2.0))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 17000000.0], N[Log[N[(N[Exp[(-N[Power[x, 1/3], $MachinePrecision])], $MachinePrecision] * N[Exp[N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(x * N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(2.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e7

   1. Initial program 79.7%

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

      1. sub-neg 79.7%

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

      2. +-commutative 79.7%

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

      3. add-log-exp 79.7%

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

      4. add-log-exp 79.7%

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

      5. sum-log 80.3%

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

   4. Applied egg-rr 80.3%

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

   if 1.7e7 < x

   1. Initial program 5.6%

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

      1. flip3-- 5.8%

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

      2. div-inv 5.8%

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

      3. rem-cube-cbrt 5.4%

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

      4. rem-cube-cbrt 8.4%

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

      5. +-commutative 8.4%

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

      6. distribute-rgt-out 8.5%

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

      7. +-commutative 8.5%

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

      8. fma-define 8.5%

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

      9. add-exp-log 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. associate-*r/ 8.4%

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

      2. *-rgt-identity 8.4%

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

      3. +-commutative 8.4%

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

      4. associate--l+ 93.1%

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

      5. +-inverses 93.1%

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

      6. metadata-eval 93.1%

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

      7. +-commutative 93.1%

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

      8. exp-prod 92.2%

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

   6. Simplified 92.2%

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

   7. Taylor expanded in x around inf 91.7%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{2 \cdot \sqrt[3]{x}}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 91.7%

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

   9. Simplified 91.7%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\sqrt[3]{x} \cdot 2}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. add-exp-log 92.0%

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

       2. log-pow 92.6%

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

       3. rem-log-exp 92.6%

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

   11. Applied egg-rr 92.6%

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

   12. Taylor expanded in x around inf 98.3%

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

       1. associate-*r/ 98.3%

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

       2. metadata-eval 98.3%

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

   14. Simplified 98.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 17000000:\\ \;\;\;\;\log \left(e^{-\sqrt[3]{x}} \cdot e^{\sqrt[3]{1 + x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{{x}^{2}}} + \sqrt[3]{\frac{1}{x}} \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 17000000:\\ \;\;\;\;\log \left(e^{\sqrt[3]{1 + x} - \sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{{x}^{2}}} + \sqrt[3]{\frac{1}{x}} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 17000000.0)
   (log (exp (- (cbrt (+ 1.0 x)) (cbrt x))))
   (/
    1.0
    (*
     x
     (+ (cbrt (+ (/ 1.0 x) (/ 2.0 (pow x 2.0)))) (* (cbrt (/ 1.0 x)) 2.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = log(exp((cbrt((1.0 + x)) - cbrt(x))));
	} else {
		tmp = 1.0 / (x * (cbrt(((1.0 / x) + (2.0 / pow(x, 2.0)))) + (cbrt((1.0 / x)) * 2.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = Math.log(Math.exp((Math.cbrt((1.0 + x)) - Math.cbrt(x))));
	} else {
		tmp = 1.0 / (x * (Math.cbrt(((1.0 / x) + (2.0 / Math.pow(x, 2.0)))) + (Math.cbrt((1.0 / x)) * 2.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 17000000.0)
		tmp = log(exp(Float64(cbrt(Float64(1.0 + x)) - cbrt(x))));
	else
		tmp = Float64(1.0 / Float64(x * Float64(cbrt(Float64(Float64(1.0 / x) + Float64(2.0 / (x ^ 2.0)))) + Float64(cbrt(Float64(1.0 / x)) * 2.0))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 17000000.0], N[Log[N[Exp[N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(x * N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(2.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e7

   1. Initial program 79.7%

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

      1. add-log-exp 80.1%

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

   4. Applied egg-rr 80.1%

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

   if 1.7e7 < x

   1. Initial program 5.6%

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

      1. flip3-- 5.8%

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

      2. div-inv 5.8%

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

      3. rem-cube-cbrt 5.4%

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

      4. rem-cube-cbrt 8.4%

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

      5. +-commutative 8.4%

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

      6. distribute-rgt-out 8.5%

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

      7. +-commutative 8.5%

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

      8. fma-define 8.5%

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

      9. add-exp-log 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. associate-*r/ 8.4%

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

      2. *-rgt-identity 8.4%

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

      3. +-commutative 8.4%

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

      4. associate--l+ 93.1%

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

      5. +-inverses 93.1%

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

      6. metadata-eval 93.1%

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

      7. +-commutative 93.1%

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

      8. exp-prod 92.2%

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

   6. Simplified 92.2%

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

   7. Taylor expanded in x around inf 91.7%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{2 \cdot \sqrt[3]{x}}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 91.7%

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

   9. Simplified 91.7%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\sqrt[3]{x} \cdot 2}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. add-exp-log 92.0%

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

       2. log-pow 92.6%

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

       3. rem-log-exp 92.6%

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

   11. Applied egg-rr 92.6%

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

   12. Taylor expanded in x around inf 98.3%

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

       1. associate-*r/ 98.3%

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

       2. metadata-eval 98.3%

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

   14. Simplified 98.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 17000000:\\ \;\;\;\;\log \left(e^{\sqrt[3]{1 + x} - \sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{{x}^{2}}} + \sqrt[3]{\frac{1}{x}} \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 25000000:\\ \;\;\;\;{\left(1 + x\right)}^{0.3333333333333333} - {\left({x}^{0.16666666666666666}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{{x}^{2}}} + \sqrt[3]{\frac{1}{x}} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 25000000.0)
   (- (pow (+ 1.0 x) 0.3333333333333333) (pow (pow x 0.16666666666666666) 2.0))
   (/
    1.0
    (*
     x
     (+ (cbrt (+ (/ 1.0 x) (/ 2.0 (pow x 2.0)))) (* (cbrt (/ 1.0 x)) 2.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 25000000.0) {
		tmp = pow((1.0 + x), 0.3333333333333333) - pow(pow(x, 0.16666666666666666), 2.0);
	} else {
		tmp = 1.0 / (x * (cbrt(((1.0 / x) + (2.0 / pow(x, 2.0)))) + (cbrt((1.0 / x)) * 2.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 25000000.0) {
		tmp = Math.pow((1.0 + x), 0.3333333333333333) - Math.pow(Math.pow(x, 0.16666666666666666), 2.0);
	} else {
		tmp = 1.0 / (x * (Math.cbrt(((1.0 / x) + (2.0 / Math.pow(x, 2.0)))) + (Math.cbrt((1.0 / x)) * 2.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 25000000.0)
		tmp = Float64((Float64(1.0 + x) ^ 0.3333333333333333) - ((x ^ 0.16666666666666666) ^ 2.0));
	else
		tmp = Float64(1.0 / Float64(x * Float64(cbrt(Float64(Float64(1.0 / x) + Float64(2.0 / (x ^ 2.0)))) + Float64(cbrt(Float64(1.0 / x)) * 2.0))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 25000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[N[Power[x, 0.16666666666666666], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(2.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5e7

   1. Initial program 79.7%

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

      1. add-sqr-sqrt 78.3%

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

      2. pow2 78.3%

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

      3. pow1/3 78.0%

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

      4. sqrt-pow1 78.0%

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

      5. metadata-eval 78.0%

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

   4. Applied egg-rr 78.0%

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

      1. pow1/3 80.1%

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

   6. Applied egg-rr 80.1%

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

   if 2.5e7 < x

   1. Initial program 5.6%

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

      1. flip3-- 5.8%

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

      2. div-inv 5.8%

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

      3. rem-cube-cbrt 5.4%

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

      4. rem-cube-cbrt 8.4%

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

      5. +-commutative 8.4%

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

      6. distribute-rgt-out 8.5%

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

      7. +-commutative 8.5%

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

      8. fma-define 8.5%

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

      9. add-exp-log 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. associate-*r/ 8.4%

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

      2. *-rgt-identity 8.4%

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

      3. +-commutative 8.4%

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

      4. associate--l+ 93.1%

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

      5. +-inverses 93.1%

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

      6. metadata-eval 93.1%

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

      7. +-commutative 93.1%

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

      8. exp-prod 92.2%

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

   6. Simplified 92.2%

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

   7. Taylor expanded in x around inf 91.7%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{2 \cdot \sqrt[3]{x}}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 91.7%

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

   9. Simplified 91.7%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\sqrt[3]{x} \cdot 2}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. add-exp-log 92.0%

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

       2. log-pow 92.6%

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

       3. rem-log-exp 92.6%

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

   11. Applied egg-rr 92.6%

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

   12. Taylor expanded in x around inf 98.3%

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

       1. associate-*r/ 98.3%

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

       2. metadata-eval 98.3%

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

   14. Simplified 98.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 25000000:\\ \;\;\;\;{\left(1 + x\right)}^{0.3333333333333333} - {\left({x}^{0.16666666666666666}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{{x}^{2}}} + \sqrt[3]{\frac{1}{x}} \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 25000000:\\ \;\;\;\;{\left(1 + x\right)}^{0.3333333333333333} - {\left({x}^{0.16666666666666666}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(3 \cdot \sqrt[3]{\frac{1}{x}} + 0.6666666666666666 \cdot \sqrt[3]{\frac{1}{{x}^{4}}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 25000000.0)
   (- (pow (+ 1.0 x) 0.3333333333333333) (pow (pow x 0.16666666666666666) 2.0))
   (/
    1.0
    (*
     x
     (+
      (* 3.0 (cbrt (/ 1.0 x)))
      (* 0.6666666666666666 (cbrt (/ 1.0 (pow x 4.0)))))))))

C

double code(double x) {
	double tmp;
	if (x <= 25000000.0) {
		tmp = pow((1.0 + x), 0.3333333333333333) - pow(pow(x, 0.16666666666666666), 2.0);
	} else {
		tmp = 1.0 / (x * ((3.0 * cbrt((1.0 / x))) + (0.6666666666666666 * cbrt((1.0 / pow(x, 4.0))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 25000000.0) {
		tmp = Math.pow((1.0 + x), 0.3333333333333333) - Math.pow(Math.pow(x, 0.16666666666666666), 2.0);
	} else {
		tmp = 1.0 / (x * ((3.0 * Math.cbrt((1.0 / x))) + (0.6666666666666666 * Math.cbrt((1.0 / Math.pow(x, 4.0))))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 25000000.0)
		tmp = Float64((Float64(1.0 + x) ^ 0.3333333333333333) - ((x ^ 0.16666666666666666) ^ 2.0));
	else
		tmp = Float64(1.0 / Float64(x * Float64(Float64(3.0 * cbrt(Float64(1.0 / x))) + Float64(0.6666666666666666 * cbrt(Float64(1.0 / (x ^ 4.0)))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 25000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[N[Power[x, 0.16666666666666666], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(N[(3.0 * N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[N[(1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5e7

   1. Initial program 79.7%

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

      1. add-sqr-sqrt 78.3%

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

      2. pow2 78.3%

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

      3. pow1/3 78.0%

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

      4. sqrt-pow1 78.0%

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

      5. metadata-eval 78.0%

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

   4. Applied egg-rr 78.0%

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

      1. pow1/3 80.1%

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

   6. Applied egg-rr 80.1%

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

   if 2.5e7 < x

   1. Initial program 5.6%

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

      1. flip3-- 5.8%

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

      2. div-inv 5.8%

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

      3. rem-cube-cbrt 5.4%

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

      4. rem-cube-cbrt 8.4%

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

      5. +-commutative 8.4%

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

      6. distribute-rgt-out 8.5%

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

      7. +-commutative 8.5%

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

      8. fma-define 8.5%

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

      9. add-exp-log 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. associate-*r/ 8.4%

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

      2. *-rgt-identity 8.4%

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

      3. +-commutative 8.4%

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

      4. associate--l+ 93.1%

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

      5. +-inverses 93.1%

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

      6. metadata-eval 93.1%

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

      7. +-commutative 93.1%

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

      8. exp-prod 92.2%

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

   6. Simplified 92.2%

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

   7. Taylor expanded in x around inf 91.7%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{2 \cdot \sqrt[3]{x}}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 91.7%

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

   9. Simplified 91.7%

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

   10. Taylor expanded in x around inf 98.3%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\sqrt[3]{\frac{1}{x}} + \left(0.6666666666666666 \cdot \sqrt[3]{\frac{1}{{x}^{4}}} + 2 \cdot \sqrt[3]{\frac{1}{x}}\right)\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 98.3%

          \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x}} + \color{blue}{\left(2 \cdot \sqrt[3]{\frac{1}{x}} + 0.6666666666666666 \cdot \sqrt[3]{\frac{1}{{x}^{4}}}\right)}\right)} \]

       2. associate-+r+ 98.3%

          \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{x}} + 2 \cdot \sqrt[3]{\frac{1}{x}}\right) + 0.6666666666666666 \cdot \sqrt[3]{\frac{1}{{x}^{4}}}\right)}} \]

       3. distribute-rgt1-in 98.3%

          \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(2 + 1\right) \cdot \sqrt[3]{\frac{1}{x}}} + 0.6666666666666666 \cdot \sqrt[3]{\frac{1}{{x}^{4}}}\right)} \]

       4. metadata-eval 98.3%

          \[\leadsto \frac{1}{x \cdot \left(\color{blue}{3} \cdot \sqrt[3]{\frac{1}{x}} + 0.6666666666666666 \cdot \sqrt[3]{\frac{1}{{x}^{4}}}\right)} \]

   12. Simplified 98.3%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(3 \cdot \sqrt[3]{\frac{1}{x}} + 0.6666666666666666 \cdot \sqrt[3]{\frac{1}{{x}^{4}}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 25000000:\\ \;\;\;\;{\left(1 + x\right)}^{0.3333333333333333} - {\left({x}^{0.16666666666666666}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(3 \cdot \sqrt[3]{\frac{1}{x}} + 0.6666666666666666 \cdot \sqrt[3]{\frac{1}{{x}^{4}}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 28000000:\\ \;\;\;\;{\left(1 + x\right)}^{0.3333333333333333} - {\left({x}^{0.16666666666666666}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(3 \cdot \sqrt[3]{\frac{1}{x}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 28000000.0)
   (- (pow (+ 1.0 x) 0.3333333333333333) (pow (pow x 0.16666666666666666) 2.0))
   (/ 1.0 (* x (* 3.0 (cbrt (/ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 28000000.0) {
		tmp = pow((1.0 + x), 0.3333333333333333) - pow(pow(x, 0.16666666666666666), 2.0);
	} else {
		tmp = 1.0 / (x * (3.0 * cbrt((1.0 / x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 28000000.0) {
		tmp = Math.pow((1.0 + x), 0.3333333333333333) - Math.pow(Math.pow(x, 0.16666666666666666), 2.0);
	} else {
		tmp = 1.0 / (x * (3.0 * Math.cbrt((1.0 / x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 28000000.0)
		tmp = Float64((Float64(1.0 + x) ^ 0.3333333333333333) - ((x ^ 0.16666666666666666) ^ 2.0));
	else
		tmp = Float64(1.0 / Float64(x * Float64(3.0 * cbrt(Float64(1.0 / x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 28000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[N[Power[x, 0.16666666666666666], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(3.0 * N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8e7

   1. Initial program 79.7%

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

      1. add-sqr-sqrt 78.3%

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

      2. pow2 78.3%

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

      3. pow1/3 78.0%

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

      4. sqrt-pow1 78.0%

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

      5. metadata-eval 78.0%

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

   4. Applied egg-rr 78.0%

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

      1. pow1/3 80.1%

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

   6. Applied egg-rr 80.1%

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

   if 2.8e7 < x

   1. Initial program 5.6%

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

      1. flip3-- 5.8%

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

      2. div-inv 5.8%

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

      3. rem-cube-cbrt 5.4%

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

      4. rem-cube-cbrt 8.4%

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

      5. +-commutative 8.4%

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

      6. distribute-rgt-out 8.5%

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

      7. +-commutative 8.5%

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

      8. fma-define 8.5%

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

      9. add-exp-log 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. associate-*r/ 8.4%

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

      2. *-rgt-identity 8.4%

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

      3. +-commutative 8.4%

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

      4. associate--l+ 93.1%

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

      5. +-inverses 93.1%

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

      6. metadata-eval 93.1%

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

      7. +-commutative 93.1%

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

      8. exp-prod 92.2%

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

   6. Simplified 92.2%

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

   7. Taylor expanded in x around inf 91.7%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{2 \cdot \sqrt[3]{x}}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 91.7%

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

   9. Simplified 91.7%

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

   10. Taylor expanded in x around inf 98.2%

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

       1. distribute-rgt1-in 98.2%

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

       2. metadata-eval 98.2%

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

   12. Simplified 98.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 28000000:\\ \;\;\;\;{\left(1 + x\right)}^{0.3333333333333333} - {\left({x}^{0.16666666666666666}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(3 \cdot \sqrt[3]{\frac{1}{x}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 30500000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(3 \cdot \sqrt[3]{\frac{1}{x}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 30500000.0)
   (- (cbrt (+ 1.0 x)) (cbrt x))
   (/ 1.0 (* x (* 3.0 (cbrt (/ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 30500000.0) {
		tmp = cbrt((1.0 + x)) - cbrt(x);
	} else {
		tmp = 1.0 / (x * (3.0 * cbrt((1.0 / x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 30500000.0) {
		tmp = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	} else {
		tmp = 1.0 / (x * (3.0 * Math.cbrt((1.0 / x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 30500000.0)
		tmp = Float64(cbrt(Float64(1.0 + x)) - cbrt(x));
	else
		tmp = Float64(1.0 / Float64(x * Float64(3.0 * cbrt(Float64(1.0 / x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 30500000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(3.0 * N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.05e7

   1. Initial program 79.7%

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

   if 3.05e7 < x

   1. Initial program 5.6%

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

      1. flip3-- 5.8%

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

      2. div-inv 5.8%

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

      3. rem-cube-cbrt 5.4%

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

      4. rem-cube-cbrt 8.4%

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

      5. +-commutative 8.4%

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

      6. distribute-rgt-out 8.5%

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

      7. +-commutative 8.5%

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

      8. fma-define 8.5%

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

      9. add-exp-log 8.4%

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

   4. Applied egg-rr 8.4%

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

      1. associate-*r/ 8.4%

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

      2. *-rgt-identity 8.4%

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

      3. +-commutative 8.4%

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

      4. associate--l+ 93.1%

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

      5. +-inverses 93.1%

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

      6. metadata-eval 93.1%

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

      7. +-commutative 93.1%

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

      8. exp-prod 92.2%

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

   6. Simplified 92.2%

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

   7. Taylor expanded in x around inf 91.7%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{2 \cdot \sqrt[3]{x}}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 91.7%

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

   9. Simplified 91.7%

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

   10. Taylor expanded in x around inf 98.2%

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

       1. distribute-rgt1-in 98.2%

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

       2. metadata-eval 98.2%

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

   12. Simplified 98.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 30500000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(3 \cdot \sqrt[3]{\frac{1}{x}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot \left(3 \cdot \sqrt[3]{\frac{1}{x}}\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* x (* 3.0 (cbrt (/ 1.0 x))))))

C

double code(double x) {
	return 1.0 / (x * (3.0 * cbrt((1.0 / x))));
}

Java

public static double code(double x) {
	return 1.0 / (x * (3.0 * Math.cbrt((1.0 / x))));
}

Julia

function code(x)
	return Float64(1.0 / Float64(x * Float64(3.0 * cbrt(Float64(1.0 / x)))))
end

Wolfram

code[x_] := N[(1.0 / N[(x * N[(3.0 * N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.8%

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

   1. flip3-- 9.1%

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

   2. div-inv 9.1%

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

   3. rem-cube-cbrt 8.8%

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

   4. rem-cube-cbrt 12.3%

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

   5. +-commutative 12.3%

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

   6. distribute-rgt-out 12.4%

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

   7. +-commutative 12.4%

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

   8. fma-define 12.3%

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

   9. add-exp-log 12.3%

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

4. Applied egg-rr 12.2%

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

   1. associate-*r/ 12.2%

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

   2. *-rgt-identity 12.2%

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

   3. +-commutative 12.2%

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

   4. associate--l+ 93.3%

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

   5. +-inverses 93.3%

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

   6. metadata-eval 93.3%

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

   7. +-commutative 93.3%

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

   8. exp-prod 92.4%

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

6. Simplified 92.4%

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

7. Taylor expanded in x around inf 89.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{2 \cdot \sqrt[3]{x}}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)} \]
8. Step-by-step derivation

   1. *-commutative 89.5%

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

9. Simplified 89.5%

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

10. Taylor expanded in x around inf 95.7%

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

    1. distribute-rgt1-in 95.7%

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

    2. metadata-eval 95.7%

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

12. Simplified 95.7%

    \[\leadsto \frac{1}{\color{blue}{x \cdot \left(3 \cdot \sqrt[3]{\frac{1}{x}}\right)}} \]
13. Add Preprocessing

Alternative 10: 20.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{x} \cdot 0.25}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* (cbrt x) 0.25) x))

C

double code(double x) {
	return (cbrt(x) * 0.25) / x;
}

Java

public static double code(double x) {
	return (Math.cbrt(x) * 0.25) / x;
}

Julia

function code(x)
	return Float64(Float64(cbrt(x) * 0.25) / x)
end

Wolfram

code[x_] := N[(N[(N[Power[x, 1/3], $MachinePrecision] * 0.25), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 8.8%

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

   1. flip3-- 9.1%

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

   2. div-inv 9.1%

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

   3. rem-cube-cbrt 8.8%

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

   4. rem-cube-cbrt 12.3%

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

   5. +-commutative 12.3%

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

   6. distribute-rgt-out 12.4%

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

   7. +-commutative 12.4%

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

   8. fma-define 12.3%

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

   9. add-exp-log 12.3%

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

4. Applied egg-rr 12.2%

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

   1. associate-*r/ 12.2%

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

   2. *-rgt-identity 12.2%

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

   3. +-commutative 12.2%

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

   4. associate--l+ 93.3%

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

   5. +-inverses 93.3%

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

   6. metadata-eval 93.3%

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

   7. +-commutative 93.3%

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

   8. exp-prod 92.4%

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

6. Simplified 92.4%

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

7. Taylor expanded in x around inf 89.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{2 \cdot \sqrt[3]{x}}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)} \]
8. Step-by-step derivation

   1. *-commutative 89.5%

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

9. Simplified 89.5%

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

10. Taylor expanded in x around inf 20.8%

    \[\leadsto \color{blue}{\frac{-0.25 \cdot \sqrt[3]{x} + 0.5 \cdot \sqrt[3]{x}}{x}} \]
11. Step-by-step derivation

    1. distribute-rgt-out 20.8%

       \[\leadsto \frac{\color{blue}{\sqrt[3]{x} \cdot \left(-0.25 + 0.5\right)}}{x} \]

    2. metadata-eval 20.8%

       \[\leadsto \frac{\sqrt[3]{x} \cdot \color{blue}{0.25}}{x} \]

12. Simplified 20.8%

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot 0.25}{x}} \]
13. Add Preprocessing

Alternative 11: 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 8.8%

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

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

   5. fabs-neg 5.6%

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

   6. unpow1/3 5.6%

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

   7. metadata-eval 5.6%

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

   8. pow-sqr 5.6%

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

   9. fabs-sqr 5.6%

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

   10. pow-sqr 5.6%

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

   11. metadata-eval 5.6%

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

   12. unpow1/3 5.6%

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

5. Simplified 5.6%

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

6. Taylor expanded in x around inf 5.6%

   \[\leadsto \color{blue}{\sqrt[3]{x}} \]
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 2024136 
(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)))