2cbrt (problem 3.3.4)

Percentage Accurate: 53.2% → 99.2%

Time: 14.2s

Alternatives: 11

Speedup: 1.0×

Specification

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: 53.2% 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: 99.2% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (/ 1.0 (fma (cbrt x) (+ (cbrt x) (cbrt x)) (pow t_0 2.0)))
     (/ 1.0 (fma (cbrt x) (+ (cbrt x) t_0) (cbrt (pow (+ 1.0 x) 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 / fma(cbrt(x), (cbrt(x) + cbrt(x)), pow(t_0, 2.0));
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + t_0), cbrt(pow((1.0 + x), 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 / fma(cbrt(x), Float64(cbrt(x) + cbrt(x)), (t_0 ^ 2.0)));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + t_0), cbrt((Float64(1.0 + x) ^ 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[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] + N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 0.0

   1. Initial program 4.3%

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

      1. flip3-- 4.3%

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

      2. div-inv 4.3%

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

      3. rem-cube-cbrt 3.9%

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

      4. rem-cube-cbrt 4.3%

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

      5. +-commutative 4.3%

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

      6. distribute-rgt-out 4.3%

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

      7. +-commutative 4.3%

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

      8. fma-def 4.3%

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

      9. add-exp-log 4.3%

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

   4. Applied egg-rr 2.2%

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

      1. associate-*r/ 2.2%

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

      2. *-rgt-identity 2.2%

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

      3. +-commutative 2.2%

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

      4. associate--l+ 49.9%

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

      5. +-inverses 49.9%

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

      6. metadata-eval 49.9%

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

      7. +-commutative 49.9%

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

      8. exp-prod 49.4%

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

   6. Simplified 49.4%

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

      1. metadata-eval 49.4%

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

      2. prod-exp 49.4%

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

      3. pow-prod-down 50.1%

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

      4. pow2 50.1%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left({\left(e^{0.3333333333333333}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}^{2}}\right)} \]

      5. pow-exp 49.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(e^{0.3333333333333333 \cdot \mathsf{log1p}\left(x\right)}\right)}}^{2}\right)} \]

      6. log1p-udef 49.9%

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

      7. +-commutative 49.9%

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

      8. log-pow 50.0%

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

      9. add-exp-log 49.8%

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

      10. pow1/3 98.4%

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

      11. pow2 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. unpow2 98.4%

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

      2. +-commutative 98.4%

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

   10. Simplified 98.4%

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

       1. +-commutative 98.4%

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

       2. add-cube-cbrt 98.3%

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

       3. pow3 98.3%

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

   12. Applied egg-rr 98.3%

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

   13. Taylor expanded in x around inf 50.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{{x}^{0.3333333333333333}} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   14. Step-by-step derivation

       1. unpow1/3 98.4%

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

   15. Simplified 98.4%

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

   if 0.0 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 98.2%

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

      1. flip3-- 98.1%

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

      2. div-inv 98.1%

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

      3. rem-cube-cbrt 98.4%

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

      4. rem-cube-cbrt 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt-out 99.8%

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

      7. +-commutative 99.8%

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

      8. fma-def 99.8%

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

      9. add-exp-log 99.8%

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

   4. Applied egg-rr 98.3%

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

      1. associate-*r/ 98.3%

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

      2. *-rgt-identity 98.3%

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

      3. +-commutative 98.3%

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

      4. associate--l+ 98.3%

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

      5. +-inverses 98.3%

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

      6. metadata-eval 98.3%

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

      7. +-commutative 98.3%

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

      8. exp-prod 98.3%

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

   6. Simplified 98.3%

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

      1. metadata-eval 98.3%

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

      2. prod-exp 98.3%

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

      3. pow-prod-down 98.3%

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

      4. pow2 98.3%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left({\left(e^{0.3333333333333333}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}^{2}}\right)} \]

      5. pow-exp 98.3%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(e^{0.3333333333333333 \cdot \mathsf{log1p}\left(x\right)}\right)}}^{2}\right)} \]

      6. log1p-udef 98.3%

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

      7. +-commutative 98.3%

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

      8. log-pow 98.3%

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

      9. add-exp-log 98.3%

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

      10. pow1/3 99.8%

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

      11. pow2 99.8%

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

      12. cbrt-unprod 99.9%

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

      13. pow2 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.1%

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

Alternative 2: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x}, {t_0}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + {x}^{3}} \cdot \sqrt[3]{\frac{1}{\mathsf{fma}\left(x, x, 1\right) - x}} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 4e-6)
     (/ 1.0 (fma (cbrt x) (+ (cbrt x) (cbrt x)) (pow t_0 2.0)))
     (-
      (* (cbrt (+ 1.0 (pow x 3.0))) (cbrt (/ 1.0 (- (fma x x 1.0) x))))
      (cbrt x)))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 4e-6) {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt(x)), pow(t_0, 2.0));
	} else {
		tmp = (cbrt((1.0 + pow(x, 3.0))) * cbrt((1.0 / (fma(x, x, 1.0) - x)))) - cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 4e-6)
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(x)), (t_0 ^ 2.0)));
	else
		tmp = Float64(Float64(cbrt(Float64(1.0 + (x ^ 3.0))) * cbrt(Float64(1.0 / Float64(fma(x, x, 1.0) - x)))) - cbrt(x));
	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], 4e-6], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(1.0 + N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(1.0 / N[(N[(x * x + 1.0), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 3.99999999999999982e-6

   1. Initial program 5.0%

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

      1. flip3-- 5.0%

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

      2. div-inv 5.0%

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

      3. rem-cube-cbrt 4.7%

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

      4. rem-cube-cbrt 6.5%

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

      5. +-commutative 6.5%

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

      6. distribute-rgt-out 6.5%

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

      7. +-commutative 6.5%

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

      8. fma-def 6.5%

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

      9. add-exp-log 6.5%

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

   4. Applied egg-rr 4.5%

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

      1. associate-*r/ 4.5%

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

      2. *-rgt-identity 4.5%

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

      3. +-commutative 4.5%

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

      4. associate--l+ 51.0%

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

      5. +-inverses 51.0%

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

      6. metadata-eval 51.0%

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

      7. +-commutative 51.0%

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

      8. exp-prod 50.5%

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

   6. Simplified 50.5%

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

      1. metadata-eval 50.5%

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

      2. prod-exp 50.5%

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

      3. pow-prod-down 51.3%

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

      4. pow2 51.3%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left({\left(e^{0.3333333333333333}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}^{2}}\right)} \]

      5. pow-exp 51.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(e^{0.3333333333333333 \cdot \mathsf{log1p}\left(x\right)}\right)}}^{2}\right)} \]

      6. log1p-udef 51.0%

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

      7. +-commutative 51.0%

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

      8. log-pow 51.1%

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

      9. add-exp-log 50.9%

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

      10. pow1/3 98.4%

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

      11. pow2 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. unpow2 98.4%

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

      2. +-commutative 98.4%

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

   10. Simplified 98.4%

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

       1. +-commutative 98.4%

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

       2. add-cube-cbrt 98.3%

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

       3. pow3 98.3%

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

   12. Applied egg-rr 98.3%

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

   13. Taylor expanded in x around inf 51.3%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{{x}^{0.3333333333333333}} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   14. Step-by-step derivation

       1. unpow1/3 98.0%

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

   15. Simplified 98.0%

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

   if 3.99999999999999982e-6 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.6%

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

      1. pow1/3 98.4%

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

      2. flip3-+ 98.4%

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

      3. div-inv 98.4%

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

      4. unpow-prod-down 98.4%

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

      5. pow3 98.4%

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

      6. metadata-eval 98.4%

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

      7. +-commutative 98.4%

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

      8. pow3 98.4%

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

      9. metadata-eval 98.4%

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

      10. *-rgt-identity 98.4%

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

      11. associate-+r- 98.4%

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

      12. fma-def 98.4%

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

   4. Applied egg-rr 98.4%

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

      1. unpow1/3 99.6%

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

      2. unpow1/3 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + {x}^{3}} \cdot \sqrt[3]{\frac{1}{\mathsf{fma}\left(x, x, 1\right) - x}} - \sqrt[3]{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{5}}} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + {x}^{3}} \cdot \sqrt[3]{\frac{1}{\mathsf{fma}\left(x, x, 1\right) - x}} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ 1.0 x)) (cbrt x)) 4e-6)
   (+
    (* (cbrt (/ 1.0 (pow x 5.0))) -0.1111111111111111)
    (* 0.3333333333333333 (cbrt (/ 1.0 (pow x 2.0)))))
   (-
    (* (cbrt (+ 1.0 (pow x 3.0))) (cbrt (/ 1.0 (- (fma x x 1.0) x))))
    (cbrt x))))

C

double code(double x) {
	double tmp;
	if ((cbrt((1.0 + x)) - cbrt(x)) <= 4e-6) {
		tmp = (cbrt((1.0 / pow(x, 5.0))) * -0.1111111111111111) + (0.3333333333333333 * cbrt((1.0 / pow(x, 2.0))));
	} else {
		tmp = (cbrt((1.0 + pow(x, 3.0))) * cbrt((1.0 / (fma(x, x, 1.0) - x)))) - cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(1.0 + x)) - cbrt(x)) <= 4e-6)
		tmp = Float64(Float64(cbrt(Float64(1.0 / (x ^ 5.0))) * -0.1111111111111111) + Float64(0.3333333333333333 * cbrt(Float64(1.0 / (x ^ 2.0)))));
	else
		tmp = Float64(Float64(cbrt(Float64(1.0 + (x ^ 3.0))) * cbrt(Float64(1.0 / Float64(fma(x, x, 1.0) - x)))) - cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 4e-6], N[(N[(N[Power[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision] + N[(0.3333333333333333 * N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(1.0 + N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(1.0 / N[(N[(x * x + 1.0), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 3.99999999999999982e-6

   1. Initial program 5.0%

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

      1. add-sqr-sqrt 3.0%

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

      2. add-sqr-sqrt 3.0%

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

      3. difference-of-squares 3.1%

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

      4. pow1/3 3.1%

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

      5. sqrt-pow1 3.1%

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

      6. metadata-eval 3.1%

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

      7. pow1/3 3.1%

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

      8. sqrt-pow1 3.1%

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

      9. metadata-eval 3.1%

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

      10. pow1/3 1.7%

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

      11. sqrt-pow1 1.7%

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

      12. metadata-eval 1.7%

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

      13. pow1/3 2.9%

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

      14. sqrt-pow1 3.0%

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

      15. metadata-eval 3.0%

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

   4. Applied egg-rr 3.0%

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

   5. Taylor expanded in x around inf 43.9%

      \[\leadsto \color{blue}{-0.1388888888888889 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{0.3333333333333333} + \left(0.027777777777777776 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{0.3333333333333333} + 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 43.9%

         \[\leadsto \color{blue}{\left(-0.1388888888888889 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{0.3333333333333333} + 0.027777777777777776 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{0.3333333333333333}\right) + 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} \]

      2. distribute-rgt-out 43.9%

         \[\leadsto \color{blue}{{\left(\frac{1}{{x}^{5}}\right)}^{0.3333333333333333} \cdot \left(-0.1388888888888889 + 0.027777777777777776\right)} + 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} \]

      3. unpow1/3 46.1%

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{5}}}} \cdot \left(-0.1388888888888889 + 0.027777777777777776\right) + 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} \]

      4. metadata-eval 46.1%

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \color{blue}{-0.1111111111111111} + 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} \]

      5. unpow1/3 49.5%

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}}} \]

   7. Simplified 49.5%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{5}}} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]

   if 3.99999999999999982e-6 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.6%

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

      1. pow1/3 98.4%

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

      2. flip3-+ 98.4%

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

      3. div-inv 98.4%

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

      4. unpow-prod-down 98.4%

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

      5. pow3 98.4%

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

      6. metadata-eval 98.4%

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

      7. +-commutative 98.4%

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

      8. pow3 98.4%

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

      9. metadata-eval 98.4%

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

      10. *-rgt-identity 98.4%

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

      11. associate-+r- 98.4%

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

      12. fma-def 98.4%

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

   4. Applied egg-rr 98.4%

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

      1. unpow1/3 99.6%

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

      2. unpow1/3 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{5}}} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + {x}^{3}} \cdot \sqrt[3]{\frac{1}{\mathsf{fma}\left(x, x, 1\right) - x}} - \sqrt[3]{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{5}}} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (cbrt (+ 1.0 x)) (cbrt x))))
   (if (<= t_0 4e-6)
     (+
      (* (cbrt (/ 1.0 (pow x 5.0))) -0.1111111111111111)
      (* 0.3333333333333333 (cbrt (/ 1.0 (pow x 2.0)))))
     t_0)))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x)) - cbrt(x);
	double tmp;
	if (t_0 <= 4e-6) {
		tmp = (cbrt((1.0 / pow(x, 5.0))) * -0.1111111111111111) + (0.3333333333333333 * cbrt((1.0 / pow(x, 2.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	double tmp;
	if (t_0 <= 4e-6) {
		tmp = (Math.cbrt((1.0 / Math.pow(x, 5.0))) * -0.1111111111111111) + (0.3333333333333333 * Math.cbrt((1.0 / Math.pow(x, 2.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(Float64(1.0 + x)) - cbrt(x))
	tmp = 0.0
	if (t_0 <= 4e-6)
		tmp = Float64(Float64(cbrt(Float64(1.0 / (x ^ 5.0))) * -0.1111111111111111) + Float64(0.3333333333333333 * cbrt(Float64(1.0 / (x ^ 2.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-6], N[(N[(N[Power[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision] + N[(0.3333333333333333 * N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 3.99999999999999982e-6

   1. Initial program 5.0%

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

      1. add-sqr-sqrt 3.0%

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

      2. add-sqr-sqrt 3.0%

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

      3. difference-of-squares 3.1%

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

      4. pow1/3 3.1%

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

      5. sqrt-pow1 3.1%

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

      6. metadata-eval 3.1%

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

      7. pow1/3 3.1%

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

      8. sqrt-pow1 3.1%

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

      9. metadata-eval 3.1%

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

      10. pow1/3 1.7%

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

      11. sqrt-pow1 1.7%

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

      12. metadata-eval 1.7%

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

      13. pow1/3 2.9%

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

      14. sqrt-pow1 3.0%

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

      15. metadata-eval 3.0%

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

   4. Applied egg-rr 3.0%

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

   5. Taylor expanded in x around inf 43.9%

      \[\leadsto \color{blue}{-0.1388888888888889 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{0.3333333333333333} + \left(0.027777777777777776 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{0.3333333333333333} + 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 43.9%

         \[\leadsto \color{blue}{\left(-0.1388888888888889 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{0.3333333333333333} + 0.027777777777777776 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{0.3333333333333333}\right) + 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} \]

      2. distribute-rgt-out 43.9%

         \[\leadsto \color{blue}{{\left(\frac{1}{{x}^{5}}\right)}^{0.3333333333333333} \cdot \left(-0.1388888888888889 + 0.027777777777777776\right)} + 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} \]

      3. unpow1/3 46.1%

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{5}}}} \cdot \left(-0.1388888888888889 + 0.027777777777777776\right) + 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} \]

      4. metadata-eval 46.1%

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \color{blue}{-0.1111111111111111} + 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} \]

      5. unpow1/3 49.5%

         \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}}} \]

   7. Simplified 49.5%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{5}}} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]

   if 3.99999999999999982e-6 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.6%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{5}}} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% 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 53.1%

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

   1. flip3-- 53.0%

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

   2. div-inv 53.0%

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

   3. rem-cube-cbrt 53.0%

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

   4. rem-cube-cbrt 53.9%

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

   5. +-commutative 53.9%

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

   6. distribute-rgt-out 53.9%

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

   7. +-commutative 53.9%

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

   8. fma-def 53.9%

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

   9. add-exp-log 53.9%

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

4. Applied egg-rr 52.1%

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

   1. associate-*r/ 52.1%

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

   2. *-rgt-identity 52.1%

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

   3. +-commutative 52.1%

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

   4. associate--l+ 75.1%

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

   5. +-inverses 75.1%

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

   6. metadata-eval 75.1%

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

   7. +-commutative 75.1%

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

   8. exp-prod 74.8%

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

6. Simplified 74.8%

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

   1. metadata-eval 74.8%

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

   2. prod-exp 74.8%

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

   3. pow-prod-down 75.2%

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

   4. pow2 75.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left({\left(e^{0.3333333333333333}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}^{2}}\right)} \]

   5. pow-exp 75.1%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(e^{0.3333333333333333 \cdot \mathsf{log1p}\left(x\right)}\right)}}^{2}\right)} \]

   6. log1p-udef 75.1%

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

   7. +-commutative 75.1%

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

   8. log-pow 75.1%

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

   9. add-exp-log 75.0%

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

   10. pow1/3 99.1%

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

   11. pow2 99.1%

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

8. Applied egg-rr 99.1%

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

   1. unpow2 99.1%

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

   2. +-commutative 99.1%

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

10. Simplified 99.1%

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

11. Final simplification 99.1%

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

Alternative 6: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (cbrt (+ 1.0 x)) (cbrt x))))
   (if (<= t_0 4e-6) (* 0.3333333333333333 (cbrt (/ 1.0 (pow x 2.0)))) t_0)))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x)) - cbrt(x);
	double tmp;
	if (t_0 <= 4e-6) {
		tmp = 0.3333333333333333 * cbrt((1.0 / pow(x, 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	double tmp;
	if (t_0 <= 4e-6) {
		tmp = 0.3333333333333333 * Math.cbrt((1.0 / Math.pow(x, 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(Float64(1.0 + x)) - cbrt(x))
	tmp = 0.0
	if (t_0 <= 4e-6)
		tmp = Float64(0.3333333333333333 * cbrt(Float64(1.0 / (x ^ 2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-6], N[(0.3333333333333333 * N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 3.99999999999999982e-6

   1. Initial program 5.0%

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

      1. add-sqr-sqrt 3.0%

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

      2. add-sqr-sqrt 3.0%

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

      3. difference-of-squares 3.1%

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

      4. pow1/3 3.1%

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

      5. sqrt-pow1 3.1%

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

      6. metadata-eval 3.1%

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

      7. pow1/3 3.1%

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

      8. sqrt-pow1 3.1%

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

      9. metadata-eval 3.1%

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

      10. pow1/3 1.7%

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

      11. sqrt-pow1 1.7%

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

      12. metadata-eval 1.7%

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

      13. pow1/3 2.9%

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

      14. sqrt-pow1 3.0%

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

      15. metadata-eval 3.0%

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

   4. Applied egg-rr 3.0%

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

   5. Taylor expanded in x around inf 45.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} \]
   6. Step-by-step derivation

      1. unpow1/3 49.1%

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

   7. Simplified 49.1%

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

   if 3.99999999999999982e-6 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.6%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.2% 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 53.1%

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

3. Final simplification 53.1%

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

Alternative 8: 50.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

   1. add-cube-cbrt 52.6%

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

   2. pow3 52.6%

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

4. Applied egg-rr 52.6%

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

5. Taylor expanded in x around 0 29.0%

   \[\leadsto \color{blue}{\left(1 + 0.3333333333333333 \cdot x\right) - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}} \]
6. Step-by-step derivation

   1. associate--l+ 29.0%

      \[\leadsto \color{blue}{1 + \left(0.3333333333333333 \cdot x - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}\right)} \]

   2. *-commutative 29.0%

      \[\leadsto 1 + \left(\color{blue}{x \cdot 0.3333333333333333} - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}\right) \]

   3. metadata-eval 29.0%

      \[\leadsto 1 + \left(x \cdot 0.3333333333333333 - {\color{blue}{1}}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}\right) \]

   4. pow-base-1 29.0%

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

   5. unpow1/3 50.4%

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

   6. *-lft-identity 50.4%

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

7. Simplified 50.4%

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

8. Final simplification 50.4%

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

Alternative 9: 50.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

   1. add-cube-cbrt 52.6%

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

   2. pow3 52.6%

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

4. Applied egg-rr 52.6%

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

5. Taylor expanded in x around 0 28.0%

   \[\leadsto \color{blue}{1 - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}} \]
6. Step-by-step derivation

   1. metadata-eval 28.0%

      \[\leadsto 1 - {\color{blue}{1}}^{0.1111111111111111} \cdot {x}^{0.3333333333333333} \]

   2. pow-base-1 28.0%

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

   3. unpow1/3 50.0%

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

   4. *-lft-identity 50.0%

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

7. Simplified 50.0%

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

8. Final simplification 50.0%

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

Alternative 10: 3.6% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in x around inf 3.7%

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

4. Final simplification 3.7%

   \[\leadsto 0 \]
5. Add Preprocessing

Alternative 11: 49.6% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in x around 0 48.9%

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

4. Final simplification 48.9%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))