2cbrt (problem 3.3.4)

Percentage Accurate: 53.0% → 99.2%

Time: 12.4s

Alternatives: 14

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: 53.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_1, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (+ (cbrt x) t_0)))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (/ 1.0 (* (cbrt x) (+ (cbrt x) t_1)))
     (/ 1.0 (fma (cbrt x) t_1 (cbrt (pow (+ 1.0 x) 2.0)))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = cbrt(x) + t_0;
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 1.0 / (cbrt(x) * (cbrt(x) + t_1));
	} else {
		tmp = 1.0 / fma(cbrt(x), t_1, cbrt(pow((1.0 + x), 2.0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = Float64(cbrt(x) + t_0)
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(1.0 / Float64(cbrt(x) * Float64(cbrt(x) + t_1)));
	else
		tmp = Float64(1.0 / fma(cbrt(x), t_1, 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]}, Block[{t$95$1 = N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $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] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$1 + 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. 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%

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

   3. Applied egg-rr 2.5%

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

      1. associate-*r/ 2.5%

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

      2. *-rgt-identity 2.5%

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

      3. +-commutative 2.5%

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

      4. associate--l+ 53.0%

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

      5. +-inverses 53.0%

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

      6. metadata-eval 53.0%

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

      7. +-commutative 53.0%

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

      8. exp-prod 52.7%

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

   5. Simplified 52.7%

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

   6. Taylor expanded in x around inf 43.7%

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

      1. unpow1/3 45.8%

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

   8. Simplified 45.8%

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

      1. expm1-log1p-u 45.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{x}^{2}}\right)}\right)\right)} \]

      2. expm1-udef 4.6%

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

      3. +-commutative 4.6%

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

      4. unpow2 4.6%

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

      5. cbrt-prod 4.6%

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

      6. pow2 4.6%

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

   10. Applied egg-rr 4.6%

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

       1. expm1-def 98.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {\left(\sqrt[3]{x}\right)}^{2}\right)}\right)\right)} \]

       2. expm1-log1p 98.5%

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

       3. fma-udef 98.5%

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

       4. unpow2 98.5%

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

       5. distribute-lft-out 98.5%

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

       6. +-commutative 98.5%

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

   12. Simplified 98.5%

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

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

   1. Initial program 98.3%

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

      1. flip3-- 98.5%

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

      2. div-inv 98.5%

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

      3. rem-cube-cbrt 98.2%

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

      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%

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

   3. Applied egg-rr 98.6%

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

      1. associate-*r/ 98.6%

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

      2. *-rgt-identity 98.6%

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

      3. +-commutative 98.6%

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

      4. associate--l+ 98.6%

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

      5. +-inverses 98.6%

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

      6. metadata-eval 98.6%

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

      7. +-commutative 98.6%

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

      8. exp-prod 98.6%

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

   5. Simplified 98.6%

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

      1. add-sqr-sqrt 98.6%

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

      2. unpow-prod-down 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. pow-sqr 98.6%

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

   9. Simplified 98.6%

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

       1. *-commutative 98.6%

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

       2. pow-unpow 98.5%

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

       3. pow1/2 98.5%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left({\color{blue}{\left({\left(e^{0.6666666666666666}\right)}^{0.5}\right)}}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}^{2}\right)} \]

       4. pow-exp 98.5%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left({\color{blue}{\left(e^{0.6666666666666666 \cdot 0.5}\right)}}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}^{2}\right)} \]

       5. metadata-eval 98.5%

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

       6. pow-exp 98.5%

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

       7. log1p-udef 98.5%

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

       8. log-pow 98.5%

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

       9. pow1/3 98.5%

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

       10. add-exp-log 99.8%

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

       11. pow2 99.8%

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

       12. cbrt-unprod 99.8%

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

       13. pow2 99.8%

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

       14. +-commutative 99.8%

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

   11. Applied egg-rr 99.8%

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

4. Final simplification 99.3%

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

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{t_0}^{2} + \sqrt[3]{x} \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (+ (cbrt x) t_0)))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (/ 1.0 (* (cbrt x) (+ (cbrt x) t_1)))
     (/ (- (+ 1.0 x) x) (+ (pow t_0 2.0) (* (cbrt x) t_1))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = cbrt(x) + t_0;
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 1.0 / (cbrt(x) * (cbrt(x) + t_1));
	} else {
		tmp = ((1.0 + x) - x) / (pow(t_0, 2.0) + (cbrt(x) * t_1));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double t_1 = Math.cbrt(x) + t_0;
	double tmp;
	if ((t_0 - Math.cbrt(x)) <= 0.0) {
		tmp = 1.0 / (Math.cbrt(x) * (Math.cbrt(x) + t_1));
	} else {
		tmp = ((1.0 + x) - x) / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * t_1));
	}
	return tmp;
}

Julia

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

Wolfram

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

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

   3. Applied egg-rr 2.5%

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

      1. associate-*r/ 2.5%

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

      2. *-rgt-identity 2.5%

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

      3. +-commutative 2.5%

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

      4. associate--l+ 53.0%

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

      5. +-inverses 53.0%

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

      6. metadata-eval 53.0%

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

      7. +-commutative 53.0%

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

      8. exp-prod 52.7%

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

   5. Simplified 52.7%

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

   6. Taylor expanded in x around inf 43.7%

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

      1. unpow1/3 45.8%

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

   8. Simplified 45.8%

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

      1. expm1-log1p-u 45.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{x}^{2}}\right)}\right)\right)} \]

      2. expm1-udef 4.6%

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

      3. +-commutative 4.6%

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

      4. unpow2 4.6%

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

      5. cbrt-prod 4.6%

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

      6. pow2 4.6%

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

   10. Applied egg-rr 4.6%

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

       1. expm1-def 98.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {\left(\sqrt[3]{x}\right)}^{2}\right)}\right)\right)} \]

       2. expm1-log1p 98.5%

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

       3. fma-udef 98.5%

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

       4. unpow2 98.5%

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

       5. distribute-lft-out 98.5%

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

       6. +-commutative 98.5%

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

   12. Simplified 98.5%

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

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

   1. Initial program 98.3%

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

      1. flip-+ 98.3%

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

      2. cbrt-div 98.3%

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

      3. metadata-eval 98.3%

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

      4. fma-neg 98.3%

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

      5. metadata-eval 98.3%

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

      6. sub-neg 98.3%

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

      7. metadata-eval 98.3%

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

   3. Applied egg-rr 98.3%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.3%

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

Alternative 3: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_1, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (+ (cbrt x) t_0)))
   (if (<= (- t_0 (cbrt x)) 4e-6)
     (/ 1.0 (* (cbrt x) (+ (cbrt x) t_1)))
     (/ 1.0 (fma (cbrt x) t_1 (pow (+ 1.0 x) 0.6666666666666666))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = cbrt(x) + t_0;
	double tmp;
	if ((t_0 - cbrt(x)) <= 4e-6) {
		tmp = 1.0 / (cbrt(x) * (cbrt(x) + t_1));
	} else {
		tmp = 1.0 / fma(cbrt(x), t_1, pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = Float64(cbrt(x) + t_0)
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 4e-6)
		tmp = Float64(1.0 / Float64(cbrt(x) * Float64(cbrt(x) + t_1)));
	else
		tmp = Float64(1.0 / fma(cbrt(x), t_1, (Float64(1.0 + x) ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $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] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$1 + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $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 5.9%

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

      1. flip3-- 6.4%

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

      2. div-inv 6.4%

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

      3. rem-cube-cbrt 5.5%

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

      4. rem-cube-cbrt 7.8%

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

      5. +-commutative 7.8%

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

      6. distribute-rgt-out 7.8%

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

      7. +-commutative 7.8%

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

      8. fma-def 7.8%

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

      9. add-exp-log 7.8%

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

   3. Applied egg-rr 4.2%

      \[\leadsto \color{blue}{\left(\left(x + 1\right) - x\right) \cdot \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]
   4. 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+ 52.9%

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

      5. +-inverses 52.9%

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

      6. metadata-eval 52.9%

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

      7. +-commutative 52.9%

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

      8. exp-prod 52.6%

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

   5. Simplified 52.6%

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

   6. Taylor expanded in x around inf 44.7%

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

      1. unpow1/3 46.7%

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

   8. Simplified 46.7%

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

      1. expm1-log1p-u 46.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{x}^{2}}\right)}\right)\right)} \]

      2. expm1-udef 6.8%

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

      3. +-commutative 6.8%

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

      4. unpow2 6.8%

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

      5. cbrt-prod 6.8%

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

      6. pow2 6.8%

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

   10. Applied egg-rr 6.8%

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

       1. expm1-def 97.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {\left(\sqrt[3]{x}\right)}^{2}\right)}\right)\right)} \]

       2. expm1-log1p 97.5%

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

       3. fma-udef 97.4%

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

       4. unpow2 97.4%

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

       5. distribute-lft-out 97.5%

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

       6. +-commutative 97.5%

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

   12. Simplified 97.5%

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

   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. Step-by-step derivation

      1. flip3-- 99.6%

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

      2. div-inv 99.5%

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

      3. rem-cube-cbrt 99.6%

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

      4. rem-cube-cbrt 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-rgt-out 99.9%

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

      7. +-commutative 99.9%

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

      8. fma-def 99.9%

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

      9. add-exp-log 99.8%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-inverses 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

      8. exp-prod 99.9%

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

   5. Simplified 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. unpow-prod-down 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. pow-sqr 99.9%

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

   9. Simplified 99.9%

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

       1. *-commutative 99.9%

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

       2. pow-unpow 99.9%

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

       3. pow1/2 99.9%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left({\color{blue}{\left({\left(e^{0.6666666666666666}\right)}^{0.5}\right)}}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}^{2}\right)} \]

       4. pow-exp 99.9%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left({\color{blue}{\left(e^{0.6666666666666666 \cdot 0.5}\right)}}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}^{2}\right)} \]

       5. metadata-eval 99.9%

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

       6. pow-exp 99.9%

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

       7. log1p-udef 99.9%

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

       8. log-pow 99.9%

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

       9. pow1/3 99.8%

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

       10. add-exp-log 99.9%

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

       11. pow2 99.9%

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

       12. +-commutative 99.9%

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

       13. +-commutative 99.9%

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

   11. Applied egg-rr 99.9%

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

       1. pow2 99.9%

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

       2. pow1/3 99.9%

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

       3. +-commutative 99.9%

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

       4. pow-pow 99.9%

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

       5. +-commutative 99.9%

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

       6. metadata-eval 99.9%

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

   13. Applied egg-rr 99.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]

Alternative 4: 99.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.5%

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

   1. flip3-- 60.6%

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

   2. div-inv 60.6%

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

   3. rem-cube-cbrt 60.3%

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

   4. rem-cube-cbrt 61.4%

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

   5. +-commutative 61.4%

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

   6. distribute-rgt-out 61.4%

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

   7. +-commutative 61.4%

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

   8. fma-def 61.4%

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

   9. add-exp-log 61.4%

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

3. Applied egg-rr 59.9%

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

   1. associate-*r/ 59.9%

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

   2. *-rgt-identity 59.9%

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

   3. +-commutative 59.9%

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

   4. associate--l+ 80.2%

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

   5. +-inverses 80.2%

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

   6. metadata-eval 80.2%

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

   7. +-commutative 80.2%

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

   8. exp-prod 80.1%

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

5. Simplified 80.1%

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

   1. add-sqr-sqrt 80.1%

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

   2. unpow-prod-down 80.4%

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

7. Applied egg-rr 80.4%

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

   1. pow-sqr 80.4%

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

9. Simplified 80.4%

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

    1. *-commutative 80.4%

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

    2. pow-unpow 80.4%

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

    3. pow1/2 80.4%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left({\color{blue}{\left({\left(e^{0.6666666666666666}\right)}^{0.5}\right)}}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}^{2}\right)} \]

    4. pow-exp 80.4%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left({\color{blue}{\left(e^{0.6666666666666666 \cdot 0.5}\right)}}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}^{2}\right)} \]

    5. metadata-eval 80.4%

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

    6. pow-exp 80.2%

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

    7. log1p-udef 80.2%

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

    8. log-pow 80.4%

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

    9. pow1/3 80.5%

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

    10. add-exp-log 99.3%

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

    11. pow2 99.3%

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

    12. +-commutative 99.3%

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

    13. +-commutative 99.3%

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

11. Applied egg-rr 99.3%

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

    1. cbrt-unprod 78.1%

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

    2. metadata-eval 78.1%

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

    3. sub-neg 78.1%

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

    4. flip-- 78.1%

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

    5. metadata-eval 78.1%

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

    6. fma-neg 78.1%

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

    7. metadata-eval 78.1%

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

    8. metadata-eval 78.1%

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

    9. sub-neg 78.1%

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

    10. flip-- 78.1%

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

    11. metadata-eval 78.1%

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

    12. fma-neg 78.1%

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

    13. metadata-eval 78.1%

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

    14. add-cube-cbrt 78.1%

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

    15. cbrt-undiv 78.1%

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

    16. cbrt-undiv 78.1%

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

    17. cbrt-undiv 78.1%

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

13. Applied egg-rr 99.3%

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

14. Final simplification 99.3%

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

Alternative 5: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} + t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{0.3333333333333333}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 4e-6)
     (/ 1.0 (* (cbrt x) (+ (cbrt x) (+ (cbrt x) t_0))))
     (- (pow (exp 0.3333333333333333) (log1p 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 / (cbrt(x) * (cbrt(x) + (cbrt(x) + t_0)));
	} else {
		tmp = pow(exp(0.3333333333333333), log1p(x)) - cbrt(x);
	}
	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)) <= 4e-6) {
		tmp = 1.0 / (Math.cbrt(x) * (Math.cbrt(x) + (Math.cbrt(x) + t_0)));
	} else {
		tmp = Math.pow(Math.exp(0.3333333333333333), Math.log1p(x)) - Math.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 / Float64(cbrt(x) * Float64(cbrt(x) + Float64(cbrt(x) + t_0))));
	else
		tmp = Float64((exp(0.3333333333333333) ^ log1p(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[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Exp[0.3333333333333333], $MachinePrecision], N[Log[1 + x], $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 5.9%

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

      1. flip3-- 6.4%

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

      2. div-inv 6.4%

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

      3. rem-cube-cbrt 5.5%

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

      4. rem-cube-cbrt 7.8%

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

      5. +-commutative 7.8%

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

      6. distribute-rgt-out 7.8%

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

      7. +-commutative 7.8%

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

      8. fma-def 7.8%

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

      9. add-exp-log 7.8%

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

   3. Applied egg-rr 4.2%

      \[\leadsto \color{blue}{\left(\left(x + 1\right) - x\right) \cdot \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]
   4. 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+ 52.9%

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

      5. +-inverses 52.9%

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

      6. metadata-eval 52.9%

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

      7. +-commutative 52.9%

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

      8. exp-prod 52.6%

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

   5. Simplified 52.6%

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

   6. Taylor expanded in x around inf 44.7%

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

      1. unpow1/3 46.7%

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

   8. Simplified 46.7%

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

      1. expm1-log1p-u 46.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{x}^{2}}\right)}\right)\right)} \]

      2. expm1-udef 6.8%

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

      3. +-commutative 6.8%

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

      4. unpow2 6.8%

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

      5. cbrt-prod 6.8%

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

      6. pow2 6.8%

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

   10. Applied egg-rr 6.8%

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

       1. expm1-def 97.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {\left(\sqrt[3]{x}\right)}^{2}\right)}\right)\right)} \]

       2. expm1-log1p 97.5%

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

       3. fma-udef 97.4%

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

       4. unpow2 97.4%

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

       5. distribute-lft-out 97.5%

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

       6. +-commutative 97.5%

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

   12. Simplified 97.5%

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

   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. Step-by-step derivation

      1. add-exp-log 99.6%

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

      2. pow1/3 99.6%

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

      3. log-pow 99.6%

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

      4. +-commutative 99.6%

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

      5. log1p-udef 99.6%

         \[\leadsto e^{0.3333333333333333 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - \sqrt[3]{x} \]

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{e^{0.3333333333333333 \cdot \mathsf{log1p}\left(x\right)}} - \sqrt[3]{x} \]
   4. Step-by-step derivation

      1. exp-prod 99.7%

         \[\leadsto \color{blue}{{\left(e^{0.3333333333333333}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}} - \sqrt[3]{x} \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{{\left(e^{0.3333333333333333}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}} - \sqrt[3]{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{0.3333333333333333}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)} - \sqrt[3]{x}\\ \end{array} \]

Alternative 6: 60.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{0.3333333333333333 \cdot \left(3 \cdot \log t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (- t_0 (cbrt x))))
   (if (<= t_1 0.0)
     (/ 1.0 (fma (cbrt x) (+ (cbrt x) t_0) 1.0))
     (exp (* 0.3333333333333333 (* 3.0 (log t_1)))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = t_0 - cbrt(x);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + t_0), 1.0);
	} else {
		tmp = exp((0.3333333333333333 * (3.0 * log(t_1))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = Float64(t_0 - cbrt(x))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + t_0), 1.0));
	else
		tmp = exp(Float64(0.3333333333333333 * Float64(3.0 * log(t_1))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[N[(0.3333333333333333 * N[(3.0 * N[Log[t$95$1], $MachinePrecision]), $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. 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%

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

   3. Applied egg-rr 2.5%

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

      1. associate-*r/ 2.5%

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

      2. *-rgt-identity 2.5%

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

      3. +-commutative 2.5%

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

      4. associate--l+ 53.0%

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

      5. +-inverses 53.0%

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

      6. metadata-eval 53.0%

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

      7. +-commutative 53.0%

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

      8. exp-prod 52.7%

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

   5. Simplified 52.7%

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

   6. Taylor expanded in x around 0 20.0%

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

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

   1. Initial program 98.3%

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

      1. add-cbrt-cube 98.2%

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

      2. pow1/3 98.3%

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

      3. pow-to-exp 98.3%

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

      4. pow3 98.3%

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

      5. log-pow 98.3%

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

   3. Applied egg-rr 98.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{0.3333333333333333 \cdot \left(3 \cdot \log \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)\right)}\\ \end{array} \]

Alternative 7: 99.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	return Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + t_0), Float64(t_0 * t_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[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 60.5%

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

   1. flip3-- 60.6%

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

   2. div-inv 60.6%

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

   3. rem-cube-cbrt 60.3%

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

   4. rem-cube-cbrt 61.4%

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

   5. +-commutative 61.4%

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

   6. distribute-rgt-out 61.4%

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

   7. +-commutative 61.4%

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

   8. fma-def 61.4%

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

   9. add-exp-log 61.4%

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

3. Applied egg-rr 59.9%

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

   1. associate-*r/ 59.9%

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

   2. *-rgt-identity 59.9%

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

   3. +-commutative 59.9%

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

   4. associate--l+ 80.2%

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

   5. +-inverses 80.2%

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

   6. metadata-eval 80.2%

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

   7. +-commutative 80.2%

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

   8. exp-prod 80.1%

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

5. Simplified 80.1%

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

   1. add-sqr-sqrt 80.1%

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

   2. unpow-prod-down 80.4%

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

7. Applied egg-rr 80.4%

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

   1. pow-sqr 80.4%

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

9. Simplified 80.4%

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

    1. *-commutative 80.4%

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

    2. pow-unpow 80.4%

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

    3. pow1/2 80.4%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left({\color{blue}{\left({\left(e^{0.6666666666666666}\right)}^{0.5}\right)}}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}^{2}\right)} \]

    4. pow-exp 80.4%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left({\color{blue}{\left(e^{0.6666666666666666 \cdot 0.5}\right)}}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}^{2}\right)} \]

    5. metadata-eval 80.4%

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

    6. pow-exp 80.2%

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

    7. log1p-udef 80.2%

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

    8. log-pow 80.4%

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

    9. pow1/3 80.5%

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

    10. add-exp-log 99.3%

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

    11. pow2 99.3%

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

    12. +-commutative 99.3%

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

    13. +-commutative 99.3%

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

11. Applied egg-rr 99.3%

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

12. Final simplification 99.3%

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

Alternative 8: 53.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (exp (* 0.3333333333333333 (* 3.0 (log (- (cbrt (+ 1.0 x)) (cbrt x)))))))

C

double code(double x) {
	return exp((0.3333333333333333 * (3.0 * log((cbrt((1.0 + x)) - cbrt(x))))));
}

Java

public static double code(double x) {
	return Math.exp((0.3333333333333333 * (3.0 * Math.log((Math.cbrt((1.0 + x)) - Math.cbrt(x))))));
}

Julia

function code(x)
	return exp(Float64(0.3333333333333333 * Float64(3.0 * log(Float64(cbrt(Float64(1.0 + x)) - cbrt(x))))))
end

Wolfram

code[x_] := N[Exp[N[(0.3333333333333333 * N[(3.0 * N[Log[N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 60.5%

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

   1. add-cbrt-cube 60.4%

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

   2. pow1/3 60.5%

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

   3. pow-to-exp 60.5%

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

   4. pow3 60.5%

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

   5. log-pow 60.5%

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

3. Applied egg-rr 60.5%

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

4. Final simplification 60.5%

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

Alternative 9: 53.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (pow (pow (- (cbrt (+ 1.0 x)) (cbrt x)) 3.0) 0.3333333333333333))

C

double code(double x) {
	return pow(pow((cbrt((1.0 + x)) - cbrt(x)), 3.0), 0.3333333333333333);
}

Java

public static double code(double x) {
	return Math.pow(Math.pow((Math.cbrt((1.0 + x)) - Math.cbrt(x)), 3.0), 0.3333333333333333);
}

Julia

function code(x)
	return (Float64(cbrt(Float64(1.0 + x)) - cbrt(x)) ^ 3.0) ^ 0.3333333333333333
end

Wolfram

code[x_] := N[Power[N[Power[N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 60.5%

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

   1. add-cbrt-cube 60.4%

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

   2. pow1/3 60.5%

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

   3. pow3 60.5%

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

3. Applied egg-rr 60.5%

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

4. Final simplification 60.5%

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

Alternative 10: 53.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.5%

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

2. Final simplification 60.5%

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

Alternative 11: 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 60.5%

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

   1. add-cube-cbrt 60.1%

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

   2. pow3 60.0%

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

3. Applied egg-rr 60.0%

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

4. Taylor expanded in x around 0 31.1%

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

   1. associate--l+ 31.1%

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

   2. *-commutative 31.1%

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

   3. metadata-eval 31.1%

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

   4. pow-base-1 31.1%

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

   5. unpow1/3 58.2%

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

   6. *-lft-identity 58.2%

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

6. Simplified 58.2%

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

7. Final simplification 58.2%

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

Alternative 12: 50.5% 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 60.5%

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

   1. add-cube-cbrt 60.1%

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

   2. pow3 60.0%

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

3. Applied egg-rr 60.0%

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

4. Taylor expanded in x around 0 30.5%

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

   1. metadata-eval 30.5%

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

   2. pow-base-1 30.5%

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

   3. unpow1/3 57.9%

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

   4. *-lft-identity 57.9%

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

6. Simplified 57.9%

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

7. Final simplification 57.9%

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

Alternative 13: 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 60.5%

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

2. Taylor expanded in x around inf 3.6%

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

3. Final simplification 3.6%

   \[\leadsto 0 \]

Alternative 14: 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 60.5%

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

2. Taylor expanded in x around 0 56.6%

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

3. Final simplification 56.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023315 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))