2cbrt (problem 3.3.4)

Percentage Accurate: 53.5% → 99.2%

Time: 19.9s

Alternatives: 16

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: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $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] * t$95$1 + N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$1 + N[Power[N[Power[N[(x + 1.0), $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 4.2%

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

      1. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 3.7%

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

      4. rem-cube-cbrt 4.2%

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

      5. reciprocal-define 4.2%

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

      6. +-commutative 4.2%

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

      7. distribute-rgt-out 4.2%

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

      8. +-commutative 4.2%

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

      9. fma-def 4.2%

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

      10. add-exp-log 4.2%

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

   4. Applied egg-rr 2.3%

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

      1. reciprocal-undefine 2.3%

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

      2. associate-*r/ 2.3%

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

      3. *-rgt-identity 2.3%

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

      4. +-commutative 2.3%

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

      5. associate--l+ 50.0%

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

      6. +-inverses 50.0%

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

      7. metadata-eval 50.0%

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

      8. reciprocal-undefine 21.3%

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

      9. +-commutative 21.3%

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

      10. exp-prod 21.3%

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

   6. Simplified 21.3%

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

      1. reciprocal-undefine 49.8%

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

      2. +-commutative 49.8%

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

      3. +-commutative 49.8%

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

   8. Applied egg-rr 49.8%

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

   9. Taylor expanded in x around inf 49.7%

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

       1. expm1-log1p-u 49.7%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({x}^{2}\right)}^{0.3333333333333333}\right)\right)}\right)} \]

       2. expm1-udef 49.7%

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

       3. unpow1/3 50.0%

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

       4. unpow2 50.0%

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

       5. cbrt-prod 93.9%

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

       6. pow2 93.9%

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

   11. Applied egg-rr 93.9%

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

       1. expm1-def 93.9%

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

       2. expm1-log1p 98.6%

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

   13. Simplified 98.6%

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

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

   1. Initial program 96.7%

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

      1. flip3-- 97.8%

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

      2. rem-cube-cbrt 97.2%

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

      3. rem-cube-cbrt 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-rgt-out 99.8%

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

      6. +-commutative 99.8%

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

      7. fma-def 99.8%

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

      8. add-exp-log 99.8%

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

      9. pow1/3 95.6%

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

      10. pow1/3 95.5%

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

      11. pow-prod-up 95.6%

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

   4. Applied egg-rr 95.6%

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

      1. *-commutative 95.6%

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

      2. log1p-udef 95.6%

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

      3. +-commutative 95.6%

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

      4. exp-to-pow 95.6%

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

      5. metadata-eval 95.6%

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

      6. pow-prod-up 95.5%

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

      7. pow1/3 95.6%

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

      8. pow1/3 99.8%

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

      9. cbrt-unprod 99.8%

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

      10. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.3%

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

Alternative 2: 99.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

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

      1. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 3.7%

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

      4. rem-cube-cbrt 4.2%

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

      5. reciprocal-define 4.2%

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

      6. +-commutative 4.2%

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

      7. distribute-rgt-out 4.2%

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

      8. +-commutative 4.2%

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

      9. fma-def 4.2%

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

      10. add-exp-log 4.2%

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

   4. Applied egg-rr 2.3%

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

      1. reciprocal-undefine 2.3%

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

      2. associate-*r/ 2.3%

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

      3. *-rgt-identity 2.3%

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

      4. +-commutative 2.3%

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

      5. associate--l+ 50.0%

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

      6. +-inverses 50.0%

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

      7. metadata-eval 50.0%

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

      8. reciprocal-undefine 21.3%

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

      9. +-commutative 21.3%

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

      10. exp-prod 21.3%

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

   6. Simplified 21.3%

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

      1. reciprocal-undefine 49.8%

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

      2. +-commutative 49.8%

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

      3. +-commutative 49.8%

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

   8. Applied egg-rr 49.8%

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

   9. Taylor expanded in x around inf 49.7%

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

       1. expm1-log1p-u 49.7%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({x}^{2}\right)}^{0.3333333333333333}\right)\right)}\right)} \]

       2. expm1-udef 49.7%

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

       3. unpow1/3 50.0%

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

       4. unpow2 50.0%

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

       5. cbrt-prod 93.9%

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

       6. pow2 93.9%

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

   11. Applied egg-rr 93.9%

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

       1. expm1-def 93.9%

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

       2. expm1-log1p 98.6%

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

   13. Simplified 98.6%

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

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

   1. Initial program 96.7%

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

      1. add-cbrt-cube 96.6%

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

      2. pow1/3 96.7%

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

      3. pow-to-exp 96.7%

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

      4. pow3 96.7%

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

      5. log-pow 96.7%

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

   4. Applied egg-rr 96.7%

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

      1. *-commutative 96.7%

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

      2. associate-*l* 96.7%

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

      3. metadata-eval 96.7%

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

      4. pow-to-exp 96.7%

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

      5. pow1 96.7%

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

      6. flip3-- 97.8%

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

      7. rem-cube-cbrt 97.2%

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

      8. rem-cube-cbrt 99.8%

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

      9. distribute-rgt-in 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.2%

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

Alternative 3: 99.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(t$95$1 + N[Power[t$95$0, 2.0], $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 4.2%

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

      1. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 3.7%

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

      4. rem-cube-cbrt 4.2%

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

      5. reciprocal-define 4.2%

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

      6. +-commutative 4.2%

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

      7. distribute-rgt-out 4.2%

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

      8. +-commutative 4.2%

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

      9. fma-def 4.2%

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

      10. add-exp-log 4.2%

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

   4. Applied egg-rr 2.3%

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

      1. reciprocal-undefine 2.3%

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

      2. associate-*r/ 2.3%

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

      3. *-rgt-identity 2.3%

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

      4. +-commutative 2.3%

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

      5. associate--l+ 50.0%

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

      6. +-inverses 50.0%

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

      7. metadata-eval 50.0%

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

      8. reciprocal-undefine 21.3%

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

      9. +-commutative 21.3%

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

      10. exp-prod 21.3%

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

   6. Simplified 21.3%

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

      1. reciprocal-undefine 49.8%

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

      2. +-commutative 49.8%

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

      3. +-commutative 49.8%

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

   8. Applied egg-rr 49.8%

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

   9. Taylor expanded in x around inf 49.7%

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

       1. fma-udef 49.7%

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

       2. +-commutative 49.7%

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

       3. unpow1/3 52.2%

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

       4. unpow2 52.2%

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

       5. cbrt-prod 98.6%

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

       6. pow2 98.6%

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

       7. +-commutative 98.6%

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

   11. Applied egg-rr 98.6%

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

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

   1. Initial program 96.7%

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

      1. add-cbrt-cube 96.6%

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

      2. pow1/3 96.7%

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

      3. pow-to-exp 96.7%

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

      4. pow3 96.7%

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

      5. log-pow 96.7%

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

   4. Applied egg-rr 96.7%

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

      1. *-commutative 96.7%

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

      2. associate-*l* 96.7%

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

      3. metadata-eval 96.7%

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

      4. pow-to-exp 96.7%

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

      5. pow1 96.7%

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

      6. flip3-- 97.8%

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

      7. rem-cube-cbrt 97.2%

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

      8. rem-cube-cbrt 99.8%

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

      9. distribute-rgt-in 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.2%

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

Alternative 4: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \left(t_0 + \sqrt[3]{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 + {x}^{3}}}{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right) - x}} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ x 1.0))))
   (if (<= (- t_0 (cbrt x)) 5e-7)
     (/ 1.0 (* (cbrt x) (+ (cbrt x) (+ t_0 (cbrt x)))))
     (- (/ (cbrt (+ 1.0 (pow x 3.0))) (cbrt (- (fma x x 1.0) x))) (cbrt x)))))

C

double code(double x) {
	double t_0 = cbrt((x + 1.0));
	double tmp;
	if ((t_0 - cbrt(x)) <= 5e-7) {
		tmp = 1.0 / (cbrt(x) * (cbrt(x) + (t_0 + cbrt(x))));
	} else {
		tmp = (cbrt((1.0 + pow(x, 3.0))) / cbrt((fma(x, x, 1.0) - x))) - cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 5e-7)
		tmp = Float64(1.0 / Float64(cbrt(x) * Float64(cbrt(x) + Float64(t_0 + cbrt(x)))));
	else
		tmp = Float64(Float64(cbrt(Float64(1.0 + (x ^ 3.0))) / cbrt(Float64(fma(x, x, 1.0) - x))) - cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-7], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(1.0 + N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(N[(x * x + 1.0), $MachinePrecision] - x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 4.99999999999999977e-7

   1. Initial program 5.9%

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

      1. flip3-- 6.9%

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

      2. div-inv 6.9%

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

      3. rem-cube-cbrt 5.8%

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

      4. rem-cube-cbrt 8.8%

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

      5. reciprocal-define 5.8%

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

      6. +-commutative 5.8%

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

      7. distribute-rgt-out 5.8%

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

      8. +-commutative 5.8%

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

      9. fma-def 5.8%

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

      10. add-exp-log 5.8%

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

   4. Applied egg-rr 2.8%

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

      1. reciprocal-undefine 3.7%

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

      2. associate-*r/ 3.7%

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

      3. *-rgt-identity 3.7%

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

      4. +-commutative 3.7%

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

      5. associate--l+ 49.1%

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

      6. +-inverses 49.1%

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

      7. metadata-eval 49.1%

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

      8. reciprocal-undefine 20.9%

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

      9. +-commutative 20.9%

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

      10. exp-prod 20.9%

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

   6. Simplified 20.9%

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

      1. reciprocal-undefine 48.9%

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

      2. +-commutative 48.9%

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

      3. +-commutative 48.9%

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

   8. Applied egg-rr 48.9%

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

   9. Taylor expanded in x around inf 51.2%

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

       1. fma-udef 51.2%

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

       2. +-commutative 51.2%

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

       3. unpow1/3 53.6%

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

       4. unpow2 53.6%

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

       5. cbrt-prod 97.7%

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

       6. pow2 97.7%

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

       7. +-commutative 97.7%

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

   11. Applied egg-rr 97.7%

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

       1. unpow2 97.7%

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

       2. distribute-lft-out 97.7%

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

       3. +-commutative 97.7%

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

       4. +-commutative 97.7%

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

   13. Simplified 97.7%

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

   if 4.99999999999999977e-7 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.2%

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

      1. flip3-+ 99.3%

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

      2. cbrt-div 99.3%

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

      3. pow3 99.2%

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

      4. metadata-eval 99.2%

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

      5. +-commutative 99.2%

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

      6. pow3 99.3%

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

      7. metadata-eval 99.3%

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

      8. *-rgt-identity 99.3%

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

      9. associate-+r- 99.3%

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

      10. fma-def 99.3%

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 98.5%

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

Alternative 5: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 + {x}^{3}}}{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right) - x}} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ x 1.0))))
   (if (<= (- t_0 (cbrt x)) 5e-7)
     (/ 1.0 (+ (pow (cbrt x) 2.0) (* (cbrt x) (+ t_0 (cbrt x)))))
     (- (/ (cbrt (+ 1.0 (pow x 3.0))) (cbrt (- (fma x x 1.0) x))) (cbrt x)))))

C

double code(double x) {
	double t_0 = cbrt((x + 1.0));
	double tmp;
	if ((t_0 - cbrt(x)) <= 5e-7) {
		tmp = 1.0 / (pow(cbrt(x), 2.0) + (cbrt(x) * (t_0 + cbrt(x))));
	} else {
		tmp = (cbrt((1.0 + pow(x, 3.0))) / cbrt((fma(x, x, 1.0) - x))) - cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 5e-7)
		tmp = Float64(1.0 / Float64((cbrt(x) ^ 2.0) + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))));
	else
		tmp = Float64(Float64(cbrt(Float64(1.0 + (x ^ 3.0))) / cbrt(Float64(fma(x, x, 1.0) - x))) - cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-7], N[(1.0 / N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(1.0 + N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(N[(x * x + 1.0), $MachinePrecision] - x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 4.99999999999999977e-7

   1. Initial program 5.9%

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

      1. flip3-- 6.9%

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

      2. div-inv 6.9%

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

      3. rem-cube-cbrt 5.8%

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

      4. rem-cube-cbrt 8.8%

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

      5. reciprocal-define 5.8%

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

      6. +-commutative 5.8%

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

      7. distribute-rgt-out 5.8%

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

      8. +-commutative 5.8%

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

      9. fma-def 5.8%

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

      10. add-exp-log 5.8%

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

   4. Applied egg-rr 2.8%

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

      1. reciprocal-undefine 3.7%

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

      2. associate-*r/ 3.7%

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

      3. *-rgt-identity 3.7%

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

      4. +-commutative 3.7%

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

      5. associate--l+ 49.1%

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

      6. +-inverses 49.1%

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

      7. metadata-eval 49.1%

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

      8. reciprocal-undefine 20.9%

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

      9. +-commutative 20.9%

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

      10. exp-prod 20.9%

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

   6. Simplified 20.9%

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

      1. reciprocal-undefine 48.9%

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

      2. +-commutative 48.9%

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

      3. +-commutative 48.9%

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

   8. Applied egg-rr 48.9%

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

   9. Taylor expanded in x around inf 51.2%

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

       1. fma-udef 51.2%

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

       2. +-commutative 51.2%

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

       3. unpow1/3 53.6%

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

       4. unpow2 53.6%

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

       5. cbrt-prod 97.7%

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

       6. pow2 97.7%

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

       7. +-commutative 97.7%

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

   11. Applied egg-rr 97.7%

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

   if 4.99999999999999977e-7 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.2%

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

      1. flip3-+ 99.3%

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

      2. cbrt-div 99.3%

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

      3. pow3 99.2%

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

      4. metadata-eval 99.2%

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

      5. +-commutative 99.2%

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

      6. pow3 99.3%

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

      7. metadata-eval 99.3%

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

      8. *-rgt-identity 99.3%

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

      9. associate-+r- 99.3%

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

      10. fma-def 99.3%

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 98.5%

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

Alternative 6: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \left(t_0 + \sqrt[3]{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ x 1.0))))
   (if (<= (- t_0 (cbrt x)) 5e-7)
     (/ 1.0 (* (cbrt x) (+ (cbrt x) (+ t_0 (cbrt x)))))
     (- (/ (cbrt (fma x x -1.0)) (cbrt (+ x -1.0))) (cbrt x)))))

C

double code(double x) {
	double t_0 = cbrt((x + 1.0));
	double tmp;
	if ((t_0 - cbrt(x)) <= 5e-7) {
		tmp = 1.0 / (cbrt(x) * (cbrt(x) + (t_0 + cbrt(x))));
	} else {
		tmp = (cbrt(fma(x, x, -1.0)) / cbrt((x + -1.0))) - cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 5e-7)
		tmp = Float64(1.0 / Float64(cbrt(x) * Float64(cbrt(x) + Float64(t_0 + cbrt(x)))));
	else
		tmp = Float64(Float64(cbrt(fma(x, x, -1.0)) / cbrt(Float64(x + -1.0))) - cbrt(x));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 4.99999999999999977e-7

   1. Initial program 5.9%

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

      1. flip3-- 6.9%

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

      2. div-inv 6.9%

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

      3. rem-cube-cbrt 5.8%

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

      4. rem-cube-cbrt 8.8%

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

      5. reciprocal-define 5.8%

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

      6. +-commutative 5.8%

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

      7. distribute-rgt-out 5.8%

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

      8. +-commutative 5.8%

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

      9. fma-def 5.8%

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

      10. add-exp-log 5.8%

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

   4. Applied egg-rr 2.8%

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

      1. reciprocal-undefine 3.7%

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

      2. associate-*r/ 3.7%

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

      3. *-rgt-identity 3.7%

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

      4. +-commutative 3.7%

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

      5. associate--l+ 49.1%

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

      6. +-inverses 49.1%

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

      7. metadata-eval 49.1%

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

      8. reciprocal-undefine 20.9%

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

      9. +-commutative 20.9%

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

      10. exp-prod 20.9%

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

   6. Simplified 20.9%

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

      1. reciprocal-undefine 48.9%

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

      2. +-commutative 48.9%

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

      3. +-commutative 48.9%

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

   8. Applied egg-rr 48.9%

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

   9. Taylor expanded in x around inf 51.2%

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

       1. fma-udef 51.2%

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

       2. +-commutative 51.2%

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

       3. unpow1/3 53.6%

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

       4. unpow2 53.6%

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

       5. cbrt-prod 97.7%

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

       6. pow2 97.7%

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

       7. +-commutative 97.7%

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

   11. Applied egg-rr 97.7%

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

       1. unpow2 97.7%

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

       2. distribute-lft-out 97.7%

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

       3. +-commutative 97.7%

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

       4. +-commutative 97.7%

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

   13. Simplified 97.7%

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

   if 4.99999999999999977e-7 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.2%

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

      1. flip-+ 99.2%

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

      2. cbrt-div 99.3%

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

      3. metadata-eval 99.3%

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

      4. fma-neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 98.5%

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

Alternative 7: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 5e-7)
   (* 0.3333333333333333 (pow (/ 1.0 (pow x 2.0)) 0.3333333333333333))
   (- (/ (cbrt (fma x x -1.0)) (cbrt (+ x -1.0))) (cbrt x))))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-7], N[(0.3333333333333333 * N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(x * x + -1.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(x + -1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 4.99999999999999977e-7

   1. Initial program 5.9%

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

      1. add-sqr-sqrt 2.7%

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

      2. add-sqr-sqrt 2.9%

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

      3. difference-of-squares 2.9%

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

      4. pow1/3 2.9%

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

      5. sqrt-pow1 2.9%

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

      6. metadata-eval 2.9%

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

      7. pow1/3 2.9%

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

      8. sqrt-pow1 2.9%

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

      9. metadata-eval 2.9%

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

      10. pow1/3 1.6%

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

      11. sqrt-pow1 1.6%

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

      12. metadata-eval 1.6%

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

      13. pow1/3 2.8%

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

      14. sqrt-pow1 2.9%

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

      15. metadata-eval 2.9%

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

   4. Applied egg-rr 2.9%

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

   5. Taylor expanded in x around inf 50.1%

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

   if 4.99999999999999977e-7 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.2%

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

      1. flip-+ 99.2%

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

      2. cbrt-div 99.3%

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

      3. metadata-eval 99.3%

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

      4. fma-neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 75.6%

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

Alternative 8: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\\ t_1 := \mathsf{reciprocal}\left(\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)\right)\right)\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -29000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 34000000:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (* 0.3333333333333333 (pow (/ 1.0 (pow x 2.0)) 0.3333333333333333)))
        (t_1
         (reciprocal (* (cbrt x) (+ (cbrt x) (+ (cbrt (+ x 1.0)) (cbrt x)))))))
   (if (<= x -1.32e+154)
     t_1
     (if (<= x -29000000.0)
       t_0
       (if (<= x 34000000.0)
         (- (/ (cbrt (fma x x -1.0)) (cbrt (+ x -1.0))) (cbrt x))
         (if (<= x 1.35e+154) t_0 t_1))))))

Derivation

1. Split input into 3 regimes

2. if x < -1.31999999999999998e154 or 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

      1. flip3-- 4.7%

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

      2. div-inv 4.7%

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

      3. rem-cube-cbrt 3.2%

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

      4. rem-cube-cbrt 4.7%

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

      5. reciprocal-define 4.7%

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

      6. +-commutative 4.7%

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

      7. distribute-rgt-out 4.7%

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

      8. +-commutative 4.7%

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

      9. fma-def 4.7%

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

      10. add-exp-log 4.7%

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

   4. Applied egg-rr 2.6%

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

      1. reciprocal-undefine 2.6%

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

      2. associate-*r/ 2.6%

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

      3. *-rgt-identity 2.6%

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

      4. +-commutative 2.6%

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

      5. associate--l+ 50.8%

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

      6. +-inverses 50.8%

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

      7. metadata-eval 50.8%

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

      8. reciprocal-undefine 21.9%

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

      9. +-commutative 21.9%

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

      10. exp-prod 21.9%

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

   6. Simplified 21.9%

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

      1. reciprocal-undefine 50.4%

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

      2. +-commutative 50.4%

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

      3. +-commutative 50.4%

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

   8. Applied egg-rr 50.4%

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

   9. Taylor expanded in x around inf 4.7%

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

       1. expm1-log1p-u 4.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, {\left({x}^{2}\right)}^{0.3333333333333333}\right)}\right)\right)} \]

       2. expm1-udef 4.7%

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

       3. reciprocal-define 4.7%

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

       4. +-commutative 4.7%

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

       5. unpow1/3 4.7%

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

       6. unpow2 4.7%

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

       7. cbrt-prod 4.7%

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

       8. pow2 4.7%

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

   11. Applied egg-rr 4.7%

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

       1. expm1-def 39.5%

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

       2. expm1-log1p 39.5%

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

       3. fma-def 39.5%

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

       4. +-commutative 39.5%

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

       5. unpow2 39.5%

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

       6. distribute-lft-out 39.5%

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

       7. +-commutative 39.5%

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

       8. +-commutative 39.5%

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

   13. Simplified 39.5%

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

   if -1.31999999999999998e154 < x < -2.9e7 or 3.4e7 < x < 1.35000000000000003e154

   1. Initial program 7.0%

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

      1. add-sqr-sqrt 3.1%

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

      2. add-sqr-sqrt 3.2%

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

      3. difference-of-squares 3.1%

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

      4. pow1/3 3.1%

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

      5. sqrt-pow1 3.1%

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

      6. metadata-eval 3.1%

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

      7. pow1/3 3.1%

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

      8. sqrt-pow1 3.1%

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

      9. metadata-eval 3.1%

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

      10. pow1/3 2.1%

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

      11. sqrt-pow1 2.1%

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

      12. metadata-eval 2.1%

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

      13. pow1/3 3.1%

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

      14. sqrt-pow1 3.1%

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

      15. metadata-eval 3.1%

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around inf 90.5%

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

   if -2.9e7 < x < 3.4e7

   1. Initial program 99.2%

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

      1. flip-+ 99.2%

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

      2. cbrt-div 99.3%

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

      3. metadata-eval 99.3%

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

      4. fma-neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq -29000000:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\\ \mathbf{elif}\;x \leq 34000000:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 4.99999999999999977e-7

   1. Initial program 5.9%

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

      1. add-sqr-sqrt 2.7%

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

      2. add-sqr-sqrt 2.9%

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

      3. difference-of-squares 2.9%

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

      4. pow1/3 2.9%

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

      5. sqrt-pow1 2.9%

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

      6. metadata-eval 2.9%

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

      7. pow1/3 2.9%

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

      8. sqrt-pow1 2.9%

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

      9. metadata-eval 2.9%

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

      10. pow1/3 1.6%

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

      11. sqrt-pow1 1.6%

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

      12. metadata-eval 1.6%

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

      13. pow1/3 2.8%

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

      14. sqrt-pow1 2.9%

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

      15. metadata-eval 2.9%

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

   4. Applied egg-rr 2.9%

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

   5. Taylor expanded in x around inf 50.1%

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

   if 4.99999999999999977e-7 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.2%

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

4. Final simplification 75.6%

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

Alternative 10: 62.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x + 1} - \sqrt[3]{x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (cbrt (+ x 1.0)) (cbrt x))))
   (if (<= t_0 2e-8)
     (* 0.3333333333333333 (cbrt (reciprocal (pow x 2.0))))
     t_0)))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 2e-8

   1. Initial program 5.5%

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

      1. add-sqr-sqrt 2.3%

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

      2. add-sqr-sqrt 2.5%

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

      3. difference-of-squares 2.5%

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

      4. pow1/3 2.5%

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

      5. sqrt-pow1 2.5%

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

      6. metadata-eval 2.5%

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

      7. pow1/3 2.5%

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

      8. sqrt-pow1 2.5%

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

      9. metadata-eval 2.5%

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

      10. pow1/3 1.2%

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

      11. sqrt-pow1 1.2%

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

      12. metadata-eval 1.2%

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

      13. pow1/3 2.4%

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

      14. sqrt-pow1 2.5%

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

      15. metadata-eval 2.5%

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

   4. Applied egg-rr 2.5%

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

   5. Taylor expanded in x around inf 49.9%

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

      1. unpow1/3 53.4%

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

      2. reciprocal-define 24.4%

         \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{\color{blue}{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

   7. Simplified 24.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

   if 2e-8 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 98.9%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \sqrt[3]{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.26:\\ \;\;\;\;1 + x \cdot -0.6666666666666666\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{reciprocal}\left(x\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (cbrt (reciprocal (pow x 2.0))))))
   (if (<= x -1.0)
     t_0
     (if (<= x 0.26)
       (+ 1.0 (* x -0.6666666666666666))
       (if (<= x 1.35e+154) t_0 (pow (cbrt (reciprocal x)) 2.0))))))

Derivation

1. Split input into 3 regimes

2. if x < -1 or 0.26000000000000001 < x < 1.35000000000000003e154

   1. Initial program 9.3%

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

      1. add-sqr-sqrt 3.9%

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

      2. add-sqr-sqrt 3.9%

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

      3. difference-of-squares 3.9%

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

      4. pow1/3 3.9%

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

      5. sqrt-pow1 3.9%

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

      6. metadata-eval 3.9%

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

      7. pow1/3 3.9%

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

      8. sqrt-pow1 3.9%

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

      9. metadata-eval 3.9%

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

      10. pow1/3 3.2%

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

      11. sqrt-pow1 3.2%

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

      12. metadata-eval 3.2%

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

      13. pow1/3 3.9%

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

      14. sqrt-pow1 4.0%

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

      15. metadata-eval 4.0%

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

   4. Applied egg-rr 4.0%

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

   5. Taylor expanded in x around inf 65.1%

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

      1. unpow1/3 69.6%

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

      2. reciprocal-define 31.9%

         \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{\color{blue}{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

   7. Simplified 31.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

   if -1 < x < 0.26000000000000001

   1. Initial program 100.0%

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

      1. flip3-- 99.9%

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

      2. div-inv 99.9%

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

      3. rem-cube-cbrt 99.9%

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

      4. rem-cube-cbrt 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. reciprocal-define 95.7%

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

      6. +-commutative 95.7%

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

      7. distribute-rgt-out 95.7%

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

      8. +-commutative 95.7%

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

      9. fma-def 95.7%

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

      10. add-exp-log 95.7%

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

   4. Applied egg-rr 95.7%

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

      1. reciprocal-undefine 99.9%

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

      2. associate-*r/ 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. +-commutative 99.9%

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

      5. associate--l+ 99.9%

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

      6. +-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. reciprocal-undefine 95.7%

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

      9. +-commutative 95.7%

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

      10. exp-prod 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in x around 0 95.3%

      \[\leadsto \color{blue}{1 + -0.6666666666666666 \cdot x} \]

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

      1. flip3-- 4.7%

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

      2. div-inv 4.7%

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

      3. rem-cube-cbrt 3.2%

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

      4. rem-cube-cbrt 4.7%

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

      5. reciprocal-define 4.7%

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

      6. +-commutative 4.7%

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

      7. distribute-rgt-out 4.7%

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

      8. +-commutative 4.7%

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

      9. fma-def 4.7%

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

      10. add-exp-log 4.7%

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

   4. Applied egg-rr 4.7%

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

      1. reciprocal-undefine 4.7%

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

      2. associate-*r/ 4.7%

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

      3. *-rgt-identity 4.7%

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

      4. +-commutative 4.7%

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

      5. associate--l+ 92.1%

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

      6. +-inverses 92.1%

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

      7. metadata-eval 92.1%

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

      8. reciprocal-undefine 39.8%

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

      9. +-commutative 39.8%

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

      10. exp-prod 39.8%

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

   6. Simplified 39.8%

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

   7. Taylor expanded in x around inf 4.7%

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

      1. unpow1/3 4.7%

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

      2. reciprocal-define 4.7%

         \[\leadsto \sqrt[3]{\color{blue}{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

   9. Simplified 4.7%

      \[\leadsto \color{blue}{\sqrt[3]{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 4.7%

          \[\leadsto \sqrt[3]{\color{blue}{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)} \cdot \sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}}} \]

       2. cbrt-prod 4.7%

          \[\leadsto \color{blue}{\sqrt[3]{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}}} \]

       3. reciprocal-undefine 4.7%

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

       4. sqrt-div 4.7%

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

       5. metadata-eval 4.7%

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

       6. unpow2 4.7%

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

       7. sqrt-prod 4.7%

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

       8. add-sqr-sqrt 4.7%

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

       9. reciprocal-define 4.7%

          \[\leadsto \sqrt[3]{\color{blue}{\mathsf{reciprocal}\left(x\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

       10. reciprocal-undefine 4.7%

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

       11. sqrt-div 4.7%

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

       12. metadata-eval 4.7%

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

       13. unpow2 4.7%

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

       14. sqrt-prod 17.7%

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

       15. add-sqr-sqrt 17.7%

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

       16. reciprocal-define 17.7%

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

   11. Applied egg-rr 17.7%

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

       1. unpow2 17.7%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{reciprocal}\left(x\right)}\right)}^{2}} \]

   13. Simplified 17.7%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}\\ \mathbf{elif}\;x \leq 0.26:\\ \;\;\;\;1 + x \cdot -0.6666666666666666\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{reciprocal}\left(x\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.38:\\ \;\;\;\;\mathsf{reciprocal}\left(\left({\left(\sqrt[3]{x}\right)}^{2}\right)\right)\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;1 + x \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;e^{\log x \cdot \left(-0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.38)
   (reciprocal (pow (cbrt x) 2.0))
   (if (<= x 1.4)
     (+ 1.0 (* x -0.6666666666666666))
     (exp (* (log x) (- 0.6666666666666666))))))

Derivation

1. Split input into 3 regimes

2. if x < -0.38

   1. Initial program 10.1%

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

      1. flip3-- 12.6%

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

      2. div-inv 12.6%

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

      3. rem-cube-cbrt 10.9%

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

      4. rem-cube-cbrt 15.0%

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

      5. reciprocal-define 8.3%

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

      6. +-commutative 8.3%

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

      7. distribute-rgt-out 8.3%

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

      8. +-commutative 8.3%

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

      9. fma-def 8.3%

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

      10. add-exp-log 8.3%

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

   4. Applied egg-rr 0.8%

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

      1. reciprocal-undefine 1.6%

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

      2. associate-*r/ 1.6%

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

      3. *-rgt-identity 1.6%

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

      4. +-commutative 1.6%

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

      5. associate--l+ 1.6%

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

      6. +-inverses 1.6%

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

      7. metadata-eval 1.6%

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

      8. reciprocal-undefine 0.8%

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

      9. +-commutative 0.8%

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

      10. exp-prod 0.8%

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

   6. Simplified 0.8%

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

   7. Taylor expanded in x around inf 12.3%

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

      1. unpow1/3 12.3%

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

      2. reciprocal-define 12.3%

         \[\leadsto \sqrt[3]{\color{blue}{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

   9. Simplified 12.3%

      \[\leadsto \color{blue}{\sqrt[3]{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]
   10. Step-by-step derivation

       1. reciprocal-undefine 12.3%

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

       2. cbrt-div 12.3%

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

       3. metadata-eval 12.3%

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

       4. unpow2 12.3%

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

       5. cbrt-prod 17.8%

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

       6. pow2 17.8%

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

   11. Applied egg-rr 17.8%

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

       1. reciprocal-define 17.8%

          \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({\left(\sqrt[3]{x}\right)}^{2}\right)\right)} \]

   13. Simplified 17.8%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({\left(\sqrt[3]{x}\right)}^{2}\right)\right)} \]

   if -0.38 < x < 1.3999999999999999

   1. Initial program 100.0%

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

      1. flip3-- 99.9%

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

      2. div-inv 99.9%

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

      3. rem-cube-cbrt 99.9%

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

      4. rem-cube-cbrt 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. reciprocal-define 96.1%

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

      6. +-commutative 96.1%

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

      7. distribute-rgt-out 96.1%

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

      8. +-commutative 96.1%

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

      9. fma-def 96.1%

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

      10. add-exp-log 96.1%

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

   4. Applied egg-rr 96.1%

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

      1. reciprocal-undefine 99.9%

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

      2. associate-*r/ 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. +-commutative 99.9%

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

      5. associate--l+ 99.9%

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

      6. +-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. reciprocal-undefine 96.1%

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

      9. +-commutative 96.1%

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

      10. exp-prod 96.1%

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

   6. Simplified 96.1%

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

   7. Taylor expanded in x around 0 95.8%

      \[\leadsto \color{blue}{1 + -0.6666666666666666 \cdot x} \]

   if 1.3999999999999999 < x

   1. Initial program 7.7%

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

      1. flip3-- 7.8%

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

      2. div-inv 7.8%

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

      3. rem-cube-cbrt 7.2%

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

      4. rem-cube-cbrt 9.9%

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

      5. reciprocal-define 6.3%

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

      6. +-commutative 6.3%

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

      7. distribute-rgt-out 6.3%

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

      8. +-commutative 6.3%

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

      9. fma-def 6.3%

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

      10. add-exp-log 6.3%

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

   4. Applied egg-rr 6.3%

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

      1. reciprocal-undefine 9.8%

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

      2. associate-*r/ 9.8%

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

      3. *-rgt-identity 9.8%

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

      4. +-commutative 9.8%

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

      5. associate--l+ 93.2%

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

      6. +-inverses 93.2%

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

      7. metadata-eval 93.2%

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

      8. reciprocal-undefine 39.5%

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

      9. +-commutative 39.5%

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

      10. exp-prod 39.5%

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

   6. Simplified 39.5%

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

      1. add-exp-log 39.5%

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

      2. reciprocal-undefine 91.6%

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

      3. log-rec 91.6%

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

      4. +-commutative 91.6%

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

      5. +-commutative 91.6%

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

   8. Applied egg-rr 91.6%

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

   9. Taylor expanded in x around inf 17.7%

      \[\leadsto \color{blue}{e^{-\log \left({\left(\frac{1}{x}\right)}^{-0.6666666666666666}\right)}} \]
   10. Step-by-step derivation

       1. reciprocal-define 17.7%

          \[\leadsto e^{-\log \left({\color{blue}{\mathsf{reciprocal}\left(x\right)}}^{-0.6666666666666666}\right)} \]

       2. log-pow 17.7%

          \[\leadsto e^{-\color{blue}{-0.6666666666666666 \cdot \log \mathsf{reciprocal}\left(x\right)}} \]

       3. reciprocal-define 17.7%

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

       4. distribute-lft-neg-in 17.7%

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

       5. metadata-eval 17.7%

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

       6. log-rec 17.7%

          \[\leadsto e^{0.6666666666666666 \cdot \color{blue}{\left(-\log x\right)}} \]

   11. Simplified 17.7%

       \[\leadsto \color{blue}{e^{0.6666666666666666 \cdot \left(-\log x\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.38:\\ \;\;\;\;\mathsf{reciprocal}\left(\left({\left(\sqrt[3]{x}\right)}^{2}\right)\right)\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;1 + x \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;e^{\log x \cdot \left(-0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.38 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;\mathsf{reciprocal}\left(\left({\left(\sqrt[3]{x}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot -0.6666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.38) (not (<= x 1.4)))
   (reciprocal (pow (cbrt x) 2.0))
   (+ 1.0 (* x -0.6666666666666666))))

Derivation

1. Split input into 2 regimes

2. if x < -0.38 or 1.3999999999999999 < x

   1. Initial program 8.9%

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

      1. flip3-- 10.1%

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

      2. div-inv 10.1%

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

      3. rem-cube-cbrt 9.0%

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

      4. rem-cube-cbrt 12.3%

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

      5. reciprocal-define 7.2%

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

      6. +-commutative 7.2%

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

      7. distribute-rgt-out 7.2%

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

      8. +-commutative 7.2%

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

      9. fma-def 7.2%

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

      10. add-exp-log 7.2%

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

   4. Applied egg-rr 3.7%

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

      1. reciprocal-undefine 5.9%

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

      2. associate-*r/ 5.9%

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

      3. *-rgt-identity 5.9%

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

      4. +-commutative 5.9%

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

      5. associate--l+ 49.5%

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

      6. +-inverses 49.5%

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

      7. metadata-eval 49.5%

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

      8. reciprocal-undefine 21.1%

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

      9. +-commutative 21.1%

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

      10. exp-prod 21.1%

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

   6. Simplified 21.1%

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

   7. Taylor expanded in x around inf 11.9%

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

      1. unpow1/3 11.9%

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

      2. reciprocal-define 11.9%

         \[\leadsto \sqrt[3]{\color{blue}{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

   9. Simplified 11.9%

      \[\leadsto \color{blue}{\sqrt[3]{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]
   10. Step-by-step derivation

       1. reciprocal-undefine 11.9%

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

       2. cbrt-div 11.9%

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

       3. metadata-eval 11.9%

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

       4. unpow2 11.9%

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

       5. cbrt-prod 17.8%

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

       6. pow2 17.8%

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

   11. Applied egg-rr 17.8%

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

       1. reciprocal-define 17.8%

          \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({\left(\sqrt[3]{x}\right)}^{2}\right)\right)} \]

   13. Simplified 17.8%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({\left(\sqrt[3]{x}\right)}^{2}\right)\right)} \]

   if -0.38 < x < 1.3999999999999999

   1. Initial program 100.0%

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

      1. flip3-- 99.9%

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

      2. div-inv 99.9%

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

      3. rem-cube-cbrt 99.9%

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

      4. rem-cube-cbrt 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. reciprocal-define 96.1%

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

      6. +-commutative 96.1%

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

      7. distribute-rgt-out 96.1%

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

      8. +-commutative 96.1%

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

      9. fma-def 96.1%

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

      10. add-exp-log 96.1%

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

   4. Applied egg-rr 96.1%

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

      1. reciprocal-undefine 99.9%

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

      2. associate-*r/ 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. +-commutative 99.9%

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

      5. associate--l+ 99.9%

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

      6. +-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. reciprocal-undefine 96.1%

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

      9. +-commutative 96.1%

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

      10. exp-prod 96.1%

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

   6. Simplified 96.1%

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

   7. Taylor expanded in x around 0 95.8%

      \[\leadsto \color{blue}{1 + -0.6666666666666666 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.38 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;\mathsf{reciprocal}\left(\left({\left(\sqrt[3]{x}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot -0.6666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.38:\\ \;\;\;\;\mathsf{reciprocal}\left(\left({\left(\sqrt[3]{x}\right)}^{2}\right)\right)\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;1 + x \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{reciprocal}\left(x\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.38)
   (reciprocal (pow (cbrt x) 2.0))
   (if (<= x 1.4)
     (+ 1.0 (* x -0.6666666666666666))
     (pow (cbrt (reciprocal x)) 2.0))))

Derivation

1. Split input into 3 regimes

2. if x < -0.38

   1. Initial program 10.1%

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

      1. flip3-- 12.6%

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

      2. div-inv 12.6%

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

      3. rem-cube-cbrt 10.9%

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

      4. rem-cube-cbrt 15.0%

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

      5. reciprocal-define 8.3%

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

      6. +-commutative 8.3%

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

      7. distribute-rgt-out 8.3%

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

      8. +-commutative 8.3%

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

      9. fma-def 8.3%

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

      10. add-exp-log 8.3%

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

   4. Applied egg-rr 0.8%

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

      1. reciprocal-undefine 1.6%

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

      2. associate-*r/ 1.6%

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

      3. *-rgt-identity 1.6%

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

      4. +-commutative 1.6%

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

      5. associate--l+ 1.6%

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

      6. +-inverses 1.6%

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

      7. metadata-eval 1.6%

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

      8. reciprocal-undefine 0.8%

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

      9. +-commutative 0.8%

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

      10. exp-prod 0.8%

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

   6. Simplified 0.8%

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

   7. Taylor expanded in x around inf 12.3%

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

      1. unpow1/3 12.3%

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

      2. reciprocal-define 12.3%

         \[\leadsto \sqrt[3]{\color{blue}{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

   9. Simplified 12.3%

      \[\leadsto \color{blue}{\sqrt[3]{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]
   10. Step-by-step derivation

       1. reciprocal-undefine 12.3%

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

       2. cbrt-div 12.3%

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

       3. metadata-eval 12.3%

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

       4. unpow2 12.3%

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

       5. cbrt-prod 17.8%

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

       6. pow2 17.8%

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

   11. Applied egg-rr 17.8%

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

       1. reciprocal-define 17.8%

          \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({\left(\sqrt[3]{x}\right)}^{2}\right)\right)} \]

   13. Simplified 17.8%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({\left(\sqrt[3]{x}\right)}^{2}\right)\right)} \]

   if -0.38 < x < 1.3999999999999999

   1. Initial program 100.0%

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

      1. flip3-- 99.9%

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

      2. div-inv 99.9%

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

      3. rem-cube-cbrt 99.9%

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

      4. rem-cube-cbrt 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. reciprocal-define 96.1%

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

      6. +-commutative 96.1%

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

      7. distribute-rgt-out 96.1%

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

      8. +-commutative 96.1%

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

      9. fma-def 96.1%

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

      10. add-exp-log 96.1%

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

   4. Applied egg-rr 96.1%

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

      1. reciprocal-undefine 99.9%

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

      2. associate-*r/ 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. +-commutative 99.9%

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

      5. associate--l+ 99.9%

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

      6. +-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. reciprocal-undefine 96.1%

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

      9. +-commutative 96.1%

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

      10. exp-prod 96.1%

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

   6. Simplified 96.1%

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

   7. Taylor expanded in x around 0 95.8%

      \[\leadsto \color{blue}{1 + -0.6666666666666666 \cdot x} \]

   if 1.3999999999999999 < x

   1. Initial program 7.7%

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

      1. flip3-- 7.8%

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

      2. div-inv 7.8%

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

      3. rem-cube-cbrt 7.2%

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

      4. rem-cube-cbrt 9.9%

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

      5. reciprocal-define 6.3%

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

      6. +-commutative 6.3%

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

      7. distribute-rgt-out 6.3%

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

      8. +-commutative 6.3%

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

      9. fma-def 6.3%

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

      10. add-exp-log 6.3%

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

   4. Applied egg-rr 6.3%

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

      1. reciprocal-undefine 9.8%

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

      2. associate-*r/ 9.8%

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

      3. *-rgt-identity 9.8%

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

      4. +-commutative 9.8%

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

      5. associate--l+ 93.2%

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

      6. +-inverses 93.2%

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

      7. metadata-eval 93.2%

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

      8. reciprocal-undefine 39.5%

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

      9. +-commutative 39.5%

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

      10. exp-prod 39.5%

          \[\leadsto \mathsf{reciprocal}\left(\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}}\right)\right)\right) \]

   6. Simplified 39.5%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)\right)} \]

   7. Taylor expanded in x around inf 11.5%

      \[\leadsto \color{blue}{{\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 11.5%

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}}} \]

      2. reciprocal-define 11.5%

         \[\leadsto \sqrt[3]{\color{blue}{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

   9. Simplified 11.5%

      \[\leadsto \color{blue}{\sqrt[3]{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 11.5%

          \[\leadsto \sqrt[3]{\color{blue}{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)} \cdot \sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}}} \]

       2. cbrt-prod 11.5%

          \[\leadsto \color{blue}{\sqrt[3]{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}}} \]

       3. reciprocal-undefine 11.5%

          \[\leadsto \sqrt[3]{\sqrt{\color{blue}{\frac{1}{{x}^{2}}}}} \cdot \sqrt[3]{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

       4. sqrt-div 11.5%

          \[\leadsto \sqrt[3]{\color{blue}{\frac{\sqrt{1}}{\sqrt{{x}^{2}}}}} \cdot \sqrt[3]{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

       5. metadata-eval 11.5%

          \[\leadsto \sqrt[3]{\frac{\color{blue}{1}}{\sqrt{{x}^{2}}}} \cdot \sqrt[3]{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

       6. unpow2 11.5%

          \[\leadsto \sqrt[3]{\frac{1}{\sqrt{\color{blue}{x \cdot x}}}} \cdot \sqrt[3]{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

       7. sqrt-prod 11.5%

          \[\leadsto \sqrt[3]{\frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \cdot \sqrt[3]{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

       8. add-sqr-sqrt 11.5%

          \[\leadsto \sqrt[3]{\frac{1}{\color{blue}{x}}} \cdot \sqrt[3]{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

       9. reciprocal-define 11.5%

          \[\leadsto \sqrt[3]{\color{blue}{\mathsf{reciprocal}\left(x\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{reciprocal}\left(\left({x}^{2}\right)\right)}} \]

       10. reciprocal-undefine 11.5%

           \[\leadsto \sqrt[3]{\mathsf{reciprocal}\left(x\right)} \cdot \sqrt[3]{\sqrt{\color{blue}{\frac{1}{{x}^{2}}}}} \]

       11. sqrt-div 11.5%

           \[\leadsto \sqrt[3]{\mathsf{reciprocal}\left(x\right)} \cdot \sqrt[3]{\color{blue}{\frac{\sqrt{1}}{\sqrt{{x}^{2}}}}} \]

       12. metadata-eval 11.5%

           \[\leadsto \sqrt[3]{\mathsf{reciprocal}\left(x\right)} \cdot \sqrt[3]{\frac{\color{blue}{1}}{\sqrt{{x}^{2}}}} \]

       13. unpow2 11.5%

           \[\leadsto \sqrt[3]{\mathsf{reciprocal}\left(x\right)} \cdot \sqrt[3]{\frac{1}{\sqrt{\color{blue}{x \cdot x}}}} \]

       14. sqrt-prod 17.7%

           \[\leadsto \sqrt[3]{\mathsf{reciprocal}\left(x\right)} \cdot \sqrt[3]{\frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

       15. add-sqr-sqrt 17.7%

           \[\leadsto \sqrt[3]{\mathsf{reciprocal}\left(x\right)} \cdot \sqrt[3]{\frac{1}{\color{blue}{x}}} \]

       16. reciprocal-define 17.7%

           \[\leadsto \sqrt[3]{\mathsf{reciprocal}\left(x\right)} \cdot \sqrt[3]{\color{blue}{\mathsf{reciprocal}\left(x\right)}} \]

   11. Applied egg-rr 17.7%

       \[\leadsto \color{blue}{\sqrt[3]{\mathsf{reciprocal}\left(x\right)} \cdot \sqrt[3]{\mathsf{reciprocal}\left(x\right)}} \]
   12. Step-by-step derivation

       1. unpow2 17.7%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{reciprocal}\left(x\right)}\right)}^{2}} \]

   13. Simplified 17.7%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{reciprocal}\left(x\right)}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.38:\\ \;\;\;\;\mathsf{reciprocal}\left(\left({\left(\sqrt[3]{x}\right)}^{2}\right)\right)\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;1 + x \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{reciprocal}\left(x\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 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 54.4%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 3.6%

   \[\leadsto \color{blue}{0} \]

4. Final simplification 3.6%

   \[\leadsto 0 \]
5. Add Preprocessing

Alternative 16: 50.1% 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 54.4%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 51.2%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 51.2%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))