2cbrt (problem 3.3.4)

Percentage Accurate: 53.0% → 99.6%

Time: 14.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 53.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = t_0 ^ 2.0
	return Float64(1.0 / Float64(t_1 + Float64(Float64(cbrt(x) / fma(cbrt(x), Float64(cbrt(x) - t_0), t_1)) * Float64(1.0 + Float64(x * 2.0)))))
end

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. pow1/3 50.3%

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

3. Applied egg-rr 50.3%

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

   1. pow1/3 52.9%

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

   2. sub-neg 52.9%

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

   3. flip3-+ 52.9%

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

   4. rem-cube-cbrt 53.1%

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

   5. +-commutative 53.1%

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

   6. pow2 53.1%

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

   7. +-commutative 53.1%

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

   8. +-commutative 53.1%

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

5. Applied egg-rr 53.1%

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

   1. associate-+l+ 59.8%

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

   2. cube-neg 59.8%

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

   3. rem-cube-cbrt 99.1%

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

   4. unsub-neg 99.1%

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

   5. +-inverses 99.1%

      \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(\left(-\sqrt[3]{x}\right) \cdot \left(-\sqrt[3]{x}\right) - \sqrt[3]{1 + x} \cdot \left(-\sqrt[3]{x}\right)\right)} \]

   6. metadata-eval 99.1%

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

   7. sqr-neg 99.1%

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

   8. unpow2 99.1%

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

   9. *-commutative 99.1%

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

   10. cancel-sign-sub 99.1%

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

   11. unpow2 99.1%

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

7. Simplified 99.1%

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

   1. flip3-+ 99.1%

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

   2. associate-*r/ 86.8%

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

   3. rem-cube-cbrt 87.0%

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

   4. rem-cube-cbrt 87.1%

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

   5. +-commutative 87.1%

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

   6. +-commutative 87.1%

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

   7. pow2 87.1%

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

   8. +-commutative 87.1%

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

   9. distribute-rgt-out-- 87.1%

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

   10. +-commutative 87.1%

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

9. Applied egg-rr 87.1%

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

    1. associate-/l* 99.6%

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

    2. associate-/r/ 99.6%

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

    3. +-commutative 99.6%

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

    4. fma-def 99.6%

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

    5. associate-+r+ 99.6%

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

    6. +-commutative 99.6%

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

    7. count-2 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 2: 99.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. 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 3.8%

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

      4. rem-cube-cbrt 4.2%

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

      5. +-commutative 4.2%

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

      6. distribute-rgt-out 4.2%

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

      7. +-commutative 4.2%

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

      8. fma-def 4.2%

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

      9. add-exp-log 4.2%

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

   3. Applied egg-rr 1.8%

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

      1. associate-*r/ 1.8%

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

      2. *-rgt-identity 1.8%

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

      3. +-commutative 1.8%

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

      4. associate--l+ 40.1%

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

      5. +-inverses 40.1%

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

      6. metadata-eval 40.1%

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

      7. +-commutative 40.1%

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

      8. exp-prod 39.6%

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

   5. Simplified 39.6%

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

   6. Taylor expanded in x around inf 48.3%

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

      1. unpow1/3 50.6%

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

   8. Simplified 50.6%

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

      1. expm1-log1p-u 50.6%

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

      2. expm1-udef 5.6%

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

      3. +-commutative 5.6%

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

      4. unpow2 5.6%

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

      5. cbrt-prod 5.6%

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

      6. pow2 5.6%

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

   10. Applied egg-rr 5.6%

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

       1. expm1-def 98.4%

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

       2. expm1-log1p 98.4%

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

       3. fma-udef 98.3%

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

       4. unpow2 98.3%

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

       5. distribute-lft-out 98.4%

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

       6. +-commutative 98.4%

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

   12. Simplified 98.4%

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

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

   1. Initial program 97.9%

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

      1. pow1/3 96.0%

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

   3. Applied egg-rr 96.0%

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

      1. pow1/3 97.9%

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

      2. sub-neg 97.9%

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

      3. flip3-+ 97.9%

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

      4. rem-cube-cbrt 98.8%

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

      5. +-commutative 98.8%

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

      6. pow2 98.8%

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

      7. +-commutative 98.8%

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

      8. +-commutative 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. associate-+l+ 98.8%

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

      2. cube-neg 98.8%

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

      3. rem-cube-cbrt 99.9%

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

      4. unsub-neg 99.9%

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

      5. +-inverses 99.9%

         \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(\left(-\sqrt[3]{x}\right) \cdot \left(-\sqrt[3]{x}\right) - \sqrt[3]{1 + x} \cdot \left(-\sqrt[3]{x}\right)\right)} \]

      6. metadata-eval 99.9%

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

      7. sqr-neg 99.9%

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

      8. unpow2 99.9%

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

      9. *-commutative 99.9%

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

      10. cancel-sign-sub 99.9%

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

      11. unpow2 99.9%

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

   7. Simplified 99.8%

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

      1. pow1/3 96.9%

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

      2. +-commutative 96.9%

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

      3. pow-pow 96.9%

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

      4. metadata-eval 96.9%

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

      5. sub-neg 96.9%

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

      6. flip-- 96.9%

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

      7. metadata-eval 96.9%

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

      8. fma-neg 96.9%

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

      9. metadata-eval 96.9%

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

      10. metadata-eval 96.9%

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

      11. metadata-eval 96.9%

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

      12. pow-prod-up 96.9%

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

      13. pow1/3 96.9%

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

      14. pow1/3 99.8%

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

      15. cbrt-unprod 99.9%

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

      16. pow2 99.9%

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

      17. metadata-eval 99.9%

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

      18. fma-neg 99.9%

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

      19. metadata-eval 99.9%

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

      20. flip-- 99.9%

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

      21. sub-neg 99.9%

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

      22. metadata-eval 99.9%

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

      23. +-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.1%

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

Alternative 3: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} + t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (- t_0 (cbrt x))))
   (if (<= t_1 4e-7)
     (/ 1.0 (* (cbrt x) (+ (cbrt x) (+ (cbrt x) t_0))))
     (exp (log t_1)))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = t_0 - cbrt(x);
	double tmp;
	if (t_1 <= 4e-7) {
		tmp = 1.0 / (cbrt(x) * (cbrt(x) + (cbrt(x) + t_0)));
	} else {
		tmp = exp(log(t_1));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double t_1 = t_0 - Math.cbrt(x);
	double tmp;
	if (t_1 <= 4e-7) {
		tmp = 1.0 / (Math.cbrt(x) * (Math.cbrt(x) + (Math.cbrt(x) + t_0)));
	} else {
		tmp = Math.exp(Math.log(t_1));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = Float64(t_0 - cbrt(x))
	tmp = 0.0
	if (t_1 <= 4e-7)
		tmp = Float64(1.0 / Float64(cbrt(x) * Float64(cbrt(x) + Float64(cbrt(x) + t_0))));
	else
		tmp = exp(log(t_1));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-7], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[t$95$1], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 3.9999999999999998e-7

   1. Initial program 5.3%

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

      1. flip3-- 5.3%

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

      2. div-inv 5.3%

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

      3. rem-cube-cbrt 5.9%

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

      4. rem-cube-cbrt 7.2%

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

      5. +-commutative 7.2%

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

      6. distribute-rgt-out 7.2%

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

      7. +-commutative 7.2%

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

      8. fma-def 7.2%

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

      9. add-exp-log 7.2%

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

   3. Applied egg-rr 3.3%

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

      1. associate-*r/ 3.3%

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

      2. *-rgt-identity 3.3%

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

      3. +-commutative 3.3%

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

      4. associate--l+ 40.3%

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

      5. +-inverses 40.3%

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

      6. metadata-eval 40.3%

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

      7. +-commutative 40.3%

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

      8. exp-prod 39.8%

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

   5. Simplified 39.8%

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

   6. Taylor expanded in x around inf 49.3%

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

      1. unpow1/3 51.5%

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

   8. Simplified 51.5%

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

      1. expm1-log1p-u 51.5%

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

      2. expm1-udef 7.3%

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

      3. +-commutative 7.3%

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

      4. unpow2 7.3%

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

      5. cbrt-prod 7.3%

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

      6. pow2 7.3%

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

   10. Applied egg-rr 7.3%

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

       1. expm1-def 97.8%

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

       2. expm1-log1p 97.8%

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

       3. fma-udef 97.8%

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

       4. unpow2 97.8%

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

       5. distribute-lft-out 97.8%

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

       6. +-commutative 97.8%

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

   12. Simplified 97.8%

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

   if 3.9999999999999998e-7 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.7%

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

      1. add-exp-log 99.7%

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

   3. Applied egg-rr 99.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)}\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (/ (+ 1.0 (- x x)) (fma (cbrt x) (+ (cbrt x) t_0) (pow t_0 2.0)))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	return (1.0 + (x - x)) / fma(cbrt(x), (cbrt(x) + t_0), pow(t_0, 2.0));
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	return Float64(Float64(1.0 + Float64(x - x)) / fma(cbrt(x), Float64(cbrt(x) + t_0), (t_0 ^ 2.0)))
end

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. pow1/3 24.8%

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

3. Applied egg-rr 24.8%

   \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. pow1/3 52.9%

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

   2. flip3-- 52.9%

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

   3. div-inv 52.9%

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

   4. pow3 52.8%

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

   5. add-cube-cbrt 53.1%

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

   6. rem-cube-cbrt 53.9%

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

   7. pow2 53.9%

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

   8. distribute-rgt-out 53.9%

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

   9. +-commutative 53.9%

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

5. Applied egg-rr 53.9%

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

   1. associate-*r/ 53.9%

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

   2. *-rgt-identity 53.9%

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

   3. +-commutative 53.9%

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

   4. associate--l+ 99.1%

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

   5. +-commutative 99.1%

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

   6. fma-def 99.1%

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

   7. +-commutative 99.1%

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

   8. +-commutative 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 5: 60.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

      1. pow1/3 0.8%

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

   3. Applied egg-rr 0.8%

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

      1. pow1/3 4.2%

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

      2. sub-neg 4.2%

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

      3. flip3-+ 4.2%

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

      4. rem-cube-cbrt 3.8%

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

      5. +-commutative 3.8%

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

      6. pow2 3.8%

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

      7. +-commutative 3.8%

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

      8. +-commutative 3.8%

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

   5. Applied egg-rr 3.8%

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

      1. associate-+l+ 17.7%

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

      2. cube-neg 17.7%

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

      3. rem-cube-cbrt 98.3%

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

      4. unsub-neg 98.3%

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

      5. +-inverses 98.3%

         \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(\left(-\sqrt[3]{x}\right) \cdot \left(-\sqrt[3]{x}\right) - \sqrt[3]{1 + x} \cdot \left(-\sqrt[3]{x}\right)\right)} \]

      6. metadata-eval 98.3%

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

      7. sqr-neg 98.3%

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

      8. unpow2 98.3%

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

      9. *-commutative 98.3%

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

      10. cancel-sign-sub 98.3%

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

      11. unpow2 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x around 0 20.0%

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

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

   1. Initial program 97.9%

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

      1. add-exp-log 97.9%

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

   3. Applied egg-rr 97.9%

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

4. Final simplification 60.4%

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

Alternative 6: 60.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

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. Step-by-step derivation

      1. pow1/3 0.8%

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

   3. Applied egg-rr 0.8%

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

      1. pow1/3 4.2%

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

      2. sub-neg 4.2%

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

      3. flip3-+ 4.2%

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

      4. rem-cube-cbrt 3.8%

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

      5. +-commutative 3.8%

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

      6. pow2 3.8%

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

      7. +-commutative 3.8%

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

      8. +-commutative 3.8%

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

   5. Applied egg-rr 3.8%

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

      1. associate-+l+ 17.7%

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

      2. cube-neg 17.7%

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

      3. rem-cube-cbrt 98.3%

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

      4. unsub-neg 98.3%

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

      5. +-inverses 98.3%

         \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(\left(-\sqrt[3]{x}\right) \cdot \left(-\sqrt[3]{x}\right) - \sqrt[3]{1 + x} \cdot \left(-\sqrt[3]{x}\right)\right)} \]

      6. metadata-eval 98.3%

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

      7. sqr-neg 98.3%

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

      8. unpow2 98.3%

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

      9. *-commutative 98.3%

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

      10. cancel-sign-sub 98.3%

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

      11. unpow2 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x around 0 20.0%

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

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

   1. Initial program 97.9%

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

4. Final simplification 60.4%

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

Alternative 7: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (/ 1.0 (+ (pow t_0 2.0) (* (cbrt x) (+ (cbrt x) t_0))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	return 1.0 / (pow(t_0, 2.0) + (cbrt(x) * (cbrt(x) + t_0)));
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	return 1.0 / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (Math.cbrt(x) + t_0)));
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	return Float64(1.0 / Float64((t_0 ^ 2.0) + Float64(cbrt(x) * Float64(cbrt(x) + t_0))))
end

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. pow1/3 50.3%

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

3. Applied egg-rr 50.3%

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

   1. pow1/3 52.9%

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

   2. sub-neg 52.9%

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

   3. flip3-+ 52.9%

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

   4. rem-cube-cbrt 53.1%

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

   5. +-commutative 53.1%

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

   6. pow2 53.1%

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

   7. +-commutative 53.1%

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

   8. +-commutative 53.1%

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

5. Applied egg-rr 53.1%

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

   1. associate-+l+ 59.8%

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

   2. cube-neg 59.8%

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

   3. rem-cube-cbrt 99.1%

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

   4. unsub-neg 99.1%

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

   5. +-inverses 99.1%

      \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(\left(-\sqrt[3]{x}\right) \cdot \left(-\sqrt[3]{x}\right) - \sqrt[3]{1 + x} \cdot \left(-\sqrt[3]{x}\right)\right)} \]

   6. metadata-eval 99.1%

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

   7. sqr-neg 99.1%

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

   8. unpow2 99.1%

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

   9. *-commutative 99.1%

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

   10. cancel-sign-sub 99.1%

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

   11. unpow2 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 8: 53.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

2. Final simplification 52.9%

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

Alternative 9: 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 52.9%

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

2. Taylor expanded in x around inf 3.7%

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

3. Final simplification 3.7%

   \[\leadsto 0 \]

Alternative 10: 49.4% 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 52.9%

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

2. Taylor expanded in x around 0 49.3%

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

3. Final simplification 49.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023338 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))