2cbrt (problem 3.3.4)

Percentage Accurate: 53.7% → 99.2%

Time: 9.7s

Alternatives: 13

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: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

   1. flip3-- 56.2%

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

   2. div-inv 56.2%

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

   3. rem-cube-cbrt 55.9%

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

   4. rem-cube-cbrt 57.2%

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

   5. cbrt-unprod 57.2%

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

   6. pow2 57.2%

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

   7. distribute-rgt-out 57.2%

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

   8. +-commutative 57.2%

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

3. Applied egg-rr 57.2%

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

   1. associate-*r/ 57.2%

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

   2. *-rgt-identity 57.2%

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

   3. +-commutative 57.2%

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

   4. associate--l+ 78.1%

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

   5. +-inverses 78.1%

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

   6. metadata-eval 78.1%

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

   7. +-commutative 78.1%

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

   8. fma-def 78.1%

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

   9. +-commutative 78.1%

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

   10. +-commutative 78.1%

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

5. Simplified 78.1%

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

   1. unpow2 78.1%

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

   2. +-commutative 78.1%

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

   3. add-cube-cbrt 78.0%

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

   4. +-commutative 78.0%

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

   5. add-cube-cbrt 78.0%

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

   6. swap-sqr 78.0%

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

   7. add-cbrt-cube 99.2%

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

   8. pow2 99.2%

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

   9. metadata-eval 99.2%

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

   10. sub-neg 99.2%

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

   11. flip-- 78.0%

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

   12. metadata-eval 78.0%

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

   13. fma-neg 78.0%

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

   14. metadata-eval 78.0%

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

   15. cbrt-undiv 78.1%

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

   16. pow2 78.1%

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

   17. clear-num 78.1%

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

7. Applied egg-rr 99.2%

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

   1. frac-2neg 99.2%

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

   2. div-inv 99.2%

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

   3. distribute-neg-frac 99.2%

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

   4. metadata-eval 99.2%

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

9. Applied egg-rr 99.2%

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

10. Final simplification 99.2%

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

Alternative 2: 75.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (- t_0 (cbrt x))))
   (if (<= t_1 1e-6)
     (/ 1.0 (fma (cbrt x) (+ (cbrt x) t_0) (cbrt (* x x))))
     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 <= 1e-6) {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + t_0), cbrt((x * x)));
	} 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 <= 1e-6)
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + t_0), cbrt(Float64(x * x))));
	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, 1e-6], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] + N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 9.99999999999999955e-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 4.7%

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

      4. rem-cube-cbrt 7.4%

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

      5. cbrt-unprod 7.4%

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

      6. pow2 7.4%

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

      7. distribute-rgt-out 7.4%

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

      8. +-commutative 7.4%

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

   3. Applied egg-rr 7.4%

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

      1. associate-*r/ 7.4%

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

      2. *-rgt-identity 7.4%

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

      3. +-commutative 7.4%

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

      4. associate--l+ 52.6%

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

      5. +-inverses 52.6%

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

      6. metadata-eval 52.6%

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

      7. +-commutative 52.6%

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

      8. fma-def 52.6%

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

      9. +-commutative 52.6%

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

      10. +-commutative 52.6%

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

   5. Simplified 52.6%

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

      1. add-exp-log 50.9%

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

      2. pow1/3 50.5%

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

      3. log-pow 50.5%

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

      4. log-pow 57.0%

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

      5. log1p-udef 57.0%

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

   7. Applied egg-rr 57.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{e^{0.3333333333333333 \cdot \left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 57.0%

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

      2. metadata-eval 57.0%

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

   9. Simplified 57.0%

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

   10. Taylor expanded in x around inf 49.6%

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

       1. unpow1/3 52.0%

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

       2. unpow2 52.0%

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

   12. Simplified 52.0%

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

   if 9.99999999999999955e-7 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.7%

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

4. Final simplification 77.7%

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

Alternative 3: 99.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

   1. flip3-- 56.2%

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

   2. div-inv 56.2%

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

   3. rem-cube-cbrt 55.9%

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

   4. rem-cube-cbrt 57.2%

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

   5. cbrt-unprod 57.2%

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

   6. pow2 57.2%

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

   7. distribute-rgt-out 57.2%

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

   8. +-commutative 57.2%

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

3. Applied egg-rr 57.2%

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

   1. associate-*r/ 57.2%

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

   2. *-rgt-identity 57.2%

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

   3. +-commutative 57.2%

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

   4. associate--l+ 78.1%

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

   5. +-inverses 78.1%

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

   6. metadata-eval 78.1%

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

   7. +-commutative 78.1%

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

   8. fma-def 78.1%

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

   9. +-commutative 78.1%

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

   10. +-commutative 78.1%

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

5. Simplified 78.1%

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

   1. unpow2 78.1%

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

   2. +-commutative 78.1%

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

   3. add-cube-cbrt 78.0%

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

   4. +-commutative 78.0%

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

   5. add-cube-cbrt 78.0%

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

   6. swap-sqr 78.0%

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

   7. add-cbrt-cube 99.2%

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

   8. pow2 99.2%

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

   9. metadata-eval 99.2%

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

   10. sub-neg 99.2%

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

   11. flip-- 78.0%

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

   12. metadata-eval 78.0%

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

   13. fma-neg 78.0%

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

   14. metadata-eval 78.0%

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

   15. cbrt-undiv 78.1%

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

   16. pow2 78.1%

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

   17. clear-num 78.1%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 4: 61.1% accurate, 0.3× speedup

Math

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

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 (fma (cbrt x) (+ (cbrt x) t_0) 1.0)) 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 / fma(cbrt(x), (cbrt(x) + t_0), 1.0);
	} 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 / fma(cbrt(x), Float64(cbrt(x) + t_0), 1.0));
	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[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $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. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 3.6%

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

      4. rem-cube-cbrt 5.0%

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

      5. cbrt-unprod 5.0%

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

      6. pow2 5.0%

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

      7. distribute-rgt-out 5.0%

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

      8. +-commutative 5.0%

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

   3. Applied egg-rr 5.0%

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

      1. associate-*r/ 5.0%

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

      2. *-rgt-identity 5.0%

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

      3. +-commutative 5.0%

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

      4. associate--l+ 51.4%

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

      5. +-inverses 51.4%

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

      6. metadata-eval 51.4%

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

      7. +-commutative 51.4%

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

      8. fma-def 51.4%

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

      9. +-commutative 51.4%

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

      10. +-commutative 51.4%

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

   5. Simplified 51.4%

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

   6. Taylor expanded in x around 0 20.0%

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

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

   1. Initial program 98.6%

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

4. Final simplification 63.3%

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

Alternative 5: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x} + \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (cbrt x) (cbrt (+ 1.0 x)))))
   (if (<= x 1.35e+154)
     (/ 1.0 (fma (cbrt x) t_0 (cbrt (pow (+ 1.0 x) 2.0))))
     (/ 1.0 (fma (cbrt x) t_0 (exp (* 0.6666666666666666 (log1p x))))))))

C

double code(double x) {
	double t_0 = cbrt(x) + cbrt((1.0 + x));
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / fma(cbrt(x), t_0, cbrt(pow((1.0 + x), 2.0)));
	} else {
		tmp = 1.0 / fma(cbrt(x), t_0, exp((0.6666666666666666 * log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(x) + cbrt(Float64(1.0 + x)))
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(1.0 / fma(cbrt(x), t_0, cbrt((Float64(1.0 + x) ^ 2.0))));
	else
		tmp = Float64(1.0 / fma(cbrt(x), t_0, exp(Float64(0.6666666666666666 * log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.35e+154], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0 + N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0 + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 64.6%

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

      1. flip3-- 64.6%

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

      2. div-inv 64.5%

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

      3. rem-cube-cbrt 64.5%

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

      4. rem-cube-cbrt 65.8%

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

      5. cbrt-unprod 65.8%

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

      6. pow2 65.8%

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

      7. distribute-rgt-out 65.8%

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

      8. +-commutative 65.8%

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

   3. Applied egg-rr 65.8%

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

      1. associate-*r/ 65.8%

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

      2. *-rgt-identity 65.8%

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

      3. +-commutative 65.8%

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

      4. associate--l+ 90.1%

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

      5. +-inverses 90.1%

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

      6. metadata-eval 90.1%

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

      7. +-commutative 90.1%

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

      8. fma-def 90.0%

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

      9. +-commutative 90.0%

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

      10. +-commutative 90.0%

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

   5. Simplified 90.0%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.9%

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

      1. flip3-- 4.9%

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

      2. div-inv 4.9%

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

      3. rem-cube-cbrt 2.9%

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

      4. rem-cube-cbrt 4.9%

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

      5. cbrt-unprod 4.9%

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

      6. pow2 4.9%

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

      7. distribute-rgt-out 4.9%

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

      8. +-commutative 4.9%

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

   3. Applied egg-rr 4.9%

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

      1. associate-*r/ 4.9%

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

      2. *-rgt-identity 4.9%

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

      3. +-commutative 4.9%

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

      4. associate--l+ 4.9%

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

      5. +-inverses 4.9%

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

      6. metadata-eval 4.9%

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

      7. +-commutative 4.9%

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

      8. fma-def 4.9%

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

      9. +-commutative 4.9%

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

      10. +-commutative 4.9%

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

   5. Simplified 4.9%

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

      1. add-exp-log 4.9%

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

      2. pow1/3 4.9%

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

      3. log-pow 4.9%

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

      4. log-pow 92.5%

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

      5. log1p-udef 92.5%

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

   7. Applied egg-rr 92.5%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{e^{0.3333333333333333 \cdot \left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 92.5%

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

      2. metadata-eval 92.5%

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

   9. Simplified 92.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \]

Alternative 6: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x} + \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, \sqrt[3]{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (cbrt x) (cbrt (+ 1.0 x)))))
   (if (<= x -1.0)
     (/ 1.0 (fma (cbrt x) t_0 (cbrt (* x x))))
     (/ 1.0 (fma (cbrt x) t_0 (exp (* 0.6666666666666666 (log1p x))))))))

C

double code(double x) {
	double t_0 = cbrt(x) + cbrt((1.0 + x));
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / fma(cbrt(x), t_0, cbrt((x * x)));
	} else {
		tmp = 1.0 / fma(cbrt(x), t_0, exp((0.6666666666666666 * log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(x) + cbrt(Float64(1.0 + x)))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 / fma(cbrt(x), t_0, cbrt(Float64(x * x))));
	else
		tmp = Float64(1.0 / fma(cbrt(x), t_0, exp(Float64(0.6666666666666666 * log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0 + N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0 + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 9.5%

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

      1. flip3-- 9.5%

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

      2. div-inv 9.5%

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

      3. rem-cube-cbrt 9.9%

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

      4. rem-cube-cbrt 12.0%

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

      5. cbrt-unprod 12.0%

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

      6. pow2 12.0%

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

      7. distribute-rgt-out 12.0%

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

      8. +-commutative 12.0%

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

   3. Applied egg-rr 12.0%

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

      1. associate-*r/ 12.0%

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

      2. *-rgt-identity 12.0%

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

      3. +-commutative 12.0%

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

      4. associate--l+ 55.8%

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

      5. +-inverses 55.8%

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

      6. metadata-eval 55.8%

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

      7. +-commutative 55.8%

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

      8. fma-def 55.8%

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

      9. +-commutative 55.8%

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

      10. +-commutative 55.8%

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

   5. Simplified 55.8%

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

      1. add-exp-log 54.1%

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

      2. pow1/3 53.7%

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

      3. log-pow 53.8%

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

      4. log-pow 0.0%

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

      5. log1p-udef 0.0%

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

   7. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{e^{0.3333333333333333 \cdot \left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 0.0%

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

      2. metadata-eval 0.0%

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

   9. Simplified 0.0%

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

   10. Taylor expanded in x around inf 49.5%

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

       1. unpow1/3 51.8%

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

       2. unpow2 51.8%

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

   12. Simplified 51.8%

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

   if -1 < x

   1. Initial program 67.0%

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

      1. flip3-- 66.9%

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

      2. div-inv 66.9%

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

      3. rem-cube-cbrt 66.5%

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

      4. rem-cube-cbrt 67.6%

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

      5. cbrt-unprod 67.7%

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

      6. pow2 67.7%

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

      7. distribute-rgt-out 67.7%

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

      8. +-commutative 67.7%

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

   3. Applied egg-rr 67.7%

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

      1. associate-*r/ 67.7%

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

      2. *-rgt-identity 67.7%

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

      3. +-commutative 67.7%

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

      4. associate--l+ 83.2%

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

      5. +-inverses 83.2%

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

      6. metadata-eval 83.2%

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

      7. +-commutative 83.2%

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

      8. fma-def 83.2%

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

      9. +-commutative 83.2%

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

      10. +-commutative 83.2%

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

   5. Simplified 83.2%

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

      1. add-exp-log 82.6%

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

      2. pow1/3 82.5%

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

      3. log-pow 82.5%

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

      4. log-pow 97.7%

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

      5. log1p-udef 97.7%

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

   7. Applied egg-rr 97.7%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{e^{0.3333333333333333 \cdot \left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 97.7%

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

      2. metadata-eval 97.7%

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

   9. Simplified 97.7%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \]

Alternative 7: 99.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

   1. flip3-- 56.2%

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

   2. div-inv 56.2%

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

   3. rem-cube-cbrt 55.9%

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

   4. rem-cube-cbrt 57.2%

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

   5. cbrt-unprod 57.2%

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

   6. pow2 57.2%

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

   7. distribute-rgt-out 57.2%

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

   8. +-commutative 57.2%

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

3. Applied egg-rr 57.2%

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

   1. associate-*r/ 57.2%

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

   2. *-rgt-identity 57.2%

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

   3. +-commutative 57.2%

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

   4. associate--l+ 78.1%

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

   5. +-inverses 78.1%

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

   6. metadata-eval 78.1%

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

   7. +-commutative 78.1%

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

   8. fma-def 78.1%

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

   9. +-commutative 78.1%

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

   10. +-commutative 78.1%

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

5. Simplified 78.1%

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

   1. expm1-log1p-u 77.3%

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

   2. expm1-udef 77.2%

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

7. Applied egg-rr 97.3%

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

   1. expm1-def 97.3%

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

   2. expm1-log1p 99.2%

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

9. Simplified 99.2%

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

10. Final simplification 99.2%

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

Alternative 8: 85.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 9.5%

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

      1. flip3-- 9.5%

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

      2. div-inv 9.5%

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

      3. rem-cube-cbrt 9.9%

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

      4. rem-cube-cbrt 12.0%

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

      5. cbrt-unprod 12.0%

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

      6. pow2 12.0%

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

      7. distribute-rgt-out 12.0%

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

      8. +-commutative 12.0%

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

   3. Applied egg-rr 12.0%

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

      1. associate-*r/ 12.0%

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

      2. *-rgt-identity 12.0%

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

      3. +-commutative 12.0%

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

      4. associate--l+ 55.8%

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

      5. +-inverses 55.8%

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

      6. metadata-eval 55.8%

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

      7. +-commutative 55.8%

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

      8. fma-def 55.8%

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

      9. +-commutative 55.8%

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

      10. +-commutative 55.8%

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

   5. Simplified 55.8%

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

      1. add-exp-log 54.1%

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

      2. pow1/3 53.7%

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

      3. log-pow 53.8%

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

      4. log-pow 0.0%

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

      5. log1p-udef 0.0%

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

   7. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{e^{0.3333333333333333 \cdot \left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 0.0%

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

      2. metadata-eval 0.0%

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

   9. Simplified 0.0%

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

   10. Taylor expanded in x around inf 49.5%

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

       1. unpow1/3 51.8%

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

       2. unpow2 51.8%

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

   12. Simplified 51.8%

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

   if -1 < x

   1. Initial program 67.0%

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

      1. flip3-- 66.9%

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

      2. div-inv 66.9%

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

      3. rem-cube-cbrt 66.5%

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

      4. rem-cube-cbrt 67.6%

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

      5. cbrt-unprod 67.7%

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

      6. pow2 67.7%

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

      7. distribute-rgt-out 67.7%

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

      8. +-commutative 67.7%

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

   3. Applied egg-rr 67.7%

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

      1. associate-*r/ 67.7%

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

      2. *-rgt-identity 67.7%

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

      3. +-commutative 67.7%

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

      4. associate--l+ 83.2%

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

      5. +-inverses 83.2%

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

      6. metadata-eval 83.2%

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

      7. +-commutative 83.2%

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

      8. fma-def 83.2%

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

      9. +-commutative 83.2%

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

      10. +-commutative 83.2%

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

   5. Simplified 83.2%

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

      1. pow1/3 82.4%

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

      2. pow-pow 97.4%

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

      3. metadata-eval 97.4%

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

   7. Applied egg-rr 97.4%

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

4. Final simplification 88.9%

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

Alternative 9: 53.7% 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 56.2%

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

2. Final simplification 56.2%

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

Alternative 10: 51.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

   1. add-cube-cbrt 56.0%

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

   2. pow3 56.0%

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

3. Applied egg-rr 56.0%

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

4. Taylor expanded in x around 0 30.0%

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

   1. associate--l+ 30.0%

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

   2. *-commutative 30.0%

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

   3. metadata-eval 30.0%

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

   4. pow-base-1 30.0%

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

   5. unpow1/3 53.6%

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

   6. *-lft-identity 53.6%

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

6. Simplified 53.6%

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

7. Final simplification 53.6%

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

Alternative 11: 51.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

   1. add-cube-cbrt 56.0%

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

   2. pow3 56.0%

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

3. Applied egg-rr 56.0%

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

4. Taylor expanded in x around 0 28.6%

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

   1. metadata-eval 28.6%

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

   2. pow-base-1 28.6%

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

   3. unpow1/3 52.8%

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

   4. *-lft-identity 52.8%

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

6. Simplified 52.8%

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

7. Final simplification 52.8%

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

Alternative 12: 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 56.2%

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

2. Taylor expanded in x around inf 3.6%

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

3. Final simplification 3.6%

   \[\leadsto 0 \]

Alternative 13: 50.4% 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 56.2%

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

2. Taylor expanded in x around 0 52.0%

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

3. Final simplification 52.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))