2cbrt (problem 3.3.4)

Percentage Accurate: 52.8% → 99.1%

Time: 13.7s

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: 52.8% 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: 99.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

   1. add-exp-log 55.4%

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

3. Applied egg-rr 55.4%

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

5. Applied egg-rr 55.5%

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

   1. div-sub 56.9%

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

   2. +-commutative 56.9%

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

   3. associate--l+ 99.2%

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

   4. +-commutative 99.2%

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

   5. fma-def 99.2%

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

   6. +-commutative 99.2%

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

   7. +-commutative 99.2%

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

7. Simplified 99.2%

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

   1. +-commutative 99.2%

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

   2. add-cbrt-cube 99.2%

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

   3. cbrt-prod 99.2%

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

   4. pow2 99.2%

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

9. Applied egg-rr 99.2%

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

10. Final simplification 99.2%

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

Alternative 2: 75.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

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

      1. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 3.8%

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

      4. rem-cube-cbrt 4.2%

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

      5. +-commutative 4.2%

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

      6. distribute-rgt-out 4.2%

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

      7. +-commutative 4.2%

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

      8. fma-def 4.2%

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

      9. add-exp-log 4.2%

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

   3. Applied egg-rr 2.3%

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

      1. associate-*r/ 2.3%

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

      2. *-rgt-identity 2.3%

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

      3. +-commutative 2.3%

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

      4. associate--l+ 48.5%

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

      5. +-inverses 48.5%

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

      6. metadata-eval 48.5%

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

      7. +-commutative 48.5%

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

      8. exp-prod 48.0%

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

   5. Simplified 48.0%

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

   6. Taylor expanded in x around inf 42.1%

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

      1. unpow1/3 44.0%

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

   8. Simplified 44.0%

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

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

   1. Initial program 97.0%

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

      1. add-exp-log 97.0%

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

   3. Applied egg-rr 97.0%

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

      1. rem-exp-log 97.0%

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

      2. add-sqr-sqrt 43.5%

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

      3. cbrt-prod 43.5%

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

      4. pow1/3 43.5%

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

      5. sqrt-pow2 43.5%

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

      6. metadata-eval 43.5%

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

      7. flip3-- 43.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 74.7%

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

Alternative 3: 61.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;e^{-\mathsf{log1p}\left(t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{t_0}^{2} + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (* (cbrt x) (+ (cbrt x) t_0))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (exp (- (log1p t_1)))
     (/ (- (+ 1.0 x) x) (+ (pow t_0 2.0) t_1)))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = cbrt(x) * (cbrt(x) + t_0);
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = exp(-log1p(t_1));
	} else {
		tmp = ((1.0 + x) - x) / (pow(t_0, 2.0) + t_1);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double t_1 = Math.cbrt(x) * (Math.cbrt(x) + t_0);
	double tmp;
	if ((t_0 - Math.cbrt(x)) <= 0.0) {
		tmp = Math.exp(-Math.log1p(t_1));
	} else {
		tmp = ((1.0 + x) - x) / (Math.pow(t_0, 2.0) + t_1);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = Float64(cbrt(x) * Float64(cbrt(x) + t_0))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = exp(Float64(-log1p(t_1)));
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64((t_0 ^ 2.0) + t_1));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

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

      1. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 3.8%

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

      4. rem-cube-cbrt 4.2%

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

      5. +-commutative 4.2%

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

      6. distribute-rgt-out 4.2%

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

      7. +-commutative 4.2%

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

      8. fma-def 4.2%

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

      9. add-exp-log 4.2%

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

   3. Applied egg-rr 2.3%

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

      1. associate-*r/ 2.3%

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

      2. *-rgt-identity 2.3%

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

      3. +-commutative 2.3%

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

      4. associate--l+ 48.5%

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

      5. +-inverses 48.5%

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

      6. metadata-eval 48.5%

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

      7. +-commutative 48.5%

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

      8. exp-prod 48.0%

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

   5. Simplified 48.0%

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

   6. Taylor expanded in x around 0 19.9%

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

      1. add-exp-log 19.9%

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

      2. log-rec 19.9%

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

      3. fma-udef 19.9%

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

      4. +-commutative 19.9%

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

      5. log1p-udef 19.9%

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

      6. +-commutative 19.9%

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

   8. Applied egg-rr 19.9%

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

      1. +-commutative 19.9%

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

   10. Simplified 19.9%

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

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

   1. Initial program 97.0%

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

      1. add-exp-log 97.0%

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

   3. Applied egg-rr 97.0%

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

      1. rem-exp-log 97.0%

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

      2. add-sqr-sqrt 43.5%

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

      3. cbrt-prod 43.5%

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

      4. pow1/3 43.5%

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

      5. sqrt-pow2 43.5%

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

      6. metadata-eval 43.5%

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

      7. flip3-- 43.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;e^{-\mathsf{log1p}\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \end{array} \]

Alternative 4: 60.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;e^{-\mathsf{log1p}\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 + {x}^{3}}}{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right) - x}} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (exp (- (log1p (* (cbrt x) (+ (cbrt x) t_0)))))
     (- (/ (cbrt (+ 1.0 (pow x 3.0))) (cbrt (- (fma x x 1.0) x))) (cbrt x)))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = exp(-log1p((cbrt(x) * (cbrt(x) + t_0))));
	} else {
		tmp = (cbrt((1.0 + pow(x, 3.0))) / cbrt((fma(x, x, 1.0) - x))) - cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = exp(Float64(-log1p(Float64(cbrt(x) * Float64(cbrt(x) + t_0)))));
	else
		tmp = Float64(Float64(cbrt(Float64(1.0 + (x ^ 3.0))) / cbrt(Float64(fma(x, x, 1.0) - x))) - cbrt(x));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

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

      1. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 3.8%

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

      4. rem-cube-cbrt 4.2%

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

      5. +-commutative 4.2%

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

      6. distribute-rgt-out 4.2%

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

      7. +-commutative 4.2%

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

      8. fma-def 4.2%

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

      9. add-exp-log 4.2%

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

   3. Applied egg-rr 2.3%

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

      1. associate-*r/ 2.3%

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

      2. *-rgt-identity 2.3%

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

      3. +-commutative 2.3%

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

      4. associate--l+ 48.5%

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

      5. +-inverses 48.5%

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

      6. metadata-eval 48.5%

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

      7. +-commutative 48.5%

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

      8. exp-prod 48.0%

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

   5. Simplified 48.0%

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

   6. Taylor expanded in x around 0 19.9%

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

      1. add-exp-log 19.9%

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

      2. log-rec 19.9%

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

      3. fma-udef 19.9%

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

      4. +-commutative 19.9%

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

      5. log1p-udef 19.9%

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

      6. +-commutative 19.9%

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

   8. Applied egg-rr 19.9%

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

      1. +-commutative 19.9%

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

   10. Simplified 19.9%

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

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

   1. Initial program 97.0%

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

      1. flip3-+ 97.0%

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

      2. cbrt-div 97.0%

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

      3. pow3 97.0%

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

      4. metadata-eval 97.0%

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

      5. +-commutative 97.0%

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

      6. pow3 97.0%

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

      7. metadata-eval 97.0%

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

      8. *-rgt-identity 97.0%

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

      9. associate-+r- 97.0%

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

      10. fma-def 97.0%

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

   3. Applied egg-rr 97.0%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;e^{-\mathsf{log1p}\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 + {x}^{3}}}{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right) - x}} - \sqrt[3]{x}\\ \end{array} \]

Alternative 5: 60.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;e^{-\mathsf{log1p}\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({t_1}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (- t_0 (cbrt x))))
   (if (<= t_1 0.0)
     (exp (- (log1p (* (cbrt x) (+ (cbrt x) t_0)))))
     (pow (pow t_1 3.0) 0.3333333333333333))))

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 = exp(-log1p((cbrt(x) * (cbrt(x) + t_0))));
	} else {
		tmp = pow(pow(t_1, 3.0), 0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double t_1 = t_0 - Math.cbrt(x);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.exp(-Math.log1p((Math.cbrt(x) * (Math.cbrt(x) + t_0))));
	} else {
		tmp = Math.pow(Math.pow(t_1, 3.0), 0.3333333333333333);
	}
	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 = exp(Float64(-log1p(Float64(cbrt(x) * Float64(cbrt(x) + t_0)))));
	else
		tmp = (t_1 ^ 3.0) ^ 0.3333333333333333;
	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[Exp[(-N[Log[1 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])], $MachinePrecision], N[Power[N[Power[t$95$1, 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

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

      1. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 3.8%

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

      4. rem-cube-cbrt 4.2%

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

      5. +-commutative 4.2%

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

      6. distribute-rgt-out 4.2%

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

      7. +-commutative 4.2%

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

      8. fma-def 4.2%

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

      9. add-exp-log 4.2%

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

   3. Applied egg-rr 2.3%

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

      1. associate-*r/ 2.3%

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

      2. *-rgt-identity 2.3%

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

      3. +-commutative 2.3%

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

      4. associate--l+ 48.5%

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

      5. +-inverses 48.5%

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

      6. metadata-eval 48.5%

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

      7. +-commutative 48.5%

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

      8. exp-prod 48.0%

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

   5. Simplified 48.0%

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

   6. Taylor expanded in x around 0 19.9%

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

      1. add-exp-log 19.9%

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

      2. log-rec 19.9%

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

      3. fma-udef 19.9%

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

      4. +-commutative 19.9%

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

      5. log1p-udef 19.9%

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

      6. +-commutative 19.9%

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

   8. Applied egg-rr 19.9%

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

      1. +-commutative 19.9%

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

   10. Simplified 19.9%

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

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

   1. Initial program 97.0%

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

      1. add-cbrt-cube 96.9%

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

      2. pow1/3 97.0%

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

      3. pow3 97.0%

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

   3. Applied egg-rr 97.0%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;e^{-\mathsf{log1p}\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternative 6: 87.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = Float64(cbrt(x) + cbrt(Float64(1.0 + x)))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * t_0)));
	elseif (x <= -1.0)
		tmp = Float64(1.0 / fma(cbrt(x), t_0, cbrt((x ^ 2.0))));
	else
		tmp = Float64(1.0 / fma(cbrt(x), t_0, exp(Float64(log1p(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, -1.35e+154], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.0], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0 + N[Power[N[Power[x, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0 + N[Exp[N[(N[Log[1 + x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 4.7%

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

      1. flip3-- 4.7%

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

      2. div-inv 4.7%

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

      3. rem-cube-cbrt 3.5%

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

      4. rem-cube-cbrt 4.7%

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

      5. +-commutative 4.7%

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

      6. distribute-rgt-out 4.7%

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

      7. +-commutative 4.7%

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

      8. fma-def 4.7%

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

      9. add-exp-log 4.7%

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

   3. Applied egg-rr 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-rgt-identity 0.0%

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

      3. +-commutative 0.0%

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

      4. associate--l+ 0.0%

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

      5. +-inverses 0.0%

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

      6. metadata-eval 0.0%

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

      7. +-commutative 0.0%

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

      8. exp-prod 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around 0 19.9%

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

      1. fma-udef 19.9%

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

      2. +-commutative 19.9%

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

   8. Applied egg-rr 19.9%

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

   if -1.35000000000000003e154 < x < -1

   1. Initial program 16.5%

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

      1. flip3-- 17.8%

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

      2. div-inv 17.8%

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

      3. rem-cube-cbrt 17.6%

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

      4. rem-cube-cbrt 23.8%

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

      5. +-commutative 23.8%

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

      6. distribute-rgt-out 23.8%

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

      7. +-commutative 23.8%

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

      8. fma-def 23.8%

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

      9. add-exp-log 23.6%

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

   3. Applied egg-rr 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-rgt-identity 0.0%

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

      3. +-commutative 0.0%

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

      4. associate--l+ 0.0%

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

      5. +-inverses 0.0%

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

      6. metadata-eval 0.0%

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

      7. +-commutative 0.0%

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

      8. exp-prod 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in x around inf 86.3%

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

      1. unpow1/3 89.9%

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

   8. Simplified 89.9%

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

   if -1 < x

   1. Initial program 69.5%

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

      1. flip3-- 69.5%

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

      2. div-inv 69.5%

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

      3. rem-cube-cbrt 69.3%

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

      4. rem-cube-cbrt 70.3%

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

      5. +-commutative 70.3%

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

      6. distribute-rgt-out 70.3%

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

      7. +-commutative 70.3%

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

      8. fma-def 70.3%

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

      9. add-exp-log 70.3%

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

   3. Applied egg-rr 70.3%

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

      1. associate-*r/ 70.3%

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

      2. *-rgt-identity 70.3%

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

      3. +-commutative 70.3%

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

      4. associate--l+ 97.7%

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

      5. +-inverses 97.7%

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

      6. metadata-eval 97.7%

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

      7. +-commutative 97.7%

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

      8. exp-prod 97.4%

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

   5. Simplified 97.4%

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

      1. add-exp-log 97.5%

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

      2. log-pow 97.7%

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

      3. rem-log-exp 97.7%

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

   7. Applied egg-rr 97.7%

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

4. Final simplification 87.9%

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

Alternative 7: 60.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double t_1 = t_0 - Math.cbrt(x);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (Math.cbrt(x) + t_0)));
	} else {
		tmp = Math.pow(Math.pow(t_1, 3.0), 0.3333333333333333);
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

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

      1. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 3.8%

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

      4. rem-cube-cbrt 4.2%

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

      5. +-commutative 4.2%

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

      6. distribute-rgt-out 4.2%

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

      7. +-commutative 4.2%

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

      8. fma-def 4.2%

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

      9. add-exp-log 4.2%

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

   3. Applied egg-rr 2.3%

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

      1. associate-*r/ 2.3%

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

      2. *-rgt-identity 2.3%

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

      3. +-commutative 2.3%

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

      4. associate--l+ 48.5%

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

      5. +-inverses 48.5%

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

      6. metadata-eval 48.5%

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

      7. +-commutative 48.5%

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

      8. exp-prod 48.0%

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

   5. Simplified 48.0%

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

   6. Taylor expanded in x around 0 19.9%

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

      1. fma-udef 19.9%

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

      2. +-commutative 19.9%

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

   8. Applied egg-rr 19.9%

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

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

   1. Initial program 97.0%

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

      1. add-cbrt-cube 96.9%

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

      2. pow1/3 97.0%

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

      3. pow3 97.0%

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

   3. Applied egg-rr 97.0%

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

4. Final simplification 62.4%

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

Alternative 8: 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 55.4%

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

   1. pow1/3 52.0%

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

3. Applied egg-rr 52.0%

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

   1. pow1/3 55.4%

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

   2. flip3-- 55.5%

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

   3. div-inv 55.5%

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

   4. rem-cube-cbrt 55.2%

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

   5. rem-cube-cbrt 56.9%

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

   6. unpow2 56.9%

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

   7. +-commutative 56.9%

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

   8. distribute-rgt-in 56.9%

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

5. Applied egg-rr 56.9%

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

   1. associate-*r/ 56.9%

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

   2. *-rgt-identity 56.9%

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

   3. +-commutative 56.9%

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

   4. associate--l+ 99.2%

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

   5. +-inverses 99.2%

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

   6. metadata-eval 99.2%

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

   7. +-commutative 99.2%

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

   8. fma-def 99.2%

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

   9. +-commutative 99.2%

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

   10. +-commutative 99.2%

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

7. Simplified 99.2%

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

8. Final simplification 99.2%

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

Alternative 9: 60.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

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

      1. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 3.8%

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

      4. rem-cube-cbrt 4.2%

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

      5. +-commutative 4.2%

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

      6. distribute-rgt-out 4.2%

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

      7. +-commutative 4.2%

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

      8. fma-def 4.2%

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

      9. add-exp-log 4.2%

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

   3. Applied egg-rr 2.3%

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

      1. associate-*r/ 2.3%

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

      2. *-rgt-identity 2.3%

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

      3. +-commutative 2.3%

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

      4. associate--l+ 48.5%

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

      5. +-inverses 48.5%

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

      6. metadata-eval 48.5%

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

      7. +-commutative 48.5%

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

      8. exp-prod 48.0%

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

   5. Simplified 48.0%

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

   6. Taylor expanded in x around 0 19.9%

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

      1. fma-udef 19.9%

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

      2. +-commutative 19.9%

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

   8. Applied egg-rr 19.9%

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

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

   1. Initial program 97.0%

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

4. Final simplification 62.4%

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

Alternative 10: 52.8% 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 55.4%

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

2. Final simplification 55.4%

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

Alternative 11: 49.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(1 + x \cdot \left(0.3333333333333333 + x \cdot -0.1111111111111111\right)\right) - \sqrt[3]{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (+ 1.0 (* x (+ 0.3333333333333333 (* x -0.1111111111111111)))) (cbrt x)))

C

double code(double x) {
	return (1.0 + (x * (0.3333333333333333 + (x * -0.1111111111111111)))) - cbrt(x);
}

Java

public static double code(double x) {
	return (1.0 + (x * (0.3333333333333333 + (x * -0.1111111111111111)))) - Math.cbrt(x);
}

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(x * Float64(0.3333333333333333 + Float64(x * -0.1111111111111111)))) - cbrt(x))
end

Wolfram

code[x_] := N[(N[(1.0 + N[(x * N[(0.3333333333333333 + N[(x * -0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

   1. pow1/3 52.0%

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

3. Applied egg-rr 52.0%

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

4. Taylor expanded in x around 0 51.5%

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

   1. *-commutative 51.5%

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

   2. unpow2 51.5%

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

   3. associate-*l* 51.5%

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

   4. *-commutative 51.5%

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

   5. distribute-lft-out 51.5%

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

6. Simplified 51.5%

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

7. Final simplification 51.5%

   \[\leadsto \left(1 + x \cdot \left(0.3333333333333333 + x \cdot -0.1111111111111111\right)\right) - \sqrt[3]{x} \]

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 55.4%

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

2. Taylor expanded in x around inf 3.6%

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

3. Final simplification 3.6%

   \[\leadsto 0 \]

Alternative 13: 49.4% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 55.4%

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

2. Taylor expanded in x around 0 50.1%

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

3. Final simplification 50.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))