2cbrt (problem 3.3.4)

Average Error: 29.8 → 0.2

Time: 6.9s

Precision: binary64

Cost: 52928

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Java

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 29.8

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

2. Applied egg-rr 29.2

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

3. Simplified 0.5

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}} \]
   Proof
   (/.f64 1 (fma.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 (+.f64 1 x)) (cbrt.f64 x)) (pow.f64 (cbrt.f64 (+.f64 1 x)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= metadata-eval (+.f64 1 0)) (fma.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 (+.f64 1 x)) (cbrt.f64 x)) (pow.f64 (cbrt.f64 (+.f64 1 x)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (Rewrite<= +-inverses_binary64 (-.f64 x x))) (fma.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 (+.f64 1 x)) (cbrt.f64 x)) (pow.f64 (cbrt.f64 (+.f64 1 x)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 1 x) x)) (fma.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 (+.f64 1 x)) (cbrt.f64 x)) (pow.f64 (cbrt.f64 (+.f64 1 x)) 2))): 114 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1)) x) (fma.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 (+.f64 1 x)) (cbrt.f64 x)) (pow.f64 (cbrt.f64 (+.f64 1 x)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (+.f64 x 1) x) (fma.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1))) (cbrt.f64 x)) (pow.f64 (cbrt.f64 (+.f64 1 x)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (+.f64 x 1) x) (fma.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) (pow.f64 (cbrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1))) 2))): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (+.f64 x 1) x) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))) (pow.f64 (cbrt.f64 (+.f64 x 1)) 2)))): 1 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (+.f64 x 1) x) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (cbrt.f64 (+.f64 x 1)) 2) (*.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (-.f64 (+.f64 x 1) x) 1)) (+.f64 (pow.f64 (cbrt.f64 (+.f64 x 1)) 2) (*.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 (+.f64 x 1) x) (/.f64 1 (+.f64 (pow.f64 (cbrt.f64 (+.f64 x 1)) 2) (*.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))))))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 17.3

   \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)\right)\right)}} \]

5. Applied egg-rr 0.5

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

6. Simplified 0.5

   \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}} \]
   Proof
   (+.f64 (pow.f64 (cbrt.f64 (+.f64 1 x)) 2) (*.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 x) (cbrt.f64 (+.f64 1 x))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (cbrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1))) 2) (*.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 x) (cbrt.f64 (+.f64 1 x))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> unpow2_binary64 (*.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 (+.f64 x 1)))) (*.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 x) (cbrt.f64 (+.f64 1 x))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 (+.f64 x 1))) (*.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 x) (cbrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-udef_binary64 (fma.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 (+.f64 x 1)) (*.f64 (cbrt.f64 x) (+.f64 (cbrt.f64 x) (cbrt.f64 (+.f64 x 1)))))): 12 points increase in error, 8 points decrease in error

7. Applied egg-rr 7.6

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

8. Simplified 0.2

   \[\leadsto \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \color{blue}{\frac{\sqrt[3]{x}}{\frac{{\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}{x + \left(x + 1\right)}}}} \]
   Proof
   (/.f64 (cbrt.f64 x) (/.f64 (+.f64 (pow.f64 (cbrt.f64 x) 2) (*.f64 (cbrt.f64 (+.f64 x 1)) (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)))) (+.f64 x (+.f64 x 1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (cbrt.f64 x) (+.f64 x (+.f64 x 1))) (+.f64 (pow.f64 (cbrt.f64 x) 2) (*.f64 (cbrt.f64 (+.f64 x 1)) (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)))))): 47 points increase in error, 18 points decrease in error
   (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 x (+.f64 x 1)) (cbrt.f64 x))) (+.f64 (pow.f64 (cbrt.f64 x) 2) (*.f64 (cbrt.f64 (+.f64 x 1)) (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))))): 0 points increase in error, 0 points decrease in error

9. Final simplification 0.2

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

Alternatives

Alternative 1
Error	25.0
Cost	39172

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

Alternative 2
Error	0.5
Cost	39168

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

Alternative 3
Error	7.3
Cost	33160

\[\begin{array}{l} t_0 := \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + t_0}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_0 + \sqrt[3]{{\left(1 + x\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \]

Alternative 4
Error	25.0
Cost	33092

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

Alternative 5
Error	13.9
Cost	32964

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

Alternative 6
Error	0.5
Cost	32896

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

Alternative 7
Error	27.8
Cost	26692

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

Alternative 8
Error	13.9
Cost	26628

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

Alternative 9
Error	27.7
Cost	26308

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

Alternative 10
Error	31.7
Cost	6976

\[\frac{1}{1 + \sqrt{x \cdot \left(x \cdot 0.4444444444444444\right)}} \]

Alternative 11
Error	61.7
Cost	64

\[0 \]

Alternative 12
Error	32.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022328 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))