Octave 3.8, jcobi/1

Average Error: 16.8 → 0.1

Time: 13.7s

Precision: binary64

Cost: 14404

\[\alpha > -1 \land \beta > -1\]

Math

\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\frac{\beta - \left(-2 - \beta\right)}{\alpha} - \frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha} \cdot \frac{-2 - \beta}{-\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.995)
   (/
    (-
     (/ (- beta (- -2.0 beta)) alpha)
     (* (/ (fma 2.0 beta 2.0) alpha) (/ (- -2.0 beta) (- alpha))))
    2.0)
   (/
    (pow
     (pow (+ (/ (- beta alpha) (+ beta (+ alpha 2.0))) 1.0) 3.0)
     0.3333333333333333)
    2.0)))

C

double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

↓

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.995) {
		tmp = (((beta - (-2.0 - beta)) / alpha) - ((fma(2.0, beta, 2.0) / alpha) * ((-2.0 - beta) / -alpha))) / 2.0;
	} else {
		tmp = pow(pow((((beta - alpha) / (beta + (alpha + 2.0))) + 1.0), 3.0), 0.3333333333333333) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
end

↓

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.995)
		tmp = Float64(Float64(Float64(Float64(beta - Float64(-2.0 - beta)) / alpha) - Float64(Float64(fma(2.0, beta, 2.0) / alpha) * Float64(Float64(-2.0 - beta) / Float64(-alpha)))) / 2.0);
	else
		tmp = Float64(((Float64(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))) + 1.0) ^ 3.0) ^ 0.3333333333333333) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

↓

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.995], N[(N[(N[(N[(beta - N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(N[(2.0 * beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(-2.0 - beta), $MachinePrecision] / (-alpha)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Power[N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.994999999999999996

   1. Initial program 58.9

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]

   2. Simplified 58.9

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]
      Proof

      [Start] 58.9	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 58.9	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around -inf 2.9

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + -1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}}{2} \]

   4. Simplified 2.9

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \left(2 + \beta\right)}{\alpha \cdot \alpha}\right)}}{2} \]
      Proof

      [Start] 2.9	\[ \frac{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + -1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      distribute-lft-out [=>] 2.9	\[ \frac{\color{blue}{-1 \cdot \left(\frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}}{2} \]
      mul-1-neg [=>] 2.9	\[ \frac{-1 \cdot \left(\frac{\color{blue}{\left(-\beta\right)} - \left(\beta + 2\right)}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 2.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \color{blue}{\left(2 + \beta\right)}}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 2.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\color{blue}{\left(2 + \beta\right)}}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 2.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \color{blue}{\left(2 + \beta\right)}}{{\alpha}^{2}}\right)}{2} \]
      unpow2 [=>] 2.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \left(2 + \beta\right)}{\color{blue}{\alpha \cdot \alpha}}\right)}{2} \]

   5. Applied egg-rr 2.9

      \[\leadsto \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \color{blue}{\left(-\left(\beta + 2\right) \cdot \left(2 + \left(\beta + \beta\right)\right)\right) \cdot \frac{1}{\alpha \cdot \left(-\alpha\right)}}\right)}{2} \]

   6. Simplified 0.4

      \[\leadsto \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \color{blue}{\frac{2 + \beta}{-\alpha} \cdot \left(-\frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha}\right)}\right)}{2} \]
      Proof

      [Start] 2.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \left(-\left(\beta + 2\right) \cdot \left(2 + \left(\beta + \beta\right)\right)\right) \cdot \frac{1}{\alpha \cdot \left(-\alpha\right)}\right)}{2} \]
      associate-*r/ [=>] 2.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \color{blue}{\frac{\left(-\left(\beta + 2\right) \cdot \left(2 + \left(\beta + \beta\right)\right)\right) \cdot 1}{\alpha \cdot \left(-\alpha\right)}}\right)}{2} \]
      *-rgt-identity [=>] 2.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\color{blue}{-\left(\beta + 2\right) \cdot \left(2 + \left(\beta + \beta\right)\right)}}{\alpha \cdot \left(-\alpha\right)}\right)}{2} \]
      distribute-rgt-neg-in [=>] 2.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\color{blue}{\left(\beta + 2\right) \cdot \left(-\left(2 + \left(\beta + \beta\right)\right)\right)}}{\alpha \cdot \left(-\alpha\right)}\right)}{2} \]
      *-commutative [=>] 2.9	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\left(\beta + 2\right) \cdot \left(-\left(2 + \left(\beta + \beta\right)\right)\right)}{\color{blue}{\left(-\alpha\right) \cdot \alpha}}\right)}{2} \]
      times-frac [=>] 0.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \color{blue}{\frac{\beta + 2}{-\alpha} \cdot \frac{-\left(2 + \left(\beta + \beta\right)\right)}{\alpha}}\right)}{2} \]
      +-commutative [=>] 0.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\color{blue}{2 + \beta}}{-\alpha} \cdot \frac{-\left(2 + \left(\beta + \beta\right)\right)}{\alpha}\right)}{2} \]
      distribute-neg-frac [<=] 0.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{2 + \beta}{-\alpha} \cdot \color{blue}{\left(-\frac{2 + \left(\beta + \beta\right)}{\alpha}\right)}\right)}{2} \]
      count-2 [=>] 0.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{2 + \beta}{-\alpha} \cdot \left(-\frac{2 + \color{blue}{2 \cdot \beta}}{\alpha}\right)\right)}{2} \]
      +-commutative [=>] 0.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{2 + \beta}{-\alpha} \cdot \left(-\frac{\color{blue}{2 \cdot \beta + 2}}{\alpha}\right)\right)}{2} \]
      fma-def [=>] 0.4	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{2 + \beta}{-\alpha} \cdot \left(-\frac{\color{blue}{\mathsf{fma}\left(2, \beta, 2\right)}}{\alpha}\right)\right)}{2} \]

   if -0.994999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.0

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]
      Proof

      [Start] 0.0	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 0.0	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.995:\\ \;\;\;\;\frac{\frac{\beta - \left(-2 - \beta\right)}{\alpha} - \frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha} \cdot \frac{-2 - \beta}{-\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	8452

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.995:\\ \;\;\;\;\frac{\frac{\beta - \left(-2 - \beta\right)}{\alpha} - \frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha} \cdot \frac{-2 - \beta}{-\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 3
Error	4.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 165000000000:\\ \;\;\;\;\frac{1 + \beta \cdot \frac{1}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	7.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 375000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	4.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 350000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	18.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{1 - \beta \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	18.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{1 - \beta \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 8
Error	18.9
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	32.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023041 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))