Octave 3.8, jcobi/1

Average Error: 15.7 → 0.1

Time: 10.1s

Precision: binary64

Cost: 8452

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.95:\\ \;\;\;\;\frac{\frac{\beta - \left(-2 - \beta\right)}{\alpha} + \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{-\alpha} \cdot \frac{-2 - \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= (/ (- beta alpha) t_0) -0.95)
     (/
      (+
       (/ (- beta (- -2.0 beta)) alpha)
       (* (/ (fma -2.0 beta -2.0) (- alpha)) (/ (- -2.0 beta) alpha)))
      2.0)
     (/ (- 1.0 (/ (- alpha beta) t_0)) 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 t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.95) {
		tmp = (((beta - (-2.0 - beta)) / alpha) + ((fma(-2.0, beta, -2.0) / -alpha) * ((-2.0 - beta) / alpha))) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / t_0)) / 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)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.95)
		tmp = Float64(Float64(Float64(Float64(beta - Float64(-2.0 - beta)) / alpha) + Float64(Float64(fma(-2.0, beta, -2.0) / Float64(-alpha)) * Float64(Float64(-2.0 - beta) / alpha))) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(alpha - beta) / t_0)) / 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_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.95], 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[(1.0 - N[(N[(alpha - beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.94999999999999996

   1. Initial program 58.6

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

   2. Simplified 58.6

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

      [Start] 58.6	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 58.6	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around -inf 4.0

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

   4. Simplified 4.0

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

      [Start] 4.0	\[ \frac{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + -1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      distribute-lft-out [=>] 4.0	\[ \frac{\color{blue}{-1 \cdot \left(\frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}}{2} \]
      mul-1-neg [=>] 4.0	\[ \frac{-1 \cdot \left(\frac{\color{blue}{\left(-\beta\right)} - \left(\beta + 2\right)}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 4.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \color{blue}{\left(2 + \beta\right)}}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 4.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\color{blue}{\left(2 + \beta\right)}}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 4.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \color{blue}{\left(2 + \beta\right)}}{{\alpha}^{2}}\right)}{2} \]
      unpow2 [=>] 4.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \left(2 + \beta\right)}{\color{blue}{\alpha \cdot \alpha}}\right)}{2} \]

   5. Applied egg-rr 4.0

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

   6. Simplified 0.5

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

      [Start] 4.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \left(-\left(2 + \beta\right) \cdot \left(2 + \left(\beta + \beta\right)\right)\right) \cdot \frac{1}{\alpha \cdot \left(-\alpha\right)}\right)}{2} \]
      associate-*r/ [=>] 4.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \color{blue}{\frac{\left(-\left(2 + \beta\right) \cdot \left(2 + \left(\beta + \beta\right)\right)\right) \cdot 1}{\alpha \cdot \left(-\alpha\right)}}\right)}{2} \]
      *-rgt-identity [=>] 4.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\color{blue}{-\left(2 + \beta\right) \cdot \left(2 + \left(\beta + \beta\right)\right)}}{\alpha \cdot \left(-\alpha\right)}\right)}{2} \]
      distribute-rgt-neg-in [=>] 4.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\color{blue}{\left(2 + \beta\right) \cdot \left(-\left(2 + \left(\beta + \beta\right)\right)\right)}}{\alpha \cdot \left(-\alpha\right)}\right)}{2} \]
      times-frac [=>] 0.5	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \color{blue}{\frac{2 + \beta}{\alpha} \cdot \frac{-\left(2 + \left(\beta + \beta\right)\right)}{-\alpha}}\right)}{2} \]
      +-commutative [=>] 0.5	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{2 + \beta}{\alpha} \cdot \frac{-\color{blue}{\left(\left(\beta + \beta\right) + 2\right)}}{-\alpha}\right)}{2} \]
      distribute-neg-in [=>] 0.5	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{2 + \beta}{\alpha} \cdot \frac{\color{blue}{\left(-\left(\beta + \beta\right)\right) + \left(-2\right)}}{-\alpha}\right)}{2} \]
      count-2 [=>] 0.5	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{2 + \beta}{\alpha} \cdot \frac{\left(-\color{blue}{2 \cdot \beta}\right) + \left(-2\right)}{-\alpha}\right)}{2} \]
      distribute-lft-neg-in [=>] 0.5	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{2 + \beta}{\alpha} \cdot \frac{\color{blue}{\left(-2\right) \cdot \beta} + \left(-2\right)}{-\alpha}\right)}{2} \]
      metadata-eval [=>] 0.5	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{2 + \beta}{\alpha} \cdot \frac{\color{blue}{-2} \cdot \beta + \left(-2\right)}{-\alpha}\right)}{2} \]
      sub-neg [<=] 0.5	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{2 + \beta}{\alpha} \cdot \frac{\color{blue}{-2 \cdot \beta - 2}}{-\alpha}\right)}{2} \]
      fma-neg [=>] 0.5	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{2 + \beta}{\alpha} \cdot \frac{\color{blue}{\mathsf{fma}\left(-2, \beta, -2\right)}}{-\alpha}\right)}{2} \]
      metadata-eval [=>] 0.5	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{2 + \beta}{\alpha} \cdot \frac{\mathsf{fma}\left(-2, \beta, \color{blue}{-2}\right)}{-\alpha}\right)}{2} \]

   if -0.94999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{\frac{\beta - \left(-2 - \beta\right)}{\alpha} + \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{-\alpha} \cdot \frac{-2 - \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	1476

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.95:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \]

Alternative 2
Error	4.2
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 22:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Error	18.7
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq 3 \cdot 10^{-158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 3.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	18.5
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq 3 \cdot 10^{-158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 6 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 5
Error	8.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 22:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{+272}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \end{array} \]

Alternative 6
Error	4.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 22:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	19.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{-158}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 6 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	17.9
Cost	196

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

Alternative 9
Error	31.9
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023237 
(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))