Octave 3.8, jcobi/1

Average Error: 16.1 → 0.1

Time: 10.2s

Precision: binary64

Cost: 7876

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\beta - \left(-2 - \beta\right)}{\alpha} + \frac{2 + \left(\beta + \beta\right)}{\alpha} \cdot \frac{-2 - \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \]

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)) -0.999996)
   (/
    (+
     (/ (- beta (- -2.0 beta)) alpha)
     (* (/ (+ 2.0 (+ beta beta)) alpha) (/ (- -2.0 beta) alpha)))
    2.0)
   (/ (fma (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))) 1.0) 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 tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = (((beta - (-2.0 - beta)) / alpha) + (((2.0 + (beta + beta)) / alpha) * ((-2.0 - beta) / alpha))) / 2.0;
	} else {
		tmp = fma((beta - alpha), (1.0 / (beta + (alpha + 2.0))), 1.0) / 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)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999996)
		tmp = Float64(Float64(Float64(Float64(beta - Float64(-2.0 - beta)) / alpha) + Float64(Float64(Float64(2.0 + Float64(beta + beta)) / alpha) * Float64(Float64(-2.0 - beta) / alpha))) / 2.0);
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), 1.0) / 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_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999996], N[(N[(N[(N[(beta - N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(N[(2.0 + N[(beta + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(-2.0 - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999995999999999996

   1. Initial program 59.4

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

   2. Simplified 59.4

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

      [Start] 59.4	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 59.4	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around -inf 3.0

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

   4. Simplified 3.0

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

      [Start] 3.0	\[ \frac{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + -1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      distribute-lft-out [=>] 3.0	\[ \frac{\color{blue}{-1 \cdot \left(\frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}}{2} \]
      mul-1-neg [=>] 3.0	\[ \frac{-1 \cdot \left(\frac{\color{blue}{\left(-\beta\right)} - \left(\beta + 2\right)}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \color{blue}{\left(2 + \beta\right)}}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\color{blue}{\left(2 + \beta\right)}}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \color{blue}{\left(2 + \beta\right)}}{{\alpha}^{2}}\right)}{2} \]
      unpow2 [=>] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \left(2 + \beta\right)}{\color{blue}{\alpha \cdot \alpha}}\right)}{2} \]

   5. Taylor expanded in alpha around 0 3.0

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

   6. Simplified 0.1

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

      [Start] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [<=] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\color{blue}{\left(2 + \beta\right)}}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [<=] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \color{blue}{\left(2 + \beta\right)}}{{\alpha}^{2}}\right)}{2} \]
      unpow2 [=>] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\color{blue}{\left(2 + \beta\right) \cdot \left(2 + \beta\right)} + \beta \cdot \left(2 + \beta\right)}{{\alpha}^{2}}\right)}{2} \]
      distribute-rgt-in [<=] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\color{blue}{\left(2 + \beta\right) \cdot \left(\left(2 + \beta\right) + \beta\right)}}{{\alpha}^{2}}\right)}{2} \]
      associate-+r+ [<=] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\left(2 + \beta\right) \cdot \color{blue}{\left(2 + \left(\beta + \beta\right)\right)}}{{\alpha}^{2}}\right)}{2} \]
      *-commutative [=>] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\color{blue}{\left(2 + \left(\beta + \beta\right)\right) \cdot \left(2 + \beta\right)}}{{\alpha}^{2}}\right)}{2} \]
      unpow2 [=>] 3.0	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{\left(2 + \left(\beta + \beta\right)\right) \cdot \left(2 + \beta\right)}{\color{blue}{\alpha \cdot \alpha}}\right)}{2} \]
      times-frac [=>] 0.1	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \color{blue}{\frac{2 + \left(\beta + \beta\right)}{\alpha} \cdot \frac{2 + \beta}{\alpha}}\right)}{2} \]

   if -0.999995999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 0.1	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\beta - \left(-2 - \beta\right)}{\alpha} + \frac{2 + \left(\beta + \beta\right)}{\alpha} \cdot \frac{-2 - \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	2116

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta - \left(-2 - \beta\right)}{\alpha} + \frac{2 + \left(\beta + \beta\right)}{\alpha} \cdot \frac{-2 - \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1860

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.9999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha} - \frac{\beta}{\alpha} \cdot \frac{\beta}{\frac{\alpha}{2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1796

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.9999995:\\ \;\;\;\;\frac{\frac{\left(\left(-2 - \beta\right) - \beta\right) - \frac{\beta}{\frac{\alpha}{\beta}} \cdot -2}{-\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \]

Alternative 4
Error	0.2
Cost	1476

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.9999995:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \]

Alternative 5
Error	18.2
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq 5.5 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2.25 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	18.0
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq 5.5 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2.25 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 7
Error	4.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right)}{2}\\ \end{array} \]

Alternative 8
Error	7.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 76000000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	4.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 70000000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 - \beta \cdot -2}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Error	18.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{-190}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.25 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	18.2
Cost	196

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

Alternative 12
Error	32.4
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023060 
(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))