Octave 3.8, jcobi/1

Average Error: 16.4 → 0.2

Time: 13.4s

Precision: binary64

Cost: 8260

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\beta}{\beta + \left(\alpha + 2\right)}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999995:\\ \;\;\;\;\frac{t_0 + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - \mathsf{fma}\left(\alpha, \frac{1}{\alpha + \left(\beta + 2\right)}, -1\right)}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ beta (+ beta (+ alpha 2.0)))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999999995)
     (/ (+ t_0 (/ (+ beta 2.0) alpha)) 2.0)
     (/ (- t_0 (fma alpha (/ 1.0 (+ alpha (+ beta 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 t_0 = beta / (beta + (alpha + 2.0));
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999995) {
		tmp = (t_0 + ((beta + 2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_0 - fma(alpha, (1.0 / (alpha + (beta + 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)
	t_0 = Float64(beta / Float64(beta + Float64(alpha + 2.0)))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999999995)
		tmp = Float64(Float64(t_0 + Float64(Float64(beta + 2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(t_0 - fma(alpha, Float64(1.0 / Float64(alpha + Float64(beta + 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_] := Block[{t$95$0 = N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999999995], N[(N[(t$95$0 + N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 - N[(alpha * N[(1.0 / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999999500000003

   1. Initial program 59.9

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

   2. Simplified 59.9

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

      [Start] 59.9	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 59.9	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 58.0

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

   4. Applied egg-rr 60.7

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

   5. Taylor expanded in alpha around inf 0.4

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

   6. Simplified 0.4

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

      [Start] 0.4	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - -1 \cdot \frac{\beta + 2}{\alpha}}{2} \]
      associate-*r/ [=>] 0.4	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\frac{-1 \cdot \left(\beta + 2\right)}{\alpha}}}{2} \]
      distribute-lft-in [=>] 0.4	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\color{blue}{-1 \cdot \beta + -1 \cdot 2}}{\alpha}}{2} \]
      metadata-eval [=>] 0.4	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-1 \cdot \beta + \color{blue}{-2}}{\alpha}}{2} \]
      +-commutative [=>] 0.4	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\color{blue}{-2 + -1 \cdot \beta}}{\alpha}}{2} \]
      mul-1-neg [=>] 0.4	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{-2 + \color{blue}{\left(-\beta\right)}}{\alpha}}{2} \]
      sub-neg [<=] 0.4	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\color{blue}{-2 - \beta}}{\alpha}}{2} \]

   if -0.99999999500000003 < (/.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

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

   4. Applied egg-rr 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999995:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \mathsf{fma}\left(\alpha, \frac{1}{\alpha + \left(\beta + 2\right)}, -1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	7876

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999995:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{t_0}, 1\right)}{2}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1860

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999995:\\ \;\;\;\;\frac{t_1 + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t_1 + 1\right) - \frac{\alpha}{t_0}}{2}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1604

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

Alternative 4
Error	0.2
Cost	1476

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

Alternative 5
Error	4.2
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 720000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	19.5
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -8 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -9 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	19.3
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -8 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -9 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 8
Error	4.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 350000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	8.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6200:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.05 \cdot 10^{+271}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Error	4.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 330000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 11
Error	19.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq -9.5 \cdot 10^{-133}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -8.3 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	18.6
Cost	196

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

Alternative 13
Error	32.3
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023045 
(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))