Octave 3.8, jcobi/1

Average Error: 15.8 → 0.3

Time: 29.6s

Precision: binary64

Cost: 44548

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

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \frac{t_0}{\alpha}\\ t_2 := \frac{\alpha}{t_0}\\ t_3 := {t_2}^{3}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -1:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \frac{1}{t_0}, \frac{1 - {t_3}^{3}}{\left(1 + \left(t_2 + {t_2}^{2}\right)\right) \cdot \left(1 + \left(t_3 + t_3 \cdot \frac{1}{t_1 \cdot \left(t_1 \cdot t_1\right)}\right)\right)}\right)}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0)))
        (t_1 (/ t_0 alpha))
        (t_2 (/ alpha t_0))
        (t_3 (pow t_2 3.0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -1.0)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/
      (fma
       beta
       (/ 1.0 t_0)
       (/
        (- 1.0 (pow t_3 3.0))
        (*
         (+ 1.0 (+ t_2 (pow t_2 2.0)))
         (+ 1.0 (+ t_3 (* t_3 (/ 1.0 (* t_1 (* t_1 t_1)))))))))
      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 = alpha + (beta + 2.0);
	double t_1 = t_0 / alpha;
	double t_2 = alpha / t_0;
	double t_3 = pow(t_2, 3.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = fma(beta, (1.0 / t_0), ((1.0 - pow(t_3, 3.0)) / ((1.0 + (t_2 + pow(t_2, 2.0))) * (1.0 + (t_3 + (t_3 * (1.0 / (t_1 * (t_1 * t_1))))))))) / 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(alpha + Float64(beta + 2.0))
	t_1 = Float64(t_0 / alpha)
	t_2 = Float64(alpha / t_0)
	t_3 = t_2 ^ 3.0
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -1.0)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(fma(beta, Float64(1.0 / t_0), Float64(Float64(1.0 - (t_3 ^ 3.0)) / Float64(Float64(1.0 + Float64(t_2 + (t_2 ^ 2.0))) * Float64(1.0 + Float64(t_3 + Float64(t_3 * Float64(1.0 / Float64(t_1 * Float64(t_1 * t_1))))))))) / 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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / alpha), $MachinePrecision]}, Block[{t$95$2 = N[(alpha / t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 3.0], $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta * N[(1.0 / t$95$0), $MachinePrecision] + N[(N[(1.0 - N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(t$95$2 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$3 + N[(t$95$3 * N[(1.0 / N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -1

   1. Initial program 60.6

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

   2. Simplified 60.6

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

      [Start] 60.6	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 60.6	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 0.0

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.5

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

   2. Simplified 0.5

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

      [Start] 0.5	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 0.5	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 0.4

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

   4. Applied egg-rr 0.5

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

   5. Applied egg-rr 0.5

      \[\leadsto \frac{\mathsf{fma}\left(\beta, \frac{1}{\alpha + \left(\beta + 2\right)}, \color{blue}{\frac{1 - {\left({\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right)}^{3}}{\left(1 + \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2}\right)\right) \cdot \left(1 + \left({\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} + {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3} \cdot {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right)\right)}}\right)}{2} \]

   6. Applied egg-rr 0.5

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

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	16452

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \frac{\alpha}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -1:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \frac{1}{t_0}, \frac{1 + {\left(\frac{-\alpha}{t_0}\right)}^{3}}{1 + \left(t_1 + t_1 \cdot t_1\right)}\right)}{2}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	8260

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

Alternative 3
Error	0.3
Cost	1860

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

Alternative 4
Error	0.4
Cost	1732

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

Alternative 5
Error	0.4
Cost	1604

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

Alternative 6
Error	0.3
Cost	1476

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

Alternative 7
Error	17.0
Cost	972

\[\begin{array}{l} \mathbf{if}\;\alpha \leq -9.4 \cdot 10^{-225}:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{elif}\;\alpha \leq -6 \cdot 10^{-254}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.35 \cdot 10^{+19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} \cdot \left(\beta + 1\right)}{2}\\ \end{array} \]

Alternative 8
Error	18.5
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -1.5 \cdot 10^{-210}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -2.45 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	18.5
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -1.5 \cdot 10^{-210}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -5.2 \cdot 10^{-258}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 10
Error	4.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.35 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \beta \cdot \frac{1}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 11
Error	4.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} \cdot \left(\beta + 1\right)}{2}\\ \end{array} \]

Alternative 12
Error	4.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.35 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 13
Error	18.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq -1.5 \cdot 10^{-210}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -2.6 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	18.0
Cost	196

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

Alternative 15
Error	32.0
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023018 
(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))