Octave 3.8, jcobi/1

Average Error: 16.6 → 0.2

Time: 13.5s

Precision: binary64

Cost: 8260

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\beta}{\beta + \left(\alpha + 2\right)}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{t_0 + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - \mathsf{fma}\left(\alpha, \frac{1}{\alpha + \left(\beta + 2\right)}, -1\right)}{2}\\ \end{array} \]

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)) -0.99999)
     (/ (+ 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)) <= -0.99999) {
		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)) <= -0.99999)
		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], -0.99999], 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)) < -0.999990000000000046

   1. Initial program 59.7

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

   2. Simplified 59.7

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

      [Start] 59.7	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 59.7	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 57.8

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

   4. Taylor expanded in alpha around inf 0.6

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

   5. Simplified 0.6

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

      [Start] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - -1 \cdot \frac{\beta + 2}{\alpha}}{2} \]
      mul-1-neg [=>] 0.6	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\left(-\frac{\beta + 2}{\alpha}\right)}}{2} \]

   if -0.999990000000000046 < (/.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

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

   4. Applied egg-rr 0.0

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \mathsf{fma}\left(\alpha, \frac{1}{\alpha + \left(\beta + 2\right)}, -1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	7876

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{t_0}, 1\right)}{2}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1604

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{t_0}}{2}\\ \end{array} \]

Alternative 4
Error	0.2
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 5
Error	17.0
Cost	972

\[\begin{array}{l} t_0 := \frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{if}\;\alpha \leq -3.4 \cdot 10^{-260}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq -2.05 \cdot 10^{-305}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} \cdot \left(\beta + 1\right)}{2}\\ \end{array} \]

Alternative 6
Error	19.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{-215}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 54:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 7
Error	4.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} \cdot \left(\beta + 1\right)}{2}\\ \end{array} \]

Alternative 8
Error	4.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	19.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{-215}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{-182}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 54:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	18.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 54:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	32.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023067 
(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))