Octave 3.8, jcobi/1

Average Error: 16.0 → 0.4

Time: 8.3s

Precision: binary64

Cost: 14724

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

Math

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

↓

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

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (if (<= (/ (- beta alpha) t_0) -1.0)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/
      (fma
       beta
       (/ 1.0 t_0)
       (- (fma (/ 1.0 (+ alpha (+ beta 2.0))) alpha -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 = 2.0 + (beta + alpha);
	double tmp;
	if (((beta - alpha) / t_0) <= -1.0) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = fma(beta, (1.0 / t_0), -fma((1.0 / (alpha + (beta + 2.0))), alpha, -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(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_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(-fma(Float64(1.0 / Float64(alpha + Float64(beta + 2.0))), alpha, -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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * alpha + -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)) < -1

   1. Initial program 60.6

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

   2. Simplified 60.6

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)} + 1}{2}} \]
      Proof
      (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 alpha (+.f64 beta 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) 2))) 1) 2): 0 points increase in error, 0 points decrease in error

   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

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)} + 1}{2}} \]
      Proof
      (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 alpha (+.f64 beta 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) 2))) 1) 2): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.5

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

   4. Applied egg-rr 0.5

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

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	1476

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

Alternative 2
Error	18.8
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq -5.8 \cdot 10^{-211}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -3.6 \cdot 10^{-227}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 300000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \alpha \cdot \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 3
Error	18.6
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq -1.88 \cdot 10^{-208}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -9 \cdot 10^{-226}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 300000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 4
Error	8.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	4.9
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.6 \cdot 10^{+45}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	18.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq -6.8 \cdot 10^{-211}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -9 \cdot 10^{-226}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 300000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	18.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 300000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	31.9
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022318 
(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))