Octave 3.8, jcobi/1

Average Error: 16.4 → 0.2

Time: 11.6s

Precision: binary64

Cost: 41924

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-1, \mathsf{fma}\left(2, \beta, -2\right), -4\right)\\ t_1 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_1} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{t_0}{\alpha}, \mathsf{fma}\left(-1, \frac{\beta}{\frac{\alpha}{\frac{\beta + -2}{\alpha}}}, \frac{4}{\alpha \cdot \alpha}\right)\right) + \left(4 \cdot \frac{t_0}{\alpha \cdot \alpha} - \frac{\beta}{\frac{\alpha \cdot \alpha}{\beta}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_1}}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (fma -1.0 (fma 2.0 beta -2.0) -4.0)) (t_1 (+ (+ beta alpha) 2.0)))
   (if (<= (/ (- beta alpha) t_1) -0.5)
     (/
      (+
       (fma
        -1.0
        (/ t_0 alpha)
        (fma
         -1.0
         (/ beta (/ alpha (/ (+ beta -2.0) alpha)))
         (/ 4.0 (* alpha alpha))))
       (- (* 4.0 (/ t_0 (* alpha alpha))) (/ beta (/ (* alpha alpha) beta))))
      2.0)
     (/ (- 1.0 (/ (- alpha beta) 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 = fma(-1.0, fma(2.0, beta, -2.0), -4.0);
	double t_1 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_1) <= -0.5) {
		tmp = (fma(-1.0, (t_0 / alpha), fma(-1.0, (beta / (alpha / ((beta + -2.0) / alpha))), (4.0 / (alpha * alpha)))) + ((4.0 * (t_0 / (alpha * alpha))) - (beta / ((alpha * alpha) / beta)))) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / 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 = fma(-1.0, fma(2.0, beta, -2.0), -4.0)
	t_1 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_1) <= -0.5)
		tmp = Float64(Float64(fma(-1.0, Float64(t_0 / alpha), fma(-1.0, Float64(beta / Float64(alpha / Float64(Float64(beta + -2.0) / alpha))), Float64(4.0 / Float64(alpha * alpha)))) + Float64(Float64(4.0 * Float64(t_0 / Float64(alpha * alpha))) - Float64(beta / Float64(Float64(alpha * alpha) / beta)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(alpha - beta) / 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[(-1.0 * N[(2.0 * beta + -2.0), $MachinePrecision] + -4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$1), $MachinePrecision], -0.5], N[(N[(N[(-1.0 * N[(t$95$0 / alpha), $MachinePrecision] + N[(-1.0 * N[(beta / N[(alpha / N[(N[(beta + -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(t$95$0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(beta / N[(N[(alpha * alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(N[(alpha - beta), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.5

   1. Initial program 58.3

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

   2. Simplified 58.3

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

      [Start] 58.3	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 58.3	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 59.5

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

   4. Taylor expanded in alpha around -inf 3.8

      \[\leadsto \frac{\color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(2 \cdot \beta - 2\right) - 4}{\alpha} + \left(4 \cdot \frac{1}{{\alpha}^{2}} + -1 \cdot \frac{\beta \cdot \left(\beta - 2\right)}{{\alpha}^{2}}\right)\right) - \left(\frac{{\beta}^{2}}{{\alpha}^{2}} + -4 \cdot \frac{-1 \cdot \left(2 \cdot \beta - 2\right) - 4}{{\alpha}^{2}}\right)}}{2} \]

   5. Simplified 0.8

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(2, \beta, -2\right), -4\right)}{\alpha}, \mathsf{fma}\left(-1, \frac{\beta}{\frac{\alpha}{\frac{\beta + -2}{\alpha}}}, \frac{4}{\alpha \cdot \alpha}\right)\right) - \left(\frac{\beta}{\frac{\alpha \cdot \alpha}{\beta}} + -4 \cdot \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(2, \beta, -2\right), -4\right)}{\alpha \cdot \alpha}\right)}}{2} \]
      Proof

      [Start] 3.8	\[ \frac{\left(-1 \cdot \frac{-1 \cdot \left(2 \cdot \beta - 2\right) - 4}{\alpha} + \left(4 \cdot \frac{1}{{\alpha}^{2}} + -1 \cdot \frac{\beta \cdot \left(\beta - 2\right)}{{\alpha}^{2}}\right)\right) - \left(\frac{{\beta}^{2}}{{\alpha}^{2}} + -4 \cdot \frac{-1 \cdot \left(2 \cdot \beta - 2\right) - 4}{{\alpha}^{2}}\right)}{2} \]

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.0

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(2, \beta, -2\right), -4\right)}{\alpha}, \mathsf{fma}\left(-1, \frac{\beta}{\frac{\alpha}{\frac{\beta + -2}{\alpha}}}, \frac{4}{\alpha \cdot \alpha}\right)\right) + \left(4 \cdot \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(2, \beta, -2\right), -4\right)}{\alpha \cdot \alpha} - \frac{\beta}{\frac{\alpha \cdot \alpha}{\beta}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	23108

\[\begin{array}{l} t_0 := \beta + \left(\beta + -2\right)\\ t_1 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_1} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{-1}{\frac{\alpha}{t_0}}, \mathsf{fma}\left(-1, \frac{\beta}{\frac{\alpha}{\frac{\beta + -2}{\alpha}}}, \frac{4}{\alpha \cdot \alpha}\right) + \frac{4}{\alpha}\right) - \mathsf{fma}\left(4, \frac{4 + t_0}{\alpha \cdot \alpha}, \frac{\beta}{\frac{\alpha \cdot \alpha}{\beta}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_1}}{2}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1476

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.99999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \]

Alternative 3
Error	21.2
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{if}\;\alpha \leq 7 \cdot 10^{-217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.7 \cdot 10^{-182}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.96:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 4
Error	8.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 64000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 5
Error	4.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 260000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	21.5
Cost	588

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.6 \cdot 10^{-217}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.7 \cdot 10^{-182}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.9:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 7
Error	18.3
Cost	580

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

Alternative 8
Error	20.9
Cost	324

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 9
Error	33.2
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023011 
(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))