Octave 3.8, jcobi/1

Average Error: 16.2 → 1.0

Time: 13.7s

Precision: binary64

Cost: 8580

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.98:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(\frac{\beta + 2}{\alpha} - \frac{{\left(\beta + 2\right)}^{2}}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{t_0}, 1\right)}{2}\\ \end{array} \]

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 (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.98)
     (/
      (+
       (/ beta t_0)
       (- (/ (+ beta 2.0) alpha) (/ (pow (+ beta 2.0) 2.0) (* alpha alpha))))
      2.0)
     (/ (fma (- beta alpha) (/ 1.0 t_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 + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.98) {
		tmp = ((beta / t_0) + (((beta + 2.0) / alpha) - (pow((beta + 2.0), 2.0) / (alpha * alpha)))) / 2.0;
	} else {
		tmp = fma((beta - alpha), (1.0 / t_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(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.98)
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(Float64(Float64(beta + 2.0) / alpha) - Float64((Float64(beta + 2.0) ^ 2.0) / Float64(alpha * alpha)))) / 2.0);
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(1.0 / t_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[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.98], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[Power[N[(beta + 2.0), $MachinePrecision], 2.0], $MachinePrecision] / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.97999999999999998

   1. Initial program 58.6

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

   2. Simplified 58.6

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]
      Proof
      (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 beta alpha) 2)) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha beta)) 2)) 1) 2): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 56.8

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

   4. Taylor expanded in alpha around inf 3.5

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

   5. Simplified 3.5

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\left(\frac{{\left(\beta + 2\right)}^{2}}{\alpha \cdot \alpha} - \frac{\beta + 2}{\alpha}\right)}}{2} \]
      Proof
      (/.f64 (-.f64 (/.f64 beta (+.f64 beta (+.f64 alpha 2))) (-.f64 (/.f64 (pow.f64 (+.f64 beta 2) 2) (*.f64 alpha alpha)) (/.f64 (+.f64 beta 2) alpha))) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 beta (+.f64 beta (+.f64 alpha 2))) (-.f64 (/.f64 (pow.f64 (+.f64 beta 2) 2) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2))) (/.f64 (+.f64 beta 2) alpha))) 2): 5 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 beta (+.f64 beta (+.f64 alpha 2))) (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2)) (neg.f64 (/.f64 (+.f64 beta 2) alpha))))) 2): 5 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 beta (+.f64 beta (+.f64 alpha 2))) (+.f64 (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha))))) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 beta (+.f64 beta (+.f64 alpha 2))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))))) 2): 0 points increase in error, 0 points decrease in error

   if -0.97999999999999998 < (/.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
      (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 beta alpha) 2)) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha beta)) 2)) 1) 2): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.0

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.98:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(\frac{\beta + 2}{\alpha} - \frac{{\left(\beta + 2\right)}^{2}}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1860

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{t_1 + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	1604

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.999998:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.999998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 4
Error	20.0
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -1.55 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -9.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	7.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1200000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	4.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 200000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	20.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq -1.55 \cdot 10^{-47}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -9.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	18.5
Cost	196

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

Alternative 9
Error	32.4
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022340 
(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))