Octave 3.8, jcobi/1

Average Error: 16.0 → 0.4

Time: 8.9s

Precision: binary64

Cost: 1604

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

Math

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

↓

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

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -1.0)
   (/ (+ (* 2.0 (/ beta alpha)) (* 2.0 (/ 1.0 alpha))) 2.0)
   (/ (+ 1.0 (/ 1.0 (/ (+ beta (+ alpha 2.0)) (- beta alpha)))) 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 tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / ((beta + (alpha + 2.0)) / (beta - alpha)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

↓

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-1.0d0)) then
        tmp = ((2.0d0 * (beta / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    else
        tmp = (1.0d0 + (1.0d0 / ((beta + (alpha + 2.0d0)) / (beta - alpha)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

↓

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / ((beta + (alpha + 2.0)) / (beta - alpha)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0

↓

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0:
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	else:
		tmp = (1.0 + (1.0 / ((beta + (alpha + 2.0)) / (beta - alpha)))) / 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)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -1.0)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(Float64(beta + Float64(alpha + 2.0)) / Float64(beta - alpha)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
end

↓

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0)
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	else
		tmp = (1.0 + (1.0 / ((beta + (alpha + 2.0)) / (beta - alpha)))) / 2.0;
	end
	tmp_2 = 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_] := 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 / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(1.0 / N[(N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $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. Taylor expanded in alpha around -inf 0.0

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

   3. Simplified 0.0

      \[\leadsto \frac{\color{blue}{\frac{\left(\beta + \beta\right) + 2}{\alpha}}}{2} \]
      Proof
      (/.f64 (+.f64 (+.f64 beta beta) 2) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (+.f64 beta (Rewrite<= *-lft-identity_binary64 (*.f64 1 beta))) 2) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (+.f64 beta (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) beta)) 2) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 beta (*.f64 -1 beta))) 2) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (-.f64 beta (*.f64 -1 beta)) (Rewrite<= metadata-eval (neg.f64 -2))) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (-.f64 beta (*.f64 -1 beta)) (neg.f64 (Rewrite<= metadata-eval (*.f64 2 -1)))) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (-.f64 beta (*.f64 -1 beta)) (*.f64 2 -1))) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 beta (+.f64 (*.f64 -1 beta) (*.f64 2 -1)))) alpha): 0 points increase in error, 1 points decrease in error
      (/.f64 (-.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 beta))) (+.f64 (*.f64 -1 beta) (*.f64 2 -1))) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 beta))) (+.f64 (*.f64 -1 beta) (*.f64 2 -1))) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (neg.f64 (*.f64 -1 beta)) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 beta -1)) (*.f64 2 -1))) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (neg.f64 (*.f64 -1 beta)) (Rewrite<= distribute-rgt-in_binary64 (*.f64 -1 (+.f64 beta 2)))) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (*.f64 -1 beta)) (neg.f64 (*.f64 -1 (+.f64 beta 2))))) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (*.f64 -1 beta) (*.f64 -1 (+.f64 beta 2))))) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (+.f64 (*.f64 -1 beta) (Rewrite=> mul-1-neg_binary64 (neg.f64 (+.f64 beta 2))))) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)))) alpha): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)))) alpha): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in beta around 0 0.0

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}}{2} \]

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

   1. Initial program 0.6

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

   2. Applied egg-rr 0.6

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

   3. Applied egg-rr 0.6

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

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	1476

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

Alternative 2
Error	4.1
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 97828207504307020:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\beta + 2}{\beta}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Error	19.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.380735074389988 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 2.3946928433048396 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 16720275.148137575:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 4
Error	4.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 97828207504307020:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\beta + 2}{\beta}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	17.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.784056229460161 \cdot 10^{-15}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 6
Error	15.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.040789765303754:\\ \;\;\;\;\frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	4.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 97828207504307020:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	20.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.380735074389988 \cdot 10^{-228}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.3946928433048396 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 16720275.148137575:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	20.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.380735074389988 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 2.3946928433048396 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 16720275.148137575:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	18.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 16720275.148137575:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	31.9
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022297 
(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))