Octave 3.8, jcobi/1

Average Error: 16.2 → 0.4

Time: 9.7s

Precision: binary64

Cost: 1476

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.6:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.6)
     (/ (+ (/ beta alpha) (/ (+ beta 2.0) alpha)) 2.0)
     (/ (+ 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) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.6) {
		tmp = ((beta / alpha) + ((beta + 2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 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) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.6d0)) then
        tmp = ((beta / alpha) + ((beta + 2.0d0) / alpha)) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 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 t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.6) {
		tmp = ((beta / alpha) + ((beta + 2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

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

↓

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.6:
		tmp = ((beta / alpha) + ((beta + 2.0) / alpha)) / 2.0
	else:
		tmp = (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(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.6)
		tmp = Float64(Float64(Float64(beta / alpha) + Float64(Float64(beta + 2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 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)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.6)
		tmp = ((beta / alpha) + ((beta + 2.0) / alpha)) / 2.0;
	else
		tmp = (t_0 + 1.0) / 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_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.6], N[(N[(N[(beta / alpha), $MachinePrecision] + N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.599999999999999978

   1. Initial program 58.5

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

   2. Simplified 58.5

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]
      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, 2 points decrease in error

   3. Taylor expanded in alpha around -inf 1.6

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

   4. Simplified 1.6

      \[\leadsto \frac{\color{blue}{-\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha}}}{2} \]
      Proof
      (/.f64 (neg.f64 (/.f64 (-.f64 (neg.f64 beta) (+.f64 2 beta)) alpha)) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (/.f64 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 beta)) (+.f64 2 beta)) alpha)) 2): 4 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (/.f64 (-.f64 (*.f64 -1 beta) (Rewrite<= +-commutative_binary64 (+.f64 beta 2))) alpha)) 2): 0 points increase in error, 4 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha))) 2): 4 points increase in error, 0 points decrease in error

   5. Applied egg-rr 1.6

      \[\leadsto \frac{-\color{blue}{\left(\frac{0}{\alpha} - \left(\frac{\beta}{\alpha} + \frac{\beta + 2}{\alpha}\right)\right)}}{2} \]

   6. Simplified 1.6

      \[\leadsto \frac{-\color{blue}{\left(\frac{-2 - \beta}{\alpha} - \frac{\beta}{\alpha}\right)}}{2} \]
      Proof
      (/.f64 (neg.f64 (-.f64 (/.f64 (-.f64 -2 beta) alpha) (/.f64 beta alpha))) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (/.f64 (-.f64 (Rewrite<= metadata-eval (-.f64 0 2)) beta) alpha) (/.f64 beta alpha))) 2): 10 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 0 (+.f64 2 beta))) alpha) (/.f64 beta alpha))) 2): 0 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (-.f64 (/.f64 (-.f64 0 (Rewrite<= +-commutative_binary64 (+.f64 beta 2))) alpha) (/.f64 beta alpha))) 2): 10 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (+.f64 beta 2))) alpha) (/.f64 beta alpha))) 2): 0 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (-.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (+.f64 beta 2) alpha))) (/.f64 beta alpha))) 2): 0 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (-.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (/.f64 (+.f64 beta 2) alpha))) (/.f64 beta alpha))) 2): 10 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= associate--r+_binary64 (-.f64 0 (+.f64 (/.f64 (+.f64 beta 2) alpha) (/.f64 beta alpha))))) 2): 0 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (-.f64 (Rewrite<= div0_binary64 (/.f64 0 alpha)) (+.f64 (/.f64 (+.f64 beta 2) alpha) (/.f64 beta alpha)))) 2): 10 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (/.f64 0 alpha) (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 beta alpha) (/.f64 (+.f64 beta 2) alpha))))) 2): 0 points increase in error, 10 points decrease in error

   if -0.599999999999999978 < (/.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 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.6:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.3:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 2
Error	7.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.3:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Error	4.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{1 + \alpha \cdot 0.5}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \]

Alternative 4
Error	4.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.3:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	18.1
Cost	580

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

Alternative 6
Error	18.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.95:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]

Alternative 7
Error	18.5
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 14500:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	32.4
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022329 
(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))