Octave 3.8, jcobi/1

Average Error: 15.8 → 0.2

Time: 5.6s

Precision: binary64

Cost: 1732

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

Math

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

↓

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

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 -0.5)
     (/
      (* (/ (+ -2.0 (* beta -2.0)) alpha) (- -1.0 (/ (- -2.0 beta) 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 <= -0.5) {
		tmp = (((-2.0 + (beta * -2.0)) / alpha) * (-1.0 - ((-2.0 - beta) / 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 <= (-0.5d0)) then
        tmp = ((((-2.0d0) + (beta * (-2.0d0))) / alpha) * ((-1.0d0) - (((-2.0d0) - beta) / 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 <= -0.5) {
		tmp = (((-2.0 + (beta * -2.0)) / alpha) * (-1.0 - ((-2.0 - beta) / 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 <= -0.5:
		tmp = (((-2.0 + (beta * -2.0)) / alpha) * (-1.0 - ((-2.0 - beta) / 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 <= -0.5)
		tmp = Float64(Float64(Float64(Float64(-2.0 + Float64(beta * -2.0)) / alpha) * Float64(-1.0 - Float64(Float64(-2.0 - beta) / 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 <= -0.5)
		tmp = (((-2.0 + (beta * -2.0)) / alpha) * (-1.0 - ((-2.0 - beta) / 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, -0.5], N[(N[(N[(N[(-2.0 + N[(beta * -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * N[(-1.0 - N[(N[(-2.0 - beta), $MachinePrecision] / alpha), $MachinePrecision]), $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)) < -0.5

   1. Initial program 58.2

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

   2. Taylor expanded in alpha around -inf 3.8

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

   3. Simplified 0.8

      \[\leadsto \frac{\color{blue}{\frac{\beta \cdot -2 + -2}{\alpha} \cdot \left(-1 - \frac{-2 - \beta}{\alpha}\right)}}{2} \]
      Proof
      (*.f64 (/.f64 (+.f64 (*.f64 beta -2) -2) alpha) (-.f64 -1 (/.f64 (-.f64 -2 beta) alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (*.f64 beta (Rewrite<= metadata-eval (+.f64 -1 -1))) -2) alpha) (-.f64 -1 (/.f64 (-.f64 -2 beta) alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -1 beta) (*.f64 -1 beta))) -2) alpha) (-.f64 -1 (/.f64 (-.f64 -2 beta) alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (+.f64 (*.f64 -1 beta) (Rewrite=> mul-1-neg_binary64 (neg.f64 beta))) -2) alpha) (-.f64 -1 (/.f64 (-.f64 -2 beta) alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 -1 beta) beta)) -2) alpha) (-.f64 -1 (/.f64 (-.f64 -2 beta) alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (-.f64 (*.f64 -1 beta) beta) (Rewrite<= metadata-eval (neg.f64 2))) alpha) (-.f64 -1 (/.f64 (-.f64 -2 beta) alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (-.f64 (*.f64 -1 beta) beta) 2)) alpha) (-.f64 -1 (/.f64 (-.f64 -2 beta) alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2))) alpha) (-.f64 -1 (/.f64 (-.f64 -2 beta) alpha))): 2 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (-.f64 -1 (/.f64 (-.f64 (Rewrite<= metadata-eval (*.f64 2 -1)) beta) alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (-.f64 -1 (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 2 -1) (neg.f64 beta))) alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (-.f64 -1 (/.f64 (+.f64 (*.f64 2 -1) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 beta))) alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (-.f64 -1 (/.f64 (+.f64 (*.f64 2 -1) (Rewrite=> *-commutative_binary64 (*.f64 beta -1))) alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (-.f64 -1 (/.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 -1 (+.f64 2 beta))) alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (-.f64 -1 (/.f64 (*.f64 -1 (Rewrite<= +-commutative_binary64 (+.f64 beta 2))) alpha))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (-.f64 -1 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (Rewrite<= unsub-neg_binary64 (+.f64 -1 (neg.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (+.f64 -1 (neg.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (/.f64 (+.f64 beta 2) alpha)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha) (+.f64 -1 (Rewrite=> remove-double-neg_binary64 (/.f64 (+.f64 beta 2) alpha)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (*.f64 (/.f64 (+.f64 beta 2) alpha) (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)))): 1 points increase in error, 2 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 beta 2) (-.f64 (*.f64 -1 beta) (+.f64 beta 2))) (*.f64 alpha alpha)))): 32 points increase in error, 16 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (/.f64 (*.f64 (+.f64 beta 2) (-.f64 (*.f64 -1 beta) (+.f64 beta 2))) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 beta 2) (pow.f64 alpha 2)) (-.f64 (*.f64 -1 beta) (+.f64 beta 2))))): 2 points increase in error, 16 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (*.f64 (/.f64 (+.f64 beta 2) (pow.f64 alpha 2)) (-.f64 (*.f64 -1 beta) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 beta 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (*.f64 (/.f64 (+.f64 beta 2) (pow.f64 alpha 2)) (-.f64 (*.f64 -1 beta) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 beta 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (*.f64 (/.f64 (+.f64 beta 2) (pow.f64 alpha 2)) (-.f64 (*.f64 -1 beta) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 -1 (+.f64 beta 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (*.f64 (/.f64 (+.f64 beta 2) (pow.f64 alpha 2)) (Rewrite=> distribute-lft-out--_binary64 (*.f64 -1 (-.f64 beta (*.f64 -1 (+.f64 beta 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (*.f64 (/.f64 (+.f64 beta 2) (pow.f64 alpha 2)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 beta (*.f64 -1 (+.f64 beta 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (/.f64 (+.f64 beta 2) (pow.f64 alpha 2)) (-.f64 beta (*.f64 -1 (+.f64 beta 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (+.f64 beta 2) (/.f64 (pow.f64 alpha 2) (-.f64 beta (*.f64 -1 (+.f64 beta 2)))))))): 4 points increase in error, 3 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (neg.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 beta 2) (-.f64 beta (*.f64 -1 (+.f64 beta 2)))) (pow.f64 alpha 2))))): 17 points increase in error, 2 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (neg.f64 (/.f64 (*.f64 (+.f64 beta 2) (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 beta (*.f64 (neg.f64 -1) (+.f64 beta 2))))) (pow.f64 alpha 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (neg.f64 (/.f64 (*.f64 (+.f64 beta 2) (+.f64 beta (*.f64 (Rewrite=> metadata-eval 1) (+.f64 beta 2)))) (pow.f64 alpha 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (neg.f64 (/.f64 (*.f64 (+.f64 beta 2) (+.f64 beta (Rewrite=> *-lft-identity_binary64 (+.f64 beta 2)))) (pow.f64 alpha 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (neg.f64 (/.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 beta (+.f64 beta 2)) (*.f64 (+.f64 beta 2) (+.f64 beta 2)))) (pow.f64 alpha 2)))): 1 points increase in error, 1 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (neg.f64 (/.f64 (+.f64 (*.f64 beta (+.f64 beta 2)) (Rewrite<= unpow2_binary64 (pow.f64 (+.f64 beta 2) 2))) (pow.f64 alpha 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (neg.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (+.f64 beta 2) 2) (*.f64 beta (+.f64 beta 2)))) (pow.f64 alpha 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 beta) (+.f64 beta 2)) alpha)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 (+.f64 beta 2) 2) (*.f64 beta (+.f64 beta 2))) (pow.f64 alpha 2))))): 0 points increase in error, 0 points decrease in error

   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

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

Alternatives

Alternative 1
Error	0.2
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9999999:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 2
Error	4.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.2581541331789086 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \]

Alternative 3
Error	4.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.4562498702661438:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 4
Error	18.1
Cost	460

\[\begin{array}{l} \mathbf{if}\;\beta \leq -5.531919580598117 \cdot 10^{-133}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -2.063566676354855 \cdot 10^{-141}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 0.4562498702661438:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	4.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.4562498702661438:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	46.8
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.991567338544697 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \end{array} \]

Alternative 7
Error	20.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 234.94453275108523:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 8
Error	58.4
Cost	192

\[\frac{\beta}{\alpha} \]

Reproduce

herbie shell --seed 2022298 
(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))