Octave 3.8, jcobi/4

Average Error: 54.0 → 9.0

Time: 26.9s

Precision: binary64

Cost: 1997

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := t_0 + 1\\ t_2 := t_0 + -1\\ \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+120} \lor \neg \left(\beta \leq 2.4 \cdot 10^{+150}\right) \land \beta \leq 3.3 \cdot 10^{+172}:\\ \;\;\;\;\frac{i}{t_1} \cdot \frac{i \cdot 0.25}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{t_1} \cdot \frac{i}{t_2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) (* i 2.0)))
        (t_1 (+ t_0 1.0))
        (t_2 (+ t_0 -1.0)))
   (if (or (<= beta 6.5e+120)
           (and (not (<= beta 2.4e+150)) (<= beta 3.3e+172)))
     (* (/ i t_1) (/ (* i 0.25) t_2))
     (* (/ (+ i alpha) t_1) (/ i t_2)))))

C

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

↓

double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = t_0 + 1.0;
	double t_2 = t_0 + -1.0;
	double tmp;
	if ((beta <= 6.5e+120) || (!(beta <= 2.4e+150) && (beta <= 3.3e+172))) {
		tmp = (i / t_1) * ((i * 0.25) / t_2);
	} else {
		tmp = ((i + alpha) / t_1) * (i / t_2);
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (beta + alpha) + (i * 2.0d0)
    t_1 = t_0 + 1.0d0
    t_2 = t_0 + (-1.0d0)
    if ((beta <= 6.5d+120) .or. (.not. (beta <= 2.4d+150)) .and. (beta <= 3.3d+172)) then
        tmp = (i / t_1) * ((i * 0.25d0) / t_2)
    else
        tmp = ((i + alpha) / t_1) * (i / t_2)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = t_0 + 1.0;
	double t_2 = t_0 + -1.0;
	double tmp;
	if ((beta <= 6.5e+120) || (!(beta <= 2.4e+150) && (beta <= 3.3e+172))) {
		tmp = (i / t_1) * ((i * 0.25) / t_2);
	} else {
		tmp = ((i + alpha) / t_1) * (i / t_2);
	}
	return tmp;
}

Python

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

↓

def code(alpha, beta, i):
	t_0 = (beta + alpha) + (i * 2.0)
	t_1 = t_0 + 1.0
	t_2 = t_0 + -1.0
	tmp = 0
	if (beta <= 6.5e+120) or (not (beta <= 2.4e+150) and (beta <= 3.3e+172)):
		tmp = (i / t_1) * ((i * 0.25) / t_2)
	else:
		tmp = ((i + alpha) / t_1) * (i / t_2)
	return tmp

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(i * Float64(Float64(alpha + beta) + i)) * Float64(Float64(beta * alpha) + Float64(i * Float64(Float64(alpha + beta) + i)))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i)))) / Float64(Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i))) - 1.0))
end

↓

function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_1 = Float64(t_0 + 1.0)
	t_2 = Float64(t_0 + -1.0)
	tmp = 0.0
	if ((beta <= 6.5e+120) || (!(beta <= 2.4e+150) && (beta <= 3.3e+172)))
		tmp = Float64(Float64(i / t_1) * Float64(Float64(i * 0.25) / t_2));
	else
		tmp = Float64(Float64(Float64(i + alpha) / t_1) * Float64(i / t_2));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(alpha, beta, i)
	t_0 = (beta + alpha) + (i * 2.0);
	t_1 = t_0 + 1.0;
	t_2 = t_0 + -1.0;
	tmp = 0.0;
	if ((beta <= 6.5e+120) || (~((beta <= 2.4e+150)) && (beta <= 3.3e+172)))
		tmp = (i / t_1) * ((i * 0.25) / t_2);
	else
		tmp = ((i + alpha) / t_1) * (i / t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] * N[(N[(beta * alpha), $MachinePrecision] + N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + -1.0), $MachinePrecision]}, If[Or[LessEqual[beta, 6.5e+120], And[N[Not[LessEqual[beta, 2.4e+150]], $MachinePrecision], LessEqual[beta, 3.3e+172]]], N[(N[(i / t$95$1), $MachinePrecision] * N[(N[(i * 0.25), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(i / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4999999999999997e120 or 2.40000000000000003e150 < beta < 3.29999999999999983e172

   1. Initial program 49.5

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

   2. Taylor expanded in i around inf 37.4

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Simplified 37.4

      \[\leadsto \frac{\color{blue}{\left(i \cdot i\right) \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      Proof

      [Start] 37.4	\[ \frac{0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 37.4	\[ \frac{\color{blue}{{i}^{2} \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 37.4	\[ \frac{\color{blue}{\left(i \cdot i\right)} \cdot 0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Applied egg-rr 5.5

      \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{i \cdot 0.25}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + -1}} \]

   if 6.4999999999999997e120 < beta < 2.40000000000000003e150 or 3.29999999999999983e172 < beta

   1. Initial program 63.5

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

   2. Taylor expanded in beta around inf 46.2

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

   3. Applied egg-rr 16.5

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

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+120} \lor \neg \left(\beta \leq 2.4 \cdot 10^{+150}\right) \land \beta \leq 3.3 \cdot 10^{+172}:\\ \;\;\;\;\frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1} \cdot \frac{i \cdot 0.25}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1} \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	1736

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ \mathbf{if}\;\beta \leq 6.3 \cdot 10^{+134}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 2.5 \cdot 10^{+150}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{i}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 1.15 \cdot 10^{+201}:\\ \;\;\;\;\left(\left(-0.03125 \cdot \frac{\beta}{i} + -0.0078125\right) + \frac{\beta}{i} \cdot 0.03125\right) \cdot -8\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 2
Error	9.2
Cost	1732

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ \mathbf{if}\;\beta \leq 1.12 \cdot 10^{+201}:\\ \;\;\;\;\frac{i}{t_0 + 1} \cdot \frac{i \cdot 0.25}{t_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 3
Error	10.3
Cost	1357

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.3 \cdot 10^{+134}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+150} \lor \neg \left(\beta \leq 4.9 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.03125 \cdot \frac{\beta}{i} + -0.0078125\right) + \frac{\beta}{i} \cdot 0.03125\right) \cdot -8\\ \end{array} \]

Alternative 4
Error	9.8
Cost	973

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+134}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 4.3 \cdot 10^{+150} \lor \neg \left(\beta \leq 2.75 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 5
Error	17.0
Cost	844

\[\begin{array}{l} t_0 := 0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{if}\;\beta \leq 5.1 \cdot 10^{+134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2.8 \cdot 10^{+150}:\\ \;\;\;\;i \cdot \frac{i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 3.2 \cdot 10^{+248}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

Alternative 6
Error	16.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+248}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

Alternative 7
Error	16.6
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+248}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

Alternative 8
Error	18.6
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023011 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))