Octave 3.8, jcobi/4

Average Error: 54.1 → 8.7

Time: 24.5s

Precision: binary64

Cost: 21444

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := t_0 + -1\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{\beta + \left(i \cdot 2 + 1\right)}\right) \cdot \frac{0.5 \cdot \left(\beta + \left(i + \alpha\right)\right) + \left(\beta + \alpha\right) \cdot -0.25}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_0}{i + \left(\beta + \alpha\right)}}}{1 + t_0} \cdot \frac{i + \alpha}{t_1}\\ \end{array} \]

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 (fma i 2.0 (+ beta alpha))) (t_1 (+ t_0 -1.0)))
   (if (<= beta 5e+119)
     (*
      (* (/ i (+ beta (* i 2.0))) (/ (+ beta i) (+ beta (+ (* i 2.0) 1.0))))
      (/ (+ (* 0.5 (+ beta (+ i alpha))) (* (+ beta alpha) -0.25)) t_1))
     (*
      (/ (/ i (/ t_0 (+ i (+ beta alpha)))) (+ 1.0 t_0))
      (/ (+ i alpha) t_1)))))

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 = fma(i, 2.0, (beta + alpha));
	double t_1 = t_0 + -1.0;
	double tmp;
	if (beta <= 5e+119) {
		tmp = ((i / (beta + (i * 2.0))) * ((beta + i) / (beta + ((i * 2.0) + 1.0)))) * (((0.5 * (beta + (i + alpha))) + ((beta + alpha) * -0.25)) / t_1);
	} else {
		tmp = ((i / (t_0 / (i + (beta + alpha)))) / (1.0 + t_0)) * ((i + alpha) / t_1);
	}
	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 = fma(i, 2.0, Float64(beta + alpha))
	t_1 = Float64(t_0 + -1.0)
	tmp = 0.0
	if (beta <= 5e+119)
		tmp = Float64(Float64(Float64(i / Float64(beta + Float64(i * 2.0))) * Float64(Float64(beta + i) / Float64(beta + Float64(Float64(i * 2.0) + 1.0)))) * Float64(Float64(Float64(0.5 * Float64(beta + Float64(i + alpha))) + Float64(Float64(beta + alpha) * -0.25)) / t_1));
	else
		tmp = Float64(Float64(Float64(i / Float64(t_0 / Float64(i + Float64(beta + alpha)))) / Float64(1.0 + t_0)) * Float64(Float64(i + alpha) / t_1));
	end
	return 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[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + -1.0), $MachinePrecision]}, If[LessEqual[beta, 5e+119], N[(N[(N[(i / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + i), $MachinePrecision] / N[(beta + N[(N[(i * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 * N[(beta + N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + alpha), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i / N[(t$95$0 / N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9999999999999999e119

   1. Initial program 48.6

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

   2. Applied egg-rr 32.6

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

   3. Taylor expanded in i around inf 3.6

      \[\leadsto \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\color{blue}{\left(0.5 \cdot \left(\beta + \alpha\right) + 0.5 \cdot i\right) - 0.25 \cdot \left(\beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]

   4. Simplified 3.6

      \[\leadsto \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\color{blue}{0.5 \cdot \left(\left(\alpha + i\right) + \beta\right) + \left(\beta + \alpha\right) \cdot -0.25}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      Proof
      (+.f64 (*.f64 1/2 (+.f64 (+.f64 alpha i) beta)) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 i alpha)) beta)) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (Rewrite<= associate-+r+_binary64 (+.f64 i (+.f64 alpha beta)))) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (+.f64 i (Rewrite<= +-commutative_binary64 (+.f64 beta alpha)))) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 beta alpha) i))) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 1/2 (+.f64 beta alpha)) (*.f64 1/2 i))) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 1/2 (+.f64 beta alpha)) (*.f64 1/2 i)) (Rewrite<= *-commutative_binary64 (*.f64 -1/4 (+.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 1/2 (+.f64 beta alpha)) (*.f64 1/2 i)) (*.f64 (Rewrite<= metadata-eval (neg.f64 1/4)) (+.f64 beta alpha))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (+.f64 (*.f64 1/2 (+.f64 beta alpha)) (*.f64 1/2 i)) (*.f64 1/4 (+.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in alpha around 0 36.0

      \[\leadsto \color{blue}{\frac{i \cdot \left(\beta + i\right)}{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(1 + 2 \cdot i\right)\right)}} \cdot \frac{0.5 \cdot \left(\left(\alpha + i\right) + \beta\right) + \left(\beta + \alpha\right) \cdot -0.25}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]

   6. Simplified 3.6

      \[\leadsto \color{blue}{\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{\beta + \left(1 + i \cdot 2\right)}\right)} \cdot \frac{0.5 \cdot \left(\left(\alpha + i\right) + \beta\right) + \left(\beta + \alpha\right) \cdot -0.25}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      Proof
      (*.f64 (/.f64 i (+.f64 beta (*.f64 i 2))) (/.f64 (+.f64 beta i) (+.f64 beta (+.f64 1 (*.f64 i 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (+.f64 beta (Rewrite<= *-commutative_binary64 (*.f64 2 i)))) (/.f64 (+.f64 beta i) (+.f64 beta (+.f64 1 (*.f64 i 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (+.f64 beta (*.f64 2 i))) (/.f64 (+.f64 beta i) (+.f64 beta (+.f64 1 (Rewrite<= *-commutative_binary64 (*.f64 2 i)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 i (+.f64 beta i)) (*.f64 (+.f64 beta (*.f64 2 i)) (+.f64 beta (+.f64 1 (*.f64 2 i)))))): 175 points increase in error, 17 points decrease in error

   if 4.9999999999999999e119 < beta

   1. Initial program 63.4

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

   2. Applied egg-rr 47.5

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

   3. Taylor expanded in beta around inf 17.4

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

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{\beta + \left(i \cdot 2 + 1\right)}\right) \cdot \frac{0.5 \cdot \left(\beta + \left(i + \alpha\right)\right) + \left(\beta + \alpha\right) \cdot -0.25}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \left(\beta + \alpha\right)}}}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.4
Cost	9156

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.06 \cdot 10^{+134}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{\beta + \left(i \cdot 2 + 1\right)}\right) \cdot \frac{0.5 \cdot \left(\beta + \left(i + \alpha\right)\right) + \left(\beta + \alpha\right) \cdot -0.25}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 2
Error	9.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.16 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 3
Error	9.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.4 \cdot 10^{+136}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 4
Error	15.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+224}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 5
Error	11.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.6 \cdot 10^{+136}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6
Error	18.6
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022329 
(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)))