Octave 3.8, jcobi/4

Average Error: 54.2 → 8.6

Time: 24.9s

Precision: binary64

Cost: 22084

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

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

Math

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

↓

\[\begin{array}{l} t_0 := i + \left(\beta + \alpha\right)\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := \frac{\frac{i}{\frac{t_1}{t_0}}}{t_1 + 1}\\ t_3 := t_1 + -1\\ \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;t_2 \cdot \frac{0.5 \cdot t_0 + \left(\beta + \alpha\right) \cdot -0.25}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\frac{1}{t_3} \cdot \left(i + \alpha\right)\right)\\ \end{array} \]

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 (+ i (+ beta alpha)))
        (t_1 (fma i 2.0 (+ beta alpha)))
        (t_2 (/ (/ i (/ t_1 t_0)) (+ t_1 1.0)))
        (t_3 (+ t_1 -1.0)))
   (if (<= beta 1.85e+134)
     (* t_2 (/ (+ (* 0.5 t_0) (* (+ beta alpha) -0.25)) t_3))
     (* t_2 (* (/ 1.0 t_3) (+ i alpha))))))

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 = i + (beta + alpha);
	double t_1 = fma(i, 2.0, (beta + alpha));
	double t_2 = (i / (t_1 / t_0)) / (t_1 + 1.0);
	double t_3 = t_1 + -1.0;
	double tmp;
	if (beta <= 1.85e+134) {
		tmp = t_2 * (((0.5 * t_0) + ((beta + alpha) * -0.25)) / t_3);
	} else {
		tmp = t_2 * ((1.0 / t_3) * (i + alpha));
	}
	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(i + Float64(beta + alpha))
	t_1 = fma(i, 2.0, Float64(beta + alpha))
	t_2 = Float64(Float64(i / Float64(t_1 / t_0)) / Float64(t_1 + 1.0))
	t_3 = Float64(t_1 + -1.0)
	tmp = 0.0
	if (beta <= 1.85e+134)
		tmp = Float64(t_2 * Float64(Float64(Float64(0.5 * t_0) + Float64(Float64(beta + alpha) * -0.25)) / t_3));
	else
		tmp = Float64(t_2 * Float64(Float64(1.0 / t_3) * Float64(i + alpha)));
	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 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(i / N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + -1.0), $MachinePrecision]}, If[LessEqual[beta, 1.85e+134], N[(t$95$2 * N[(N[(N[(0.5 * t$95$0), $MachinePrecision] + N[(N[(beta + alpha), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(1.0 / t$95$3), $MachinePrecision] * N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.85000000000000007e134

   1. Initial program 49.1

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

   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 4.6

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

   4. Simplified 4.6

      \[\leadsto \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\color{blue}{0.5 \cdot \left(\left(\beta + \alpha\right) + i\right) + -0.25 \cdot \left(\beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      Proof
      (+.f64 (*.f64 1/2 (+.f64 (+.f64 beta alpha) i)) (*.f64 -1/4 (+.f64 beta alpha))): 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 -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

   if 1.85000000000000007e134 < beta

   1. Initial program 63.8

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

   2. Applied egg-rr 49.3

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

   3. Taylor expanded in beta around inf 16.2

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

   4. Applied egg-rr 16.2

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

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \left(\beta + \alpha\right)}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{0.5 \cdot \left(i + \left(\beta + \alpha\right)\right) + \left(\beta + \alpha\right) \cdot -0.25}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \left(\beta + \alpha\right)}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{1}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \cdot \left(i + \alpha\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	8.7
Cost	21572

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := t_0 + -1\\ \mathbf{if}\;\beta \leq 1.75 \cdot 10^{+133}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{i \cdot 2 + \left(\beta + 1\right)}\right) \cdot \frac{\beta \cdot -0.25 + 0.5 \cdot \left(\beta + i\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_0}{i + \left(\beta + \alpha\right)}}}{t_0 + 1} \cdot \left(\frac{1}{t_1} \cdot \left(i + \alpha\right)\right)\\ \end{array} \]

Alternative 2
Error	8.7
Cost	8900

\[\begin{array}{l} t_0 := \frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{i \cdot 2 + \left(\beta + 1\right)}\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1\\ \mathbf{if}\;\beta \leq 3.95 \cdot 10^{+133}:\\ \;\;\;\;t_0 \cdot \frac{\beta \cdot -0.25 + 0.5 \cdot \left(\beta + i\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{i + \alpha}{t_1}\\ \end{array} \]

Alternative 3
Error	8.9
Cost	8516

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{i \cdot 2 + \left(\beta + 1\right)}\right) \cdot \frac{i + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \end{array} \]

Alternative 4
Error	9.6
Cost	7492

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.28 \cdot 10^{+134}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5
Error	9.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+133}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 6
Error	16.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.6 \cdot 10^{+251}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \]

Alternative 7
Error	18.8
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022337 
(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)))