Octave 3.8, jcobi/2

Average Error: 23.8 → 2.5

Time: 1.7min

Precision: binary64

Cost: 24520

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\mathsf{fma}\left(2, i, 2\right) + \beta\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{t_1 + 2}\\ \mathbf{if}\;t_2 \leq -0.99999999999998:\\ \;\;\;\;\left(\frac{\beta - \left(-4 \cdot i - \beta\right)}{-\alpha} - \frac{-2}{-\alpha}\right) \cdot -0.5\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\alpha \cdot \alpha - \beta \cdot \beta}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} - t_0}{t_0} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(2 + \alpha\right)} + 1}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ alpha (+ (fma 2.0 i 2.0) beta)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ t_1 2.0))))
   (if (<= t_2 -0.99999999999998)
     (* (- (/ (- beta (- (* -4.0 i) beta)) (- alpha)) (/ -2.0 (- alpha))) -0.5)
     (if (<= t_2 2e-10)
       (*
        (/
         (-
          (/ (- (* alpha alpha) (* beta beta)) (fma 2.0 i (+ beta alpha)))
          t_0)
         t_0)
        -0.5)
       (/ (+ (/ (- beta alpha) (+ beta (+ 2.0 alpha))) 1.0) 2.0)))))

C

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

↓

double code(double alpha, double beta, double i) {
	double t_0 = alpha + (fma(2.0, i, 2.0) + beta);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0);
	double tmp;
	if (t_2 <= -0.99999999999998) {
		tmp = (((beta - ((-4.0 * i) - beta)) / -alpha) - (-2.0 / -alpha)) * -0.5;
	} else if (t_2 <= 2e-10) {
		tmp = (((((alpha * alpha) - (beta * beta)) / fma(2.0, i, (beta + alpha))) - t_0) / t_0) * -0.5;
	} else {
		tmp = (((beta - alpha) / (beta + (2.0 + alpha))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

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

↓

function code(alpha, beta, i)
	t_0 = Float64(alpha + Float64(fma(2.0, i, 2.0) + beta))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(t_1 + 2.0))
	tmp = 0.0
	if (t_2 <= -0.99999999999998)
		tmp = Float64(Float64(Float64(Float64(beta - Float64(Float64(-4.0 * i) - beta)) / Float64(-alpha)) - Float64(-2.0 / Float64(-alpha))) * -0.5);
	elseif (t_2 <= 2e-10)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(alpha * alpha) - Float64(beta * beta)) / fma(2.0, i, Float64(beta + alpha))) - t_0) / t_0) * -0.5);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(beta + Float64(2.0 + alpha))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

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

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(alpha + N[(N[(2.0 * i + 2.0), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.99999999999998], N[(N[(N[(N[(beta - N[(N[(-4.0 * i), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] / (-alpha)), $MachinePrecision] - N[(-2.0 / (-alpha)), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$2, 2e-10], N[(N[(N[(N[(N[(N[(alpha * alpha), $MachinePrecision] - N[(beta * beta), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.99999999999998002

   1. Initial program 63.2

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

   2. Simplified 63.2

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

   3. Taylor expanded in alpha around -inf 5.3

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

   4. Taylor expanded in i around 0 5.3

      \[\leadsto \frac{\color{blue}{-1 \cdot \beta + \left(-1 \cdot \left(\beta + 2\right) + -4 \cdot i\right)}}{\alpha} \cdot -0.5 \]

   5. Simplified 5.3

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot i - \left(\beta - -2\right)\right) - \beta}}{\alpha} \cdot -0.5 \]
      Proof

   6. Applied egg-rr 5.3

      \[\leadsto \color{blue}{\left(\frac{\beta - \left(-4 \cdot i - \beta\right)}{-\alpha} - \frac{-2}{-\alpha}\right)} \cdot -0.5 \]

   if -0.99999999999998002 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < 2.00000000000000007e-10

   1. Initial program 0.2

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

   2. Simplified 0.3

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

   3. Applied egg-rr 0.2

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

   4. Simplified 0.2

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

   if 2.00000000000000007e-10 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 40.9

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

   2. Taylor expanded in i around 0 5.3

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

Alternatives

Alternative 1
Error	2.5
Cost	5192

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2}\\ \mathbf{if}\;t_1 \leq -0.99999999:\\ \;\;\;\;\left(\frac{\beta - \left(-4 \cdot i - \beta\right)}{-\alpha} - \frac{-2}{-\alpha}\right) \cdot -0.5\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{t_1 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(2 + \alpha\right)} + 1}{2}\\ \end{array} \]

Alternative 2
Error	13.0
Cost	1104

\[\begin{array}{l} t_0 := \frac{-4 \cdot i - 2}{\alpha} \cdot -0.5\\ t_1 := \frac{\frac{\beta}{\beta - -2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 1.45 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 1.35 \cdot 10^{+140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.15 \cdot 10^{+171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 1.65 \cdot 10^{+193}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.0
Cost	1104

\[\begin{array}{l} t_0 := \frac{\frac{\beta}{\beta - -2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{+140}:\\ \;\;\;\;\left(\frac{-4 \cdot i}{\alpha} - \frac{2}{\alpha}\right) \cdot -0.5\\ \mathbf{elif}\;\alpha \leq 1.12 \cdot 10^{+174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.6 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot i - 2}{\alpha} \cdot -0.5\\ \end{array} \]

Alternative 4
Error	9.9
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\beta}{\beta - -2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha} + \frac{-4 \cdot i}{\alpha}\right) \cdot -0.5\\ \end{array} \]

Alternative 5
Error	10.0
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.55 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\beta}{\beta - -2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-4 \cdot i - \left(\beta - -2\right)\right) - \beta}{\alpha} \cdot -0.5\\ \end{array} \]

Alternative 6
Error	12.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.45 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{\beta}{\beta - -2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot i - 2}{\alpha} \cdot -0.5\\ \end{array} \]

Alternative 7
Error	14.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{2}{2 + \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2}\\ \end{array} \]

Alternative 8
Error	17.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2}\\ \end{array} \]

Alternative 9
Error	24.5
Cost	192

\[\frac{1}{2} \]

Reproduce

herbie shell --seed 2023033 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))