Octave 3.8, jcobi/2

Percentage Accurate: 62.4% → 97.6%

Time: 14.5s

Alternatives: 8

Speedup: 9.5×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

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
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Initial Program: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

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
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Alternative 1: 97.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.5)
     (/ (/ (+ 2.0 (- (* 2.0 (+ beta i)) (* i -2.0))) alpha) 2.0)
     (/
      (+
       (*
        (+ alpha beta)
        (/
         (/ (- beta alpha) (+ alpha (+ beta (fma 2.0 i 2.0))))
         (+ alpha (fma 2.0 i beta))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5) {
		tmp = ((2.0 + ((2.0 * (beta + i)) - (i * -2.0))) / alpha) / 2.0;
	} else {
		tmp = (((alpha + beta) * (((beta - alpha) / (alpha + (beta + fma(2.0, i, 2.0)))) / (alpha + fma(2.0, i, beta)))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.5)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(2.0 * Float64(beta + i)) - Float64(i * -2.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(alpha + beta) * Float64(Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) / Float64(alpha + fma(2.0, i, beta)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 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)) < -0.5

   1. Initial program 3.8%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 18.3%

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

   4. Taylor expanded in alpha around inf 87.7%

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

   5. Taylor expanded in beta around 0 87.7%

      \[\leadsto \frac{\frac{\color{blue}{\left(2 + \left(2 \cdot \beta + 2 \cdot i\right)\right) - -2 \cdot i}}{\alpha}}{2} \]
   6. Step-by-step derivation

      1. associate--l+ 87.7%

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

      2. distribute-lft-out 87.7%

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

   7. Simplified 87.7%

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

   if -0.5 < (/.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 80.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 79.9%

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

      2. associate-+l+ 79.9%

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

      3. associate-+l+ 79.9%

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

   3. Simplified 79.9%

      \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} + 1}{2}} \]
   4. Step-by-step derivation

      1. div-inv 79.9%

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

      2. associate-+r+ 79.9%

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

      3. fma-def 79.9%

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

      4. +-commutative 79.9%

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

      5. fma-udef 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. associate-*r/ 79.9%

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

      2. *-rgt-identity 79.9%

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

      3. associate-*r/ 84.9%

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

      4. +-commutative 84.9%

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

      5. associate-/r* 100.0%

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

      6. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 96.8%

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

Alternative 2: 97.6% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.5)
     (/ (/ (+ 2.0 (- (* 2.0 (+ beta i)) (* i -2.0))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (* (- beta alpha) (/ (+ alpha beta) (+ alpha (fma 2.0 i beta))))
        t_1))
      2.0))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 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)) < -0.5

   1. Initial program 3.8%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 18.3%

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

   4. Taylor expanded in alpha around inf 87.7%

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

   5. Taylor expanded in beta around 0 87.7%

      \[\leadsto \frac{\frac{\color{blue}{\left(2 + \left(2 \cdot \beta + 2 \cdot i\right)\right) - -2 \cdot i}}{\alpha}}{2} \]
   6. Step-by-step derivation

      1. associate--l+ 87.7%

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

      2. distribute-lft-out 87.7%

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

   7. Simplified 87.7%

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

   if -0.5 < (/.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 80.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. *-commutative 80.5%

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

      2. *-un-lft-identity 80.5%

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

      3. times-frac 100.0%

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

      4. associate-+r+ 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-udef 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 96.8%

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

Alternative 3: 96.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.5)
     (/ (/ (+ 2.0 (- (* 2.0 (+ beta i)) (* i -2.0))) alpha) 2.0)
     (/ (+ 1.0 (/ beta t_1)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5) {
		tmp = ((2.0 + ((2.0 * (beta + i)) - (i * -2.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / t_1)) / 2.0;
	}
	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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = 2.0d0 + t_0
    if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= (-0.5d0)) then
        tmp = ((2.0d0 + ((2.0d0 * (beta + i)) - (i * (-2.0d0)))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (beta / t_1)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5) {
		tmp = ((2.0 + ((2.0 * (beta + i)) - (i * -2.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / t_1)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = 2.0 + t_0
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5:
		tmp = ((2.0 + ((2.0 * (beta + i)) - (i * -2.0))) / alpha) / 2.0
	else:
		tmp = (1.0 + (beta / t_1)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.5)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(2.0 * Float64(beta + i)) - Float64(i * -2.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / t_1)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = 2.0 + t_0;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5)
		tmp = ((2.0 + ((2.0 * (beta + i)) - (i * -2.0))) / alpha) / 2.0;
	else
		tmp = (1.0 + (beta / t_1)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 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)) < -0.5

   1. Initial program 3.8%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 18.3%

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

   4. Taylor expanded in alpha around inf 87.7%

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

   5. Taylor expanded in beta around 0 87.7%

      \[\leadsto \frac{\frac{\color{blue}{\left(2 + \left(2 \cdot \beta + 2 \cdot i\right)\right) - -2 \cdot i}}{\alpha}}{2} \]
   6. Step-by-step derivation

      1. associate--l+ 87.7%

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

      2. distribute-lft-out 87.7%

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

   7. Simplified 87.7%

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

   if -0.5 < (/.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 80.5%

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

   2. Taylor expanded in beta around inf 99.2%

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

4. Final simplification 96.2%

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

Alternative 4: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.7 \cdot 10^{+101}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{+232} \lor \neg \left(\alpha \leq 5.8 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.7e+101)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (if (or (<= alpha 6.2e+232) (not (<= alpha 5.8e+259)))
     (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.7e+101) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else if ((alpha <= 6.2e+232) || !(alpha <= 5.8e+259)) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	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
    real(8) :: tmp
    if (alpha <= 2.7d+101) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else if ((alpha <= 6.2d+232) .or. (.not. (alpha <= 5.8d+259))) then
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.7e+101) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else if ((alpha <= 6.2e+232) || !(alpha <= 5.8e+259)) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.7e+101:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	elif (alpha <= 6.2e+232) or not (alpha <= 5.8e+259):
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.7e+101)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	elseif ((alpha <= 6.2e+232) || !(alpha <= 5.8e+259))
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.7e+101)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	elseif ((alpha <= 6.2e+232) || ~((alpha <= 5.8e+259)))
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.7e+101], N[(N[(1.0 + N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 6.2e+232], N[Not[LessEqual[alpha, 5.8e+259]], $MachinePrecision]], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 2.70000000000000006e101

   1. Initial program 78.6%

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

   2. Taylor expanded in beta around inf 94.3%

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

   if 2.70000000000000006e101 < alpha < 6.19999999999999966e232 or 5.7999999999999999e259 < alpha

   1. Initial program 11.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 34.7%

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

   4. Taylor expanded in alpha around inf 71.1%

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

   5. Taylor expanded in beta around 0 65.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(2 + 2 \cdot i\right) - -2 \cdot i}}{\alpha}}{2} \]
   6. Step-by-step derivation

      1. associate--l+ 65.9%

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

      2. distribute-rgt-out-- 65.9%

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

      3. metadata-eval 65.9%

         \[\leadsto \frac{\frac{2 + i \cdot \color{blue}{4}}{\alpha}}{2} \]

      4. *-commutative 65.9%

         \[\leadsto \frac{\frac{2 + \color{blue}{4 \cdot i}}{\alpha}}{2} \]

   7. Simplified 65.9%

      \[\leadsto \frac{\frac{\color{blue}{2 + 4 \cdot i}}{\alpha}}{2} \]

   if 6.19999999999999966e232 < alpha < 5.7999999999999999e259

   1. Initial program 1.3%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 0.0%

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

      2. associate-+l+ 0.0%

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

      3. associate-+l+ 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in i around 0 15.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   5. Step-by-step derivation

      1. associate-+r+ 15.7%

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

   6. Simplified 15.7%

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

   7. Taylor expanded in alpha around inf 77.1%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.7 \cdot 10^{+101}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{+232} \lor \neg \left(\alpha \leq 5.8 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 80.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.3 \cdot 10^{+232} \lor \neg \left(\alpha \leq 4.5 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.2e+101)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (or (<= alpha 2.3e+232) (not (<= alpha 4.5e+259)))
     (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.2e+101) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 2.3e+232) || !(alpha <= 4.5e+259)) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	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
    real(8) :: tmp
    if (alpha <= 1.2d+101) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if ((alpha <= 2.3d+232) .or. (.not. (alpha <= 4.5d+259))) then
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.2e+101) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 2.3e+232) || !(alpha <= 4.5e+259)) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.2e+101:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif (alpha <= 2.3e+232) or not (alpha <= 4.5e+259):
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.2e+101)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif ((alpha <= 2.3e+232) || !(alpha <= 4.5e+259))
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.2e+101)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif ((alpha <= 2.3e+232) || ~((alpha <= 4.5e+259)))
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.2e+101], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 2.3e+232], N[Not[LessEqual[alpha, 4.5e+259]], $MachinePrecision]], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.19999999999999994e101

   1. Initial program 78.6%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 78.0%

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

      2. associate-+l+ 78.0%

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

      3. associate-+l+ 78.0%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in i around 0 81.0%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   5. Step-by-step derivation

      1. associate-+r+ 81.0%

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

   6. Simplified 81.0%

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

   7. Taylor expanded in alpha around 0 86.1%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]

   if 1.19999999999999994e101 < alpha < 2.30000000000000006e232 or 4.4999999999999997e259 < alpha

   1. Initial program 11.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 34.7%

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

   4. Taylor expanded in alpha around inf 71.1%

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

   5. Taylor expanded in beta around 0 65.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(2 + 2 \cdot i\right) - -2 \cdot i}}{\alpha}}{2} \]
   6. Step-by-step derivation

      1. associate--l+ 65.9%

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

      2. distribute-rgt-out-- 65.9%

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

      3. metadata-eval 65.9%

         \[\leadsto \frac{\frac{2 + i \cdot \color{blue}{4}}{\alpha}}{2} \]

      4. *-commutative 65.9%

         \[\leadsto \frac{\frac{2 + \color{blue}{4 \cdot i}}{\alpha}}{2} \]

   7. Simplified 65.9%

      \[\leadsto \frac{\frac{\color{blue}{2 + 4 \cdot i}}{\alpha}}{2} \]

   if 2.30000000000000006e232 < alpha < 4.4999999999999997e259

   1. Initial program 1.3%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 0.0%

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

      2. associate-+l+ 0.0%

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

      3. associate-+l+ 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in i around 0 15.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   5. Step-by-step derivation

      1. associate-+r+ 15.7%

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

   6. Simplified 15.7%

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

   7. Taylor expanded in alpha around inf 77.1%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.3 \cdot 10^{+232} \lor \neg \left(\alpha \leq 4.5 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 75.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 1.1e+103) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.1e+103) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	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
    real(8) :: tmp
    if (i <= 1.1d+103) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.1e+103) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 1.1e+103:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 1.1e+103)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 1.1e+103)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 1.1e+103], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 1.09999999999999996e103

   1. Initial program 54.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 54.2%

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

      2. associate-+l+ 54.2%

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

      3. associate-+l+ 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in i around 0 70.9%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   5. Step-by-step derivation

      1. associate-+r+ 70.9%

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

   6. Simplified 70.9%

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

   7. Taylor expanded in alpha around 0 69.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]

   if 1.09999999999999996e103 < i

   1. Initial program 71.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 70.9%

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

      2. associate-+l+ 70.9%

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

      3. associate-+l+ 70.9%

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

   3. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} + 1}{2}} \]
   4. Step-by-step derivation

      1. div-inv 70.9%

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

      2. associate-+r+ 70.9%

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

      3. fma-def 70.9%

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

      4. +-commutative 70.9%

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

      5. fma-udef 70.9%

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

   5. Applied egg-rr 70.9%

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

      1. associate-*r/ 70.9%

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

      2. *-rgt-identity 70.9%

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

      3. associate-*r/ 84.8%

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

      4. +-commutative 84.8%

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

      5. associate-/r* 90.0%

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

      6. +-commutative 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in i around inf 84.8%

      \[\leadsto \frac{\color{blue}{1}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7: 72.0% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.42 \cdot 10^{+44}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 1.42e+44) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.42e+44) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	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
    real(8) :: tmp
    if (beta <= 1.42d+44) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.42e+44) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.42e+44:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.42e+44)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.42e+44)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.42e+44], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.41999999999999994e44

   1. Initial program 71.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 71.0%

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

      2. associate-+l+ 71.0%

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

      3. associate-+l+ 71.0%

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

   3. Simplified 71.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} + 1}{2}} \]
   4. Step-by-step derivation

      1. div-inv 71.0%

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

      2. associate-+r+ 71.0%

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

      3. fma-def 71.0%

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

      4. +-commutative 71.0%

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

      5. fma-udef 71.0%

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

   5. Applied egg-rr 71.0%

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

      1. associate-*r/ 71.0%

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

      2. *-rgt-identity 71.0%

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

      3. associate-*r/ 75.0%

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

      4. +-commutative 75.0%

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

      5. associate-/r* 75.1%

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

      6. +-commutative 75.1%

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

   7. Simplified 75.1%

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

   8. Taylor expanded in i around inf 73.7%

      \[\leadsto \frac{\color{blue}{1}}{2} \]

   if 1.41999999999999994e44 < beta

   1. Initial program 35.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l/ 33.8%

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

      2. associate-+l+ 33.8%

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

      3. associate-+l+ 33.8%

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

   3. Simplified 33.8%

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

   4. Taylor expanded in beta around inf 70.8%

      \[\leadsto \frac{\color{blue}{2}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.42 \cdot 10^{+44}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 60.5% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.5)

C

double code(double alpha, double beta, double i) {
	return 0.5;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.5d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.5;
}

Python

def code(alpha, beta, i):
	return 0.5

Julia

function code(alpha, beta, i)
	return 0.5
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.5;
end

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 60.4%

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
2. Step-by-step derivation

   1. associate-/l/ 59.8%

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

   2. associate-+l+ 59.8%

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

   3. associate-+l+ 59.8%

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

3. Simplified 59.8%

   \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} + 1}{2}} \]
4. Step-by-step derivation

   1. div-inv 59.8%

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

   2. associate-+r+ 59.8%

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

   3. fma-def 59.8%

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

   4. +-commutative 59.8%

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

   5. fma-udef 59.8%

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

5. Applied egg-rr 59.8%

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

   1. associate-*r/ 59.8%

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

   2. *-rgt-identity 59.8%

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

   3. associate-*r/ 67.1%

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

   4. +-commutative 67.1%

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

   5. associate-/r* 78.7%

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

   6. +-commutative 78.7%

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

7. Simplified 78.7%

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

8. Taylor expanded in i around inf 60.9%

   \[\leadsto \frac{\color{blue}{1}}{2} \]

9. Final simplification 60.9%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023322 
(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))