Octave 3.8, jcobi/2

Percentage Accurate: 63.5% → 97.5%

Time: 20.4s

Alternatives: 11

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: 63.5% 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: 97.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))) (t_1 (+ (* 2.0 i) (+ alpha beta))))
   (if (<= (/ (/ (* (- beta alpha) (+ alpha beta)) t_1) (+ 2.0 t_1)) -1.0)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0)
     (/
      (+
       (*
        (- beta alpha)
        (/
         (/ (+ alpha beta) (+ 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 = beta + (2.0 * i);
	double t_1 = (2.0 * i) + (alpha + beta);
	double tmp;
	if (((((beta - alpha) * (alpha + beta)) / t_1) / (2.0 + t_1)) <= -1.0) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (((beta - alpha) * (((alpha + beta) / (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(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(2.0 * i) + Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) * Float64(alpha + beta)) / t_1) / Float64(2.0 + t_1)) <= -1.0)
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(Float64(alpha + beta) / 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[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * i), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(N[(alpha + beta), $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)) < -1

   1. Initial program 2.0%

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

   3. Simplified 18.0%

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

   4. Taylor expanded in alpha around inf 87.9%

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

   if -1 < (/.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 83.5%

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

      1. associate-/l/ 83.0%

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

      2. associate-+l+ 83.0%

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

      3. associate-+l+ 83.0%

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

   3. Simplified 83.0%

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

      1. *-un-lft-identity 83.0%

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

      2. times-frac 89.4%

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

      3. associate-+r+ 89.4%

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

      4. fma-def 89.4%

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

      5. +-commutative 89.4%

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

      6. fma-udef 89.4%

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

   5. Applied egg-rr 89.4%

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

      1. /-rgt-identity 89.4%

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

      2. associate-*r/ 83.0%

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

      3. *-commutative 83.0%

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

      4. associate-*r/ 89.4%

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

      5. associate-/r* 100.0%

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

      6. +-commutative 100.0%

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

      7. +-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 97.3%

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

Alternative 2: 97.5% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i)))
        (t_1 (+ (* 2.0 i) (+ alpha beta)))
        (t_2 (+ 2.0 t_1)))
   (if (<= (/ (/ (* (- beta alpha) (+ alpha beta)) t_1) t_2) -1.0)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (*
         (+ alpha beta)
         (* (- beta alpha) (/ 1.0 (+ alpha (fma 2.0 i beta)))))
        t_2))
      2.0))))

C

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

Julia

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

Wolfram

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

   1. Initial program 2.0%

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

   3. Simplified 18.0%

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

   4. Taylor expanded in alpha around inf 87.9%

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

   if -1 < (/.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 83.5%

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

      1. div-inv 83.5%

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

      2. associate-+r+ 83.5%

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

      3. +-commutative 83.5%

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

      4. fma-udef 83.5%

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

   3. Applied egg-rr 83.5%

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

      1. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 97.3%

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

Alternative 3: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := 2 \cdot i + \left(\alpha + \beta\right)\\ t_2 := 2 + t_1\\ \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{t_1}}{t_2} \leq -1:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{t_2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i)))
        (t_1 (+ (* 2.0 i) (+ alpha beta)))
        (t_2 (+ 2.0 t_1)))
   (if (<= (/ (/ (* (- beta alpha) (+ alpha beta)) t_1) t_2) -1.0)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0)
     (/ (+ 1.0 (/ (- beta alpha) t_2)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (2.0 * i) + (alpha + beta);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((((beta - alpha) * (alpha + beta)) / t_1) / t_2) <= -1.0) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / t_2)) / 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) :: t_2
    real(8) :: tmp
    t_0 = beta + (2.0d0 * i)
    t_1 = (2.0d0 * i) + (alpha + beta)
    t_2 = 2.0d0 + t_1
    if (((((beta - alpha) * (alpha + beta)) / t_1) / t_2) <= (-1.0d0)) then
        tmp = ((t_0 + (2.0d0 + t_0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / t_2)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (2.0 * i) + (alpha + beta);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((((beta - alpha) * (alpha + beta)) / t_1) / t_2) <= -1.0) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / t_2)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	t_1 = (2.0 * i) + (alpha + beta)
	t_2 = 2.0 + t_1
	tmp = 0
	if ((((beta - alpha) * (alpha + beta)) / t_1) / t_2) <= -1.0:
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / t_2)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(2.0 * i) + Float64(alpha + beta))
	t_2 = Float64(2.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) * Float64(alpha + beta)) / t_1) / t_2) <= -1.0)
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / t_2)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	t_1 = (2.0 * i) + (alpha + beta);
	t_2 = 2.0 + t_1;
	tmp = 0.0;
	if (((((beta - alpha) * (alpha + beta)) / t_1) / t_2) <= -1.0)
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / t_2)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

   1. Initial program 2.0%

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

   3. Simplified 18.0%

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

   4. Taylor expanded in alpha around inf 87.9%

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

   if -1 < (/.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 83.5%

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

   2. Taylor expanded in i around 0 98.9%

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

4. Final simplification 96.5%

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

Alternative 4: 78.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{if}\;2 \cdot i \leq 5 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;2 \cdot i \leq 2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \mathbf{elif}\;2 \cdot i \leq 2 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2 \cdot i}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 2.0)))
   (if (<= (* 2.0 i) 5e-90)
     t_0
     (if (<= (* 2.0 i) 2e-78)
       (/ (/ (+ 2.0 (+ beta beta)) alpha) 2.0)
       (if (<= (* 2.0 i) 2e+46)
         t_0
         (/ (+ 1.0 (/ beta (+ beta (* 2.0 i)))) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	double tmp;
	if ((2.0 * i) <= 5e-90) {
		tmp = t_0;
	} else if ((2.0 * i) <= 2e-78) {
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	} else if ((2.0 * i) <= 2e+46) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 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) :: tmp
    t_0 = (1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) / 2.0d0
    if ((2.0d0 * i) <= 5d-90) then
        tmp = t_0
    else if ((2.0d0 * i) <= 2d-78) then
        tmp = ((2.0d0 + (beta + beta)) / alpha) / 2.0d0
    else if ((2.0d0 * i) <= 2d+46) then
        tmp = t_0
    else
        tmp = (1.0d0 + (beta / (beta + (2.0d0 * i)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	double tmp;
	if ((2.0 * i) <= 5e-90) {
		tmp = t_0;
	} else if ((2.0 * i) <= 2e-78) {
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	} else if ((2.0 * i) <= 2e+46) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0
	tmp = 0
	if (2.0 * i) <= 5e-90:
		tmp = t_0
	elif (2.0 * i) <= 2e-78:
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0
	elif (2.0 * i) <= 2e+46:
		tmp = t_0
	else:
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) / 2.0)
	tmp = 0.0
	if (Float64(2.0 * i) <= 5e-90)
		tmp = t_0;
	elseif (Float64(2.0 * i) <= 2e-78)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + beta)) / alpha) / 2.0);
	elseif (Float64(2.0 * i) <= 2e+46)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(2.0 * i)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	tmp = 0.0;
	if ((2.0 * i) <= 5e-90)
		tmp = t_0;
	elseif ((2.0 * i) <= 2e-78)
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	elseif ((2.0 * i) <= 2e+46)
		tmp = t_0;
	else
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[N[(2.0 * i), $MachinePrecision], 5e-90], t$95$0, If[LessEqual[N[(2.0 * i), $MachinePrecision], 2e-78], N[(N[(N[(2.0 + N[(beta + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[N[(2.0 * i), $MachinePrecision], 2e+46], t$95$0, N[(N[(1.0 + N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 2 i) < 5.00000000000000019e-90 or 2e-78 < (*.f64 2 i) < 2e46

   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. Step-by-step derivation

      1. associate-/l/ 62.7%

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

      2. associate-+l+ 62.7%

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

      3. associate-+l+ 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in i around 0 76.7%

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

      1. associate-+r+ 76.7%

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

   6. Simplified 76.7%

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

   if 5.00000000000000019e-90 < (*.f64 2 i) < 2e-78

   1. Initial program 1.6%

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

   3. Simplified 5.8%

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

   4. Taylor expanded in alpha around inf 100.0%

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

   5. Taylor expanded in i around 0 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. remove-double-neg 100.0%

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

   7. Simplified 100.0%

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

   if 2e46 < (*.f64 2 i)

   1. Initial program 71.9%

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

      1. div-inv 71.9%

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

      2. associate-+r+ 71.9%

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

      3. +-commutative 71.9%

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

      4. fma-udef 71.9%

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

   3. Applied egg-rr 71.9%

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

      1. associate-*l* 92.7%

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

      2. +-commutative 92.7%

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

      3. +-commutative 92.7%

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

   5. Simplified 92.7%

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

   6. Taylor expanded in alpha around 0 91.4%

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

   7. Taylor expanded in alpha around inf 89.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 5 \cdot 10^{-90}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;2 \cdot i \leq 2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \mathbf{elif}\;2 \cdot i \leq 2 \cdot 10^{+46}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2 \cdot i}}{2}\\ \end{array} \]

Alternative 5: 84.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ \mathbf{if}\;\alpha \leq 8.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.42 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{2 + t_0}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 6.8 \cdot 10^{+186}:\\ \;\;\;\;\frac{1 + \frac{\beta}{t_0}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))))
   (if (<= alpha 8.5e+21)
     (/ (+ 1.0 (/ (- beta alpha) (+ 2.0 (+ (* 2.0 i) (+ alpha beta))))) 2.0)
     (if (<= alpha 1.42e+54)
       (/ (/ (+ 2.0 t_0) alpha) 2.0)
       (if (<= alpha 6.8e+186)
         (/ (+ 1.0 (/ beta t_0)) 2.0)
         (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (alpha <= 8.5e+21) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0;
	} else if (alpha <= 1.42e+54) {
		tmp = ((2.0 + t_0) / alpha) / 2.0;
	} else if (alpha <= 6.8e+186) {
		tmp = (1.0 + (beta / t_0)) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / 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) :: t_0
    real(8) :: tmp
    t_0 = beta + (2.0d0 * i)
    if (alpha <= 8.5d+21) then
        tmp = (1.0d0 + ((beta - alpha) / (2.0d0 + ((2.0d0 * i) + (alpha + beta))))) / 2.0d0
    else if (alpha <= 1.42d+54) then
        tmp = ((2.0d0 + t_0) / alpha) / 2.0d0
    else if (alpha <= 6.8d+186) then
        tmp = (1.0d0 + (beta / t_0)) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (alpha <= 8.5e+21) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0;
	} else if (alpha <= 1.42e+54) {
		tmp = ((2.0 + t_0) / alpha) / 2.0;
	} else if (alpha <= 6.8e+186) {
		tmp = (1.0 + (beta / t_0)) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	tmp = 0
	if alpha <= 8.5e+21:
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0
	elif alpha <= 1.42e+54:
		tmp = ((2.0 + t_0) / alpha) / 2.0
	elif alpha <= 6.8e+186:
		tmp = (1.0 + (beta / t_0)) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	tmp = 0.0
	if (alpha <= 8.5e+21)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(2.0 + Float64(Float64(2.0 * i) + Float64(alpha + beta))))) / 2.0);
	elseif (alpha <= 1.42e+54)
		tmp = Float64(Float64(Float64(2.0 + t_0) / alpha) / 2.0);
	elseif (alpha <= 6.8e+186)
		tmp = Float64(Float64(1.0 + Float64(beta / t_0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	tmp = 0.0;
	if (alpha <= 8.5e+21)
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0;
	elseif (alpha <= 1.42e+54)
		tmp = ((2.0 + t_0) / alpha) / 2.0;
	elseif (alpha <= 6.8e+186)
		tmp = (1.0 + (beta / t_0)) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 8.5e+21], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(N[(2.0 * i), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.42e+54], N[(N[(N[(2.0 + t$95$0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 6.8e+186], N[(N[(1.0 + N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 8.5e21

   1. Initial program 88.4%

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

   2. Taylor expanded in i around 0 98.7%

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

   if 8.5e21 < alpha < 1.41999999999999995e54

   1. Initial program 17.2%

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

   2. Taylor expanded in alpha around inf 5.6%

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

      1. mul-1-neg 5.6%

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

   4. Simplified 5.6%

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

   5. Taylor expanded in alpha around inf 87.0%

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

   if 1.41999999999999995e54 < alpha < 6.8000000000000001e186

   1. Initial program 29.4%

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

      1. div-inv 29.6%

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

      2. associate-+r+ 29.6%

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

      3. +-commutative 29.6%

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

      4. fma-udef 29.6%

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

   3. Applied egg-rr 29.6%

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

      1. associate-*l* 63.9%

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

      2. +-commutative 63.9%

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

      3. +-commutative 63.9%

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

   5. Simplified 63.9%

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

   6. Taylor expanded in alpha around 0 64.3%

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

   7. Taylor expanded in alpha around inf 64.3%

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

   if 6.8000000000000001e186 < alpha

   1. Initial program 1.2%

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

   3. Simplified 25.7%

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

   4. Taylor expanded in alpha around inf 81.3%

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

   5. Taylor expanded in beta around 0 62.1%

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

   6. Taylor expanded in i around 0 62.1%

      \[\leadsto \frac{\frac{\color{blue}{2 + 4 \cdot i}}{\alpha}}{2} \]
   7. Step-by-step derivation

      1. *-commutative 62.1%

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

   8. Simplified 62.1%

      \[\leadsto \frac{\frac{\color{blue}{2 + i \cdot 4}}{\alpha}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.42 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + 2 \cdot i\right)}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 6.8 \cdot 10^{+186}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2 \cdot i}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 79.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{1 - \frac{\alpha}{2 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2 \cdot i}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.2)
   (/ (- 1.0 (/ alpha (+ 2.0 (+ (* 2.0 i) (+ alpha beta))))) 2.0)
   (/ (+ 1.0 (/ beta (+ beta (* 2.0 i)))) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.2) {
		tmp = (1.0 - (alpha / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 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 (beta <= 4.2d0) then
        tmp = (1.0d0 - (alpha / (2.0d0 + ((2.0d0 * i) + (alpha + beta))))) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (beta + (2.0d0 * i)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.2) {
		tmp = (1.0 - (alpha / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.2:
		tmp = (1.0 - (alpha / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0
	else:
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.2)
		tmp = Float64(Float64(1.0 - Float64(alpha / Float64(2.0 + Float64(Float64(2.0 * i) + Float64(alpha + beta))))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(2.0 * i)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.2)
		tmp = (1.0 - (alpha / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0;
	else
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 4.2], N[(N[(1.0 - N[(alpha / N[(2.0 + N[(N[(2.0 * i), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.20000000000000018

   1. Initial program 73.5%

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

   2. Taylor expanded in alpha around inf 76.4%

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

      1. mul-1-neg 76.4%

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

   4. Simplified 76.4%

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

   if 4.20000000000000018 < beta

   1. Initial program 51.9%

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

      1. div-inv 51.9%

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

      2. associate-+r+ 51.9%

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

      3. +-commutative 51.9%

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

      4. fma-udef 51.9%

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

   3. Applied egg-rr 51.9%

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

      1. associate-*l* 88.6%

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

      2. +-commutative 88.6%

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

      3. +-commutative 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in alpha around 0 87.9%

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

   7. Taylor expanded in alpha around inf 85.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{1 - \frac{\alpha}{2 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2 \cdot i}}{2}\\ \end{array} \]

Alternative 7: 75.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;i \leq 3.8 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 8 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \mathbf{elif}\;i \leq 2.2 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))
   (if (<= i 3.8e-90)
     t_0
     (if (<= i 8e-79)
       (/ (/ (+ 2.0 (+ beta beta)) alpha) 2.0)
       (if (<= i 2.2e+149) t_0 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (i <= 3.8e-90) {
		tmp = t_0;
	} else if (i <= 8e-79) {
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	} else if (i <= 2.2e+149) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    if (i <= 3.8d-90) then
        tmp = t_0
    else if (i <= 8d-79) then
        tmp = ((2.0d0 + (beta + beta)) / alpha) / 2.0d0
    else if (i <= 2.2d+149) then
        tmp = t_0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (i <= 3.8e-90) {
		tmp = t_0;
	} else if (i <= 8e-79) {
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	} else if (i <= 2.2e+149) {
		tmp = t_0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0
	tmp = 0
	if i <= 3.8e-90:
		tmp = t_0
	elif i <= 8e-79:
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0
	elif i <= 2.2e+149:
		tmp = t_0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0)
	tmp = 0.0
	if (i <= 3.8e-90)
		tmp = t_0;
	elseif (i <= 8e-79)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + beta)) / alpha) / 2.0);
	elseif (i <= 2.2e+149)
		tmp = t_0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	tmp = 0.0;
	if (i <= 3.8e-90)
		tmp = t_0;
	elseif (i <= 8e-79)
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	elseif (i <= 2.2e+149)
		tmp = t_0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[i, 3.8e-90], t$95$0, If[LessEqual[i, 8e-79], N[(N[(N[(2.0 + N[(beta + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[i, 2.2e+149], t$95$0, 0.5]]]]

Derivation

1. Split input into 3 regimes

2. if i < 3.8e-90 or 8e-79 < i < 2.2e149

   1. Initial program 64.3%

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

   2. Taylor expanded in beta around inf 62.3%

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

      1. mul-1-neg 62.3%

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

      2. sub-neg 62.3%

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

   4. Simplified 62.3%

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

   5. Taylor expanded in alpha around 0 60.3%

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

   6. Taylor expanded in i around 0 74.3%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]
   7. Step-by-step derivation

      1. +-commutative 74.3%

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

   8. Simplified 74.3%

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

   if 3.8e-90 < i < 8e-79

   1. Initial program 1.6%

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

   3. Simplified 5.8%

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

   4. Taylor expanded in alpha around inf 100.0%

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

   5. Taylor expanded in i around 0 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. remove-double-neg 100.0%

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

   7. Simplified 100.0%

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

   if 2.2e149 < i

   1. Initial program 73.7%

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

   3. Simplified 95.6%

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

      1. div-inv 95.8%

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

   5. Applied egg-rr 95.8%

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

      1. +-commutative 95.8%

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

   7. Simplified 95.8%

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

   8. Taylor expanded in i around inf 92.3%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.8 \cdot 10^{-90}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;i \leq 8 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \mathbf{elif}\;i \leq 2.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8: 78.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 73.5%

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

   3. Simplified 78.3%

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

      1. div-inv 78.5%

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

   5. Applied egg-rr 78.5%

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

      1. +-commutative 78.5%

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

   7. Simplified 78.5%

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

   8. Taylor expanded in i around inf 74.7%

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

   if 2 < beta

   1. Initial program 51.9%

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

      1. div-inv 51.9%

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

      2. associate-+r+ 51.9%

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

      3. +-commutative 51.9%

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

      4. fma-udef 51.9%

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

   3. Applied egg-rr 51.9%

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

      1. associate-*l* 88.6%

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

      2. +-commutative 88.6%

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

      3. +-commutative 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in alpha around 0 87.9%

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

   7. Taylor expanded in alpha around inf 85.1%

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

4. Final simplification 78.4%

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

Alternative 9: 76.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 2e+149) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2e+149) {
		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 <= 2d+149) 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 <= 2e+149) {
		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 <= 2e+149:
		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 <= 2e+149)
		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 <= 2e+149)
		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, 2e+149], 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 < 2.0000000000000001e149

   1. Initial program 62.6%

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

   2. Taylor expanded in beta around inf 60.8%

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

      1. mul-1-neg 60.8%

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

      2. sub-neg 60.8%

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

   4. Simplified 60.8%

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

   5. Taylor expanded in alpha around 0 58.8%

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

   6. Taylor expanded in i around 0 72.4%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]
   7. Step-by-step derivation

      1. +-commutative 72.4%

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

   8. Simplified 72.4%

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

   if 2.0000000000000001e149 < i

   1. Initial program 73.7%

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

   3. Simplified 95.6%

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

      1. div-inv 95.8%

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

   5. Applied egg-rr 95.8%

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

      1. +-commutative 95.8%

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

   7. Simplified 95.8%

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

   8. Taylor expanded in i around inf 92.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10: 72.6% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+65}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 2.05e+65) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.05e+65) {
		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 <= 2.05d+65) 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 <= 2.05e+65) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.05e+65:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.05e+65)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.05e+65)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2.05e+65], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0500000000000001e65

   1. Initial program 74.6%

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

   3. Simplified 78.8%

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

      1. div-inv 78.9%

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

   5. Applied egg-rr 78.9%

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

      1. +-commutative 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in i around inf 71.8%

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

   if 2.0500000000000001e65 < beta

   1. Initial program 40.5%

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

      1. associate-/l/ 38.8%

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

      2. associate-+l+ 38.8%

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

      3. associate-+l+ 38.8%

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

   3. Simplified 38.8%

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

   4. Taylor expanded in beta around inf 68.5%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+65}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 61.9% 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 65.7%

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

3. Simplified 82.0%

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

   1. div-inv 82.1%

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

5. Applied egg-rr 82.1%

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

   1. +-commutative 82.1%

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

7. Simplified 82.1%

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

8. Taylor expanded in i around inf 62.7%

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

9. Final simplification 62.7%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023332 
(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))