Octave 3.8, jcobi/2

Percentage Accurate: 62.8% → 97.8%

Time: 14.9s

Alternatives: 11

Speedup: 4.8×

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: 62.8% 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: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.99999995:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

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))
        -0.99999995)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)
     (/
      (fma
       (+ 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)) <= -0.99999995) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = fma((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)) <= -0.99999995)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0)))) / alpha) / 2.0);
	else
		tmp = Float64(fma(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], -0.99999995], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], 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] + 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 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999949999999971

   1. Initial program 2.8%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in alpha around inf 88.7%

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

   if -0.999999949999999971 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 78.8%

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

   3. Simplified 99.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999995:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.99999995:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} + 1}{2}\\ \end{array} \end{array} \]

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))
        -0.99999995)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)
     (/
      (+
       (*
        (/ (+ alpha beta) (+ (+ alpha beta) (fma 2.0 i 2.0)))
        (/ (- beta alpha) (+ beta (fma 2.0 i alpha))))
       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)) <= -0.99999995) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = ((((alpha + beta) / ((alpha + beta) + fma(2.0, i, 2.0))) * ((beta - alpha) / (beta + fma(2.0, i, alpha)))) + 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)) <= -0.99999995)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))) * Float64(Float64(beta - alpha) / Float64(beta + fma(2.0, i, alpha)))) + 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], -0.99999995], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $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 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999949999999971

   1. Initial program 2.8%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in alpha around inf 88.7%

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

   if -0.999999949999999971 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 78.8%

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

   3. Simplified 99.8%

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

      1. associate-*r/ 78.8%

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

      2. *-commutative 78.8%

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

      3. fma-undefine 78.8%

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

      4. +-commutative 78.8%

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

      5. fma-define 78.8%

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

      6. associate-+r+ 78.8%

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

      7. associate-/l/ 78.1%

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

      8. +-commutative 78.1%

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

      9. associate-+r+ 78.1%

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

      10. times-frac 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. fma-define 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999995:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.99999995:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{t\_1 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta \cdot \left(\left(\frac{\alpha}{\beta} + \frac{2}{\beta}\right) + 1\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.99999995)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)
     (if (<= t_1 2e-27)
       (/ (+ t_1 1.0) 2.0)
       (/
        (+
         (/ (- beta alpha) (* beta (+ (+ (/ alpha beta) (/ 2.0 beta)) 1.0)))
         1.0)
        2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.99999995) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else if (t_1 <= 2e-27) {
		tmp = (t_1 + 1.0) / 2.0;
	} else {
		tmp = (((beta - alpha) / (beta * (((alpha / beta) + (2.0 / beta)) + 1.0))) + 1.0) / 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 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)
    if (t_1 <= (-0.99999995d0)) then
        tmp = (((beta - beta) + (2.0d0 + ((beta * 2.0d0) + (i * 4.0d0)))) / alpha) / 2.0d0
    else if (t_1 <= 2d-27) then
        tmp = (t_1 + 1.0d0) / 2.0d0
    else
        tmp = (((beta - alpha) / (beta * (((alpha / beta) + (2.0d0 / beta)) + 1.0d0))) + 1.0d0) / 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 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.99999995) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else if (t_1 <= 2e-27) {
		tmp = (t_1 + 1.0) / 2.0;
	} else {
		tmp = (((beta - alpha) / (beta * (((alpha / beta) + (2.0 / beta)) + 1.0))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)
	tmp = 0
	if t_1 <= -0.99999995:
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0
	elif t_1 <= 2e-27:
		tmp = (t_1 + 1.0) / 2.0
	else:
		tmp = (((beta - alpha) / (beta * (((alpha / beta) + (2.0 / beta)) + 1.0))) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -0.99999995)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0)))) / alpha) / 2.0);
	elseif (t_1 <= 2e-27)
		tmp = Float64(Float64(t_1 + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(beta * Float64(Float64(Float64(alpha / beta) + Float64(2.0 / beta)) + 1.0))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	tmp = 0.0;
	if (t_1 <= -0.99999995)
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	elseif (t_1 <= 2e-27)
		tmp = (t_1 + 1.0) / 2.0;
	else
		tmp = (((beta - alpha) / (beta * (((alpha / beta) + (2.0 / beta)) + 1.0))) + 1.0) / 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[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.99999995], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e-27], N[(N[(t$95$1 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta * N[(N[(N[(alpha / beta), $MachinePrecision] + N[(2.0 / beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999949999999971

   1. Initial program 2.8%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in alpha around inf 88.7%

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

   if -0.999999949999999971 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 2.0000000000000001e-27

   1. Initial program 99.7%

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

   if 2.0000000000000001e-27 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 40.4%

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

   3. Simplified 99.9%

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

      1. associate-*r/ 40.4%

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

      2. *-commutative 40.4%

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

      3. fma-undefine 40.4%

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

      4. +-commutative 40.4%

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

      5. fma-define 40.4%

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

      6. associate-+r+ 40.4%

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

      7. associate-/l/ 38.5%

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

      8. +-commutative 38.5%

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

      9. associate-+r+ 38.5%

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

      10. times-frac 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

      13. fma-define 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in i around 0 95.3%

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

      1. associate-+r+ 95.3%

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

   9. Simplified 95.3%

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

   10. Taylor expanded in beta around inf 95.3%

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

       1. +-commutative 95.3%

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

       2. associate-*r/ 95.3%

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

       3. metadata-eval 95.3%

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

   12. Simplified 95.3%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999995:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta \cdot \left(\left(\frac{\alpha}{\beta} + \frac{2}{\beta}\right) + 1\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.7e+82)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.7e+82) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (((beta - beta) + (2.0 + ((beta * 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) :: tmp
    if (alpha <= 2.7d+82) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = (((beta - beta) + (2.0d0 + ((beta * 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 tmp;
	if (alpha <= 2.7e+82) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.7e+82:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.7e+82)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0)))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.7e+82)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.7e+82], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.6999999999999999e82

   1. Initial program 77.7%

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

   3. Simplified 95.6%

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

      1. associate-*r/ 77.7%

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

      2. *-commutative 77.7%

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

      3. fma-undefine 77.7%

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

      4. +-commutative 77.7%

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

      5. fma-define 77.7%

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

      6. associate-+r+ 77.7%

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

      7. associate-/l/ 77.1%

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

      8. +-commutative 77.1%

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

      9. associate-+r+ 77.1%

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

      10. times-frac 95.6%

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

      11. +-commutative 95.6%

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

      12. +-commutative 95.6%

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

      13. fma-define 95.6%

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

   6. Applied egg-rr 95.6%

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

   7. Taylor expanded in i around 0 83.7%

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

      1. associate-+r+ 83.7%

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

   9. Simplified 83.7%

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

   10. Taylor expanded in alpha around 0 86.1%

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

   if 2.6999999999999999e82 < alpha

   1. Initial program 9.9%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in alpha around inf 71.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.75 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(\frac{2}{\alpha} - -0.5 \cdot \frac{2 + \beta \cdot 2}{\alpha \cdot i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.75e+82)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (* i (- (/ 2.0 alpha) (* -0.5 (/ (+ 2.0 (* beta 2.0)) (* alpha i)))))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.75e+82) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = i * ((2.0 / alpha) - (-0.5 * ((2.0 + (beta * 2.0)) / (alpha * i))));
	}
	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 <= 2.75d+82) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = i * ((2.0d0 / alpha) - ((-0.5d0) * ((2.0d0 + (beta * 2.0d0)) / (alpha * i))))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.75e+82) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = i * ((2.0 / alpha) - (-0.5 * ((2.0 + (beta * 2.0)) / (alpha * i))));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.75e+82:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = i * ((2.0 / alpha) - (-0.5 * ((2.0 + (beta * 2.0)) / (alpha * i))))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.75e+82)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(i * Float64(Float64(2.0 / alpha) - Float64(-0.5 * Float64(Float64(2.0 + Float64(beta * 2.0)) / Float64(alpha * i)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.75e+82)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = i * ((2.0 / alpha) - (-0.5 * ((2.0 + (beta * 2.0)) / (alpha * i))));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.75e+82], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(i * N[(N[(2.0 / alpha), $MachinePrecision] - N[(-0.5 * N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.74999999999999998e82

   1. Initial program 77.7%

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

   3. Simplified 95.6%

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

      1. associate-*r/ 77.7%

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

      2. *-commutative 77.7%

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

      3. fma-undefine 77.7%

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

      4. +-commutative 77.7%

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

      5. fma-define 77.7%

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

      6. associate-+r+ 77.7%

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

      7. associate-/l/ 77.1%

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

      8. +-commutative 77.1%

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

      9. associate-+r+ 77.1%

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

      10. times-frac 95.6%

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

      11. +-commutative 95.6%

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

      12. +-commutative 95.6%

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

      13. fma-define 95.6%

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

   6. Applied egg-rr 95.6%

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

   7. Taylor expanded in i around 0 83.7%

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

      1. associate-+r+ 83.7%

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

   9. Simplified 83.7%

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

   10. Taylor expanded in alpha around 0 86.1%

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

   if 2.74999999999999998e82 < alpha

   1. Initial program 9.9%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in alpha around inf 71.4%

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

   6. Taylor expanded in i around -inf 71.1%

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(-0.5 \cdot \frac{\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}}{i} - 2 \cdot \frac{1}{\alpha}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 71.1%

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

      2. mul-1-neg 71.1%

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

      3. distribute-lft1-in 71.1%

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

      4. metadata-eval 71.1%

         \[\leadsto \left(-i\right) \cdot \left(-0.5 \cdot \frac{\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}}{i} - 2 \cdot \frac{1}{\alpha}\right) \]

      5. mul0-lft 71.1%

         \[\leadsto \left(-i\right) \cdot \left(-0.5 \cdot \frac{\color{blue}{0} - -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}}{i} - 2 \cdot \frac{1}{\alpha}\right) \]

      6. neg-sub0 71.1%

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

      7. mul-1-neg 71.1%

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

      8. remove-double-neg 71.1%

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

      9. associate-/r* 69.2%

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

      10. associate-*r/ 69.2%

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

      11. metadata-eval 69.2%

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

   8. Simplified 69.2%

      \[\leadsto \color{blue}{\left(-i\right) \cdot \left(-0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha \cdot i} - \frac{2}{\alpha}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.75 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(\frac{2}{\alpha} - -0.5 \cdot \frac{2 + \beta \cdot 2}{\alpha \cdot i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.75e+82)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.75e+82) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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) :: tmp
    if (alpha <= 1.75d+82) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 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 tmp;
	if (alpha <= 1.75e+82) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.75e+82:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.75e+82)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 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)
	tmp = 0.0;
	if (alpha <= 1.75e+82)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.75e+82], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $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 2 regimes

2. if alpha < 1.75e82

   1. Initial program 77.7%

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

   3. Simplified 95.6%

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

      1. associate-*r/ 77.7%

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

      2. *-commutative 77.7%

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

      3. fma-undefine 77.7%

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

      4. +-commutative 77.7%

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

      5. fma-define 77.7%

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

      6. associate-+r+ 77.7%

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

      7. associate-/l/ 77.1%

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

      8. +-commutative 77.1%

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

      9. associate-+r+ 77.1%

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

      10. times-frac 95.6%

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

      11. +-commutative 95.6%

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

      12. +-commutative 95.6%

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

      13. fma-define 95.6%

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

   6. Applied egg-rr 95.6%

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

   7. Taylor expanded in i around 0 83.7%

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

      1. associate-+r+ 83.7%

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

   9. Simplified 83.7%

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

   10. Taylor expanded in alpha around 0 86.1%

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

   if 1.75e82 < alpha

   1. Initial program 9.9%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in alpha around inf 71.4%

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

   6. Taylor expanded in beta around 0 54.1%

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

      1. *-commutative 54.1%

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

   8. Simplified 54.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.5e+164)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.5e+164) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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 <= 1.5d+164) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 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 <= 1.5e+164) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.5e+164:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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 <= 1.5e+164)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 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 <= 1.5e+164)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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, 1.5e+164], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $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 2 regimes

2. if alpha < 1.5e164

   1. Initial program 71.8%

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

   3. Simplified 90.4%

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

      1. associate-*r/ 71.8%

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

      2. *-commutative 71.8%

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

      3. fma-undefine 71.8%

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

      4. +-commutative 71.8%

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

      5. fma-define 71.8%

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

      6. associate-+r+ 71.8%

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

      7. associate-/l/ 71.2%

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

      8. +-commutative 71.2%

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

      9. associate-+r+ 71.2%

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

      10. times-frac 90.4%

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

      11. +-commutative 90.4%

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

      12. +-commutative 90.4%

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

      13. fma-define 90.4%

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

   6. Applied egg-rr 90.4%

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

   7. Taylor expanded in i around 0 76.8%

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

      1. associate-+r+ 76.8%

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

   9. Simplified 76.8%

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

   10. Taylor expanded in alpha around 0 81.7%

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

   if 1.5e164 < alpha

   1. Initial program 1.1%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in alpha around inf 80.8%

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

   6. Taylor expanded in i around 0 59.4%

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

      1. distribute-rgt1-in 59.4%

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

      2. metadata-eval 59.4%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot \beta - -1 \cdot \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]

      3. mul0-lft 59.4%

         \[\leadsto \frac{\frac{\color{blue}{0} - -1 \cdot \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]

      4. neg-sub0 59.4%

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

      5. mul-1-neg 59.4%

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

      6. remove-double-neg 59.4%

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

   8. Simplified 59.4%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{2 + \frac{1}{i}}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.2e+82)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (* i (/ (+ 2.0 (/ 1.0 i)) alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.2e+82) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = i * ((2.0 + (1.0 / i)) / alpha);
	}
	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 <= 2.2d+82) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = i * ((2.0d0 + (1.0d0 / i)) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.2e+82) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = i * ((2.0 + (1.0 / i)) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.2e+82:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = i * ((2.0 + (1.0 / i)) / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.2e+82)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(i * Float64(Float64(2.0 + Float64(1.0 / i)) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.2e+82)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = i * ((2.0 + (1.0 / i)) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.2e+82], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(i * N[(N[(2.0 + N[(1.0 / i), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.2000000000000001e82

   1. Initial program 77.7%

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

   3. Simplified 95.6%

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

      1. associate-*r/ 77.7%

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

      2. *-commutative 77.7%

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

      3. fma-undefine 77.7%

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

      4. +-commutative 77.7%

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

      5. fma-define 77.7%

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

      6. associate-+r+ 77.7%

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

      7. associate-/l/ 77.1%

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

      8. +-commutative 77.1%

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

      9. associate-+r+ 77.1%

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

      10. times-frac 95.6%

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

      11. +-commutative 95.6%

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

      12. +-commutative 95.6%

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

      13. fma-define 95.6%

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

   6. Applied egg-rr 95.6%

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

   7. Taylor expanded in i around 0 83.7%

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

      1. associate-+r+ 83.7%

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

   9. Simplified 83.7%

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

   10. Taylor expanded in alpha around 0 86.1%

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

   if 2.2000000000000001e82 < alpha

   1. Initial program 9.9%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in alpha around inf 71.4%

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

   6. Taylor expanded in i around -inf 71.1%

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(-0.5 \cdot \frac{\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}}{i} - 2 \cdot \frac{1}{\alpha}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 71.1%

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

      2. mul-1-neg 71.1%

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

      3. distribute-lft1-in 71.1%

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

      4. metadata-eval 71.1%

         \[\leadsto \left(-i\right) \cdot \left(-0.5 \cdot \frac{\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}}{i} - 2 \cdot \frac{1}{\alpha}\right) \]

      5. mul0-lft 71.1%

         \[\leadsto \left(-i\right) \cdot \left(-0.5 \cdot \frac{\color{blue}{0} - -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}}{i} - 2 \cdot \frac{1}{\alpha}\right) \]

      6. neg-sub0 71.1%

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

      7. mul-1-neg 71.1%

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

      8. remove-double-neg 71.1%

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

      9. associate-/r* 69.2%

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

      10. associate-*r/ 69.2%

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

      11. metadata-eval 69.2%

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

   8. Simplified 69.2%

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

   9. Taylor expanded in beta around 0 53.7%

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

       1. associate-*r/ 53.7%

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

       2. metadata-eval 53.7%

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

       3. +-commutative 53.7%

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

       4. associate-/r* 53.9%

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

   11. Simplified 53.9%

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

   12. Taylor expanded in alpha around 0 54.1%

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

       1. associate-/l* 53.8%

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

   14. Simplified 53.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{2 + \frac{1}{i}}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{+107}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2e+107) 0.5 (/ 1.0 alpha)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2e+107) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	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 <= 2d+107) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2e+107) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2e+107:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2e+107)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2e+107)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2e+107], 0.5, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.9999999999999999e107

   1. Initial program 76.4%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in i around inf 70.0%

      \[\leadsto \color{blue}{0.5} \]

   if 1.9999999999999999e107 < alpha

   1. Initial program 3.5%

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

   3. Simplified 32.2%

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

   5. Taylor expanded in alpha around inf 73.7%

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

   6. Taylor expanded in i around -inf 73.3%

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(-0.5 \cdot \frac{\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}}{i} - 2 \cdot \frac{1}{\alpha}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 73.3%

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

      2. mul-1-neg 73.3%

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

      3. distribute-lft1-in 73.3%

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

      4. metadata-eval 73.3%

         \[\leadsto \left(-i\right) \cdot \left(-0.5 \cdot \frac{\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}}{i} - 2 \cdot \frac{1}{\alpha}\right) \]

      5. mul0-lft 73.3%

         \[\leadsto \left(-i\right) \cdot \left(-0.5 \cdot \frac{\color{blue}{0} - -1 \cdot \frac{2 + 2 \cdot \beta}{\alpha}}{i} - 2 \cdot \frac{1}{\alpha}\right) \]

      6. neg-sub0 73.3%

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

      7. mul-1-neg 73.3%

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

      8. remove-double-neg 73.3%

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

      9. associate-/r* 71.2%

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

      10. associate-*r/ 71.2%

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

      11. metadata-eval 71.2%

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

   8. Simplified 71.2%

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

   9. Taylor expanded in beta around 0 52.9%

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

       1. associate-*r/ 52.9%

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

       2. metadata-eval 52.9%

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

       3. +-commutative 52.9%

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

       4. associate-/r* 53.2%

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

   11. Simplified 53.2%

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

   12. Taylor expanded in i around 0 34.4%

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

Alternative 10: 72.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+64}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 1.25e+64) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.25e+64) {
		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 <= 1.25d+64) 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 <= 1.25e+64) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.25e+64:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.25e+64)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.25e+64)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.25e+64], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.25e64

   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 78.0%

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

   5. Taylor expanded in i around inf 73.6%

      \[\leadsto \color{blue}{0.5} \]

   if 1.25e64 < beta

   1. Initial program 30.0%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in beta around inf 75.8%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+64}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.2% 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 61.6%

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

3. Simplified 68.1%

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

5. Taylor expanded in i around inf 59.8%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(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))