Octave 3.8, jcobi/2

Percentage Accurate: 63.3% → 97.7%

Time: 21.1s

Alternatives: 15

Speedup: 2.1×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.3% 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: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha} - -4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

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)) -0.999999)
     (/ (- (/ (+ beta (- beta -2.0)) alpha) (* -4.0 (/ i 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)) <= -0.999999) {
		tmp = (((beta + (beta - -2.0)) / alpha) - (-4.0 * (i / 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)) <= -0.999999)
		tmp = Float64(Float64(Float64(Float64(beta + Float64(beta - -2.0)) / alpha) - Float64(-4.0 * Float64(i / 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], -0.999999], N[(N[(N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(-4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $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))) < -0.999998999999999971

   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 15.2%

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

   5. Taylor expanded in alpha around -inf 90.0%

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

   6. Taylor expanded in i around 0 90.1%

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

      1. associate--l+ 90.1%

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

      2. distribute-lft1-in 90.1%

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

      3. metadata-eval 90.1%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      4. associate-/l* 90.1%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{\frac{0 \cdot \beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      5. metadata-eval 90.1%

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

      6. distribute-rgt1-in 90.1%

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

      7. associate-*r/ 90.1%

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

      8. div-sub 90.1%

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

   8. Simplified 90.1%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \frac{0 - \left(-\left(\left(-2 - \beta\right) - \beta\right)\right)}{\alpha}\right)}}{2} \]

   if -0.999998999999999971 < (/.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 79.1%

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

   3. Simplified 99.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha} - -4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha} - -4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \end{array} \]

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)) -0.999999)
     (/ (- (/ (+ beta (- beta -2.0)) alpha) (* -4.0 (/ i alpha))) 2.0)
     (/
      (+
       1.0
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        (+ alpha (+ beta (fma 2.0 i 2.0)))))
      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)) <= -0.999999) {
		tmp = (((beta + (beta - -2.0)) / alpha) - (-4.0 * (i / alpha))) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / (alpha + (beta + fma(2.0, i, 2.0))))) / 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)) <= -0.999999)
		tmp = Float64(Float64(Float64(Float64(beta + Float64(beta - -2.0)) / alpha) - Float64(-4.0 * Float64(i / alpha))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0))))) / 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], -0.999999], N[(N[(N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(-4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999998999999999971

   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 15.2%

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

   5. Taylor expanded in alpha around -inf 90.0%

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

   6. Taylor expanded in i around 0 90.1%

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

      1. associate--l+ 90.1%

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

      2. distribute-lft1-in 90.1%

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

      3. metadata-eval 90.1%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      4. associate-/l* 90.1%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{\frac{0 \cdot \beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      5. metadata-eval 90.1%

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

      6. distribute-rgt1-in 90.1%

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

      7. associate-*r/ 90.1%

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

      8. div-sub 90.1%

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

   8. Simplified 90.1%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \frac{0 - \left(-\left(\left(-2 - \beta\right) - \beta\right)\right)}{\alpha}\right)}}{2} \]

   if -0.999998999999999971 < (/.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 79.1%

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

   3. Simplified 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
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha} - -4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha} - -4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \end{array} \]

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)) -0.999999)
     (/ (- (/ (+ beta (- beta -2.0)) alpha) (* -4.0 (/ i alpha))) 2.0)
     (/
      (+
       1.0
       (/
        (* (- beta alpha) (/ beta (+ beta (* 2.0 i))))
        (+ alpha (+ beta (fma 2.0 i 2.0)))))
      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)) <= -0.999999) {
		tmp = (((beta + (beta - -2.0)) / alpha) - (-4.0 * (i / alpha))) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / (beta + (2.0 * i)))) / (alpha + (beta + fma(2.0, i, 2.0))))) / 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)) <= -0.999999)
		tmp = Float64(Float64(Float64(Float64(beta + Float64(beta - -2.0)) / alpha) - Float64(-4.0 * Float64(i / alpha))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(beta / Float64(beta + Float64(2.0 * i)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0))))) / 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], -0.999999], N[(N[(N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(-4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999998999999999971

   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 15.2%

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

   5. Taylor expanded in alpha around -inf 90.0%

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

   6. Taylor expanded in i around 0 90.1%

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

      1. associate--l+ 90.1%

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

      2. distribute-lft1-in 90.1%

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

      3. metadata-eval 90.1%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      4. associate-/l* 90.1%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{\frac{0 \cdot \beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      5. metadata-eval 90.1%

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

      6. distribute-rgt1-in 90.1%

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

      7. associate-*r/ 90.1%

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

      8. div-sub 90.1%

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

   8. Simplified 90.1%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \frac{0 - \left(-\left(\left(-2 - \beta\right) - \beta\right)\right)}{\alpha}\right)}}{2} \]

   if -0.999998999999999971 < (/.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 79.1%

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

   3. Simplified 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. Taylor expanded in alpha around 0 99.3%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha} - -4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha} - -4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \end{array} \]

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)) -0.5)
     (/ (- (/ (+ beta (- beta -2.0)) alpha) (* -4.0 (/ i alpha))) 2.0)
     (/ (+ 1.0 (/ beta (+ alpha (+ beta (fma 2.0 i 2.0))))) 2.0))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.5

   1. Initial program 4.0%

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

   3. Simplified 16.2%

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

   5. Taylor expanded in alpha around -inf 89.4%

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

   6. Taylor expanded in i around 0 89.4%

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

      1. associate--l+ 89.4%

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

      2. distribute-lft1-in 89.4%

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

      3. metadata-eval 89.4%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      4. associate-/l* 89.4%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{\frac{0 \cdot \beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      5. metadata-eval 89.4%

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

      6. distribute-rgt1-in 89.4%

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

      7. associate-*r/ 89.4%

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

      8. div-sub 89.4%

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

   8. Simplified 89.4%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \frac{0 - \left(-\left(\left(-2 - \beta\right) - \beta\right)\right)}{\alpha}\right)}}{2} \]

   if -0.5 < (/.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 79.1%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 97.8%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha} - -4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha} - -4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 \cdot i + \left(\beta + 2\right)}}{2}\\ \end{array} \end{array} \]

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)) -0.5)
     (/ (- (/ (+ beta (- beta -2.0)) alpha) (* -4.0 (/ i alpha))) 2.0)
     (/ (+ 1.0 (/ beta (+ (* 2.0 i) (+ beta 2.0)))) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5) {
		tmp = (((beta + (beta - -2.0)) / alpha) - (-4.0 * (i / alpha))) / 2.0;
	} else {
		tmp = (1.0 + (beta / ((2.0 * i) + (beta + 2.0)))) / 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) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.5d0)) then
        tmp = (((beta + (beta - (-2.0d0))) / alpha) - ((-4.0d0) * (i / alpha))) / 2.0d0
    else
        tmp = (1.0d0 + (beta / ((2.0d0 * i) + (beta + 2.0d0)))) / 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 tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5) {
		tmp = (((beta + (beta - -2.0)) / alpha) - (-4.0 * (i / alpha))) / 2.0;
	} else {
		tmp = (1.0 + (beta / ((2.0 * i) + (beta + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5:
		tmp = (((beta + (beta - -2.0)) / alpha) - (-4.0 * (i / alpha))) / 2.0
	else:
		tmp = (1.0 + (beta / ((2.0 * i) + (beta + 2.0)))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.5)
		tmp = Float64(Float64(Float64(Float64(beta + Float64(beta - -2.0)) / alpha) - Float64(-4.0 * Float64(i / alpha))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(2.0 * i) + Float64(beta + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5)
		tmp = (((beta + (beta - -2.0)) / alpha) - (-4.0 * (i / alpha))) / 2.0;
	else
		tmp = (1.0 + (beta / ((2.0 * i) + (beta + 2.0)))) / 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]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(-4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(N[(2.0 * i), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.5

   1. Initial program 4.0%

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

   3. Simplified 16.2%

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

   5. Taylor expanded in alpha around -inf 89.4%

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

   6. Taylor expanded in i around 0 89.4%

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

      1. associate--l+ 89.4%

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

      2. distribute-lft1-in 89.4%

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

      3. metadata-eval 89.4%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      4. associate-/l* 89.4%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{\frac{0 \cdot \beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      5. metadata-eval 89.4%

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

      6. distribute-rgt1-in 89.4%

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

      7. associate-*r/ 89.4%

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

      8. div-sub 89.4%

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

   8. Simplified 89.4%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \frac{0 - \left(-\left(\left(-2 - \beta\right) - \beta\right)\right)}{\alpha}\right)}}{2} \]

   if -0.5 < (/.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 79.1%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 97.8%

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

   6. Taylor expanded in alpha around 0 97.8%

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

      1. associate-+r+ 97.8%

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

      2. +-commutative 97.8%

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

   8. Simplified 97.8%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha} - -4 \cdot \frac{i}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 \cdot i + \left(\beta + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 \cdot i + \left(\beta + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 4 \cdot 10^{+116} \lor \neg \left(\alpha \leq 4.4 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) - i \cdot -4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.4e+54)
   (/ (+ 1.0 (/ beta (+ (* 2.0 i) (+ beta 2.0)))) 2.0)
   (if (or (<= alpha 4e+116) (not (<= alpha 4.4e+142)))
     (/ (/ (- (+ 2.0 (* beta 2.0)) (* i -4.0)) alpha) 2.0)
     (/ (+ 1.0 (/ beta (+ alpha beta))) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.4e+54) {
		tmp = (1.0 + (beta / ((2.0 * i) + (beta + 2.0)))) / 2.0;
	} else if ((alpha <= 4e+116) || !(alpha <= 4.4e+142)) {
		tmp = (((2.0 + (beta * 2.0)) - (i * -4.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (alpha + beta))) / 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 <= 4.4d+54) then
        tmp = (1.0d0 + (beta / ((2.0d0 * i) + (beta + 2.0d0)))) / 2.0d0
    else if ((alpha <= 4d+116) .or. (.not. (alpha <= 4.4d+142))) then
        tmp = (((2.0d0 + (beta * 2.0d0)) - (i * (-4.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (alpha + beta))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.4e+54) {
		tmp = (1.0 + (beta / ((2.0 * i) + (beta + 2.0)))) / 2.0;
	} else if ((alpha <= 4e+116) || !(alpha <= 4.4e+142)) {
		tmp = (((2.0 + (beta * 2.0)) - (i * -4.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.4e+54:
		tmp = (1.0 + (beta / ((2.0 * i) + (beta + 2.0)))) / 2.0
	elif (alpha <= 4e+116) or not (alpha <= 4.4e+142):
		tmp = (((2.0 + (beta * 2.0)) - (i * -4.0)) / alpha) / 2.0
	else:
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4.4e+54)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(2.0 * i) + Float64(beta + 2.0)))) / 2.0);
	elseif ((alpha <= 4e+116) || !(alpha <= 4.4e+142))
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) - Float64(i * -4.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(alpha + beta))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 4.4e+54)
		tmp = (1.0 + (beta / ((2.0 * i) + (beta + 2.0)))) / 2.0;
	elseif ((alpha <= 4e+116) || ~((alpha <= 4.4e+142)))
		tmp = (((2.0 + (beta * 2.0)) - (i * -4.0)) / alpha) / 2.0;
	else
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4.4e+54], N[(N[(1.0 + N[(beta / N[(N[(2.0 * i), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 4e+116], N[Not[LessEqual[alpha, 4.4e+142]], $MachinePrecision]], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] - N[(i * -4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 4.3999999999999998e54

   1. Initial program 81.1%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in beta around inf 96.3%

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

   6. Taylor expanded in alpha around 0 96.3%

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

      1. associate-+r+ 96.3%

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

      2. +-commutative 96.3%

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

   8. Simplified 96.3%

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

   if 4.3999999999999998e54 < alpha < 4.00000000000000006e116 or 4.39999999999999974e142 < alpha

   1. Initial program 7.5%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in alpha around -inf 75.5%

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

   6. Taylor expanded in i around 0 75.6%

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

      1. associate--l+ 75.6%

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

      2. distribute-lft1-in 75.6%

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

      3. metadata-eval 75.6%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      4. associate-/l* 75.6%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{\frac{0 \cdot \beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      5. metadata-eval 75.6%

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

      6. distribute-rgt1-in 75.6%

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

      7. associate-*r/ 75.6%

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

      8. div-sub 75.6%

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

   8. Simplified 75.6%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \frac{0 - \left(-\left(\left(-2 - \beta\right) - \beta\right)\right)}{\alpha}\right)}}{2} \]

   9. Taylor expanded in alpha around 0 75.5%

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

   if 4.00000000000000006e116 < alpha < 4.39999999999999974e142

   1. Initial program 58.5%

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

   3. Simplified 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. Taylor expanded in beta around inf 77.6%

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

   6. Taylor expanded in beta around inf 77.2%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 \cdot i + \left(\beta + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 4 \cdot 10^{+116} \lor \neg \left(\alpha \leq 4.4 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) - i \cdot -4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \beta}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.1 \cdot 10^{+54}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.8 \cdot 10^{+116} \lor \neg \left(\alpha \leq 2.1 \cdot 10^{+176}\right):\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5.1e+54)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (or (<= alpha 1.8e+116) (not (<= alpha 2.1e+176)))
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (+ 1.0 (/ beta (+ alpha beta))) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.1e+54) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 1.8e+116) || !(alpha <= 2.1e+176)) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (alpha + beta))) / 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 <= 5.1d+54) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if ((alpha <= 1.8d+116) .or. (.not. (alpha <= 2.1d+176))) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (alpha + beta))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.1e+54) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 1.8e+116) || !(alpha <= 2.1e+176)) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5.1e+54:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif (alpha <= 1.8e+116) or not (alpha <= 2.1e+176):
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5.1e+54)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif ((alpha <= 1.8e+116) || !(alpha <= 2.1e+176))
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(alpha + beta))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 5.1e+54)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif ((alpha <= 1.8e+116) || ~((alpha <= 2.1e+176)))
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5.1e+54], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 1.8e+116], N[Not[LessEqual[alpha, 2.1e+176]], $MachinePrecision]], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 5.10000000000000009e54

   1. Initial program 81.1%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in i around 0 80.8%

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

      1. associate-+r+ 80.8%

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

      2. +-commutative 80.8%

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

   7. Simplified 80.8%

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

   8. Taylor expanded in alpha around 0 88.1%

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

      1. +-commutative 88.1%

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

   10. Simplified 88.1%

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

   if 5.10000000000000009e54 < alpha < 1.79999999999999985e116 or 2.0999999999999999e176 < alpha

   1. Initial program 8.8%

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

   3. Simplified 26.1%

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

   5. Taylor expanded in i around 0 13.9%

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

      1. associate-+r+ 13.9%

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

      2. +-commutative 13.9%

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

   7. Simplified 13.9%

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

   8. Taylor expanded in alpha around inf 56.0%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 56.0%

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

   10. Simplified 56.0%

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

   if 1.79999999999999985e116 < alpha < 2.0999999999999999e176

   1. Initial program 21.0%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in beta around inf 56.6%

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

   6. Taylor expanded in beta around inf 56.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.1 \cdot 10^{+54}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.8 \cdot 10^{+116} \lor \neg \left(\alpha \leq 2.1 \cdot 10^{+176}\right):\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \beta}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 6.1 \cdot 10^{+54}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.45 \cdot 10^{+117} \lor \neg \left(\alpha \leq 8.2 \cdot 10^{+143}\right):\\ \;\;\;\;\frac{\frac{2 - i \cdot -4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 6.1e+54)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (or (<= alpha 2.45e+117) (not (<= alpha 8.2e+143)))
     (/ (/ (- 2.0 (* i -4.0)) alpha) 2.0)
     (/ (+ 1.0 (/ beta (+ alpha beta))) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 6.1e+54) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 2.45e+117) || !(alpha <= 8.2e+143)) {
		tmp = ((2.0 - (i * -4.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (alpha + beta))) / 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 <= 6.1d+54) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if ((alpha <= 2.45d+117) .or. (.not. (alpha <= 8.2d+143))) then
        tmp = ((2.0d0 - (i * (-4.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (alpha + beta))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 6.1e+54) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 2.45e+117) || !(alpha <= 8.2e+143)) {
		tmp = ((2.0 - (i * -4.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 6.1e+54:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif (alpha <= 2.45e+117) or not (alpha <= 8.2e+143):
		tmp = ((2.0 - (i * -4.0)) / alpha) / 2.0
	else:
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 6.1e+54)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif ((alpha <= 2.45e+117) || !(alpha <= 8.2e+143))
		tmp = Float64(Float64(Float64(2.0 - Float64(i * -4.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(alpha + beta))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 6.1e+54)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif ((alpha <= 2.45e+117) || ~((alpha <= 8.2e+143)))
		tmp = ((2.0 - (i * -4.0)) / alpha) / 2.0;
	else
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 6.1e+54], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 2.45e+117], N[Not[LessEqual[alpha, 8.2e+143]], $MachinePrecision]], N[(N[(N[(2.0 - N[(i * -4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 6.0999999999999998e54

   1. Initial program 81.1%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in i around 0 80.8%

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

      1. associate-+r+ 80.8%

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

      2. +-commutative 80.8%

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

   7. Simplified 80.8%

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

   8. Taylor expanded in alpha around 0 88.1%

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

      1. +-commutative 88.1%

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

   10. Simplified 88.1%

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

   if 6.0999999999999998e54 < alpha < 2.45e117 or 8.2000000000000007e143 < alpha

   1. Initial program 7.5%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in alpha around -inf 75.5%

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

   6. Taylor expanded in i around 0 75.6%

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

      1. associate--l+ 75.6%

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

      2. distribute-lft1-in 75.6%

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

      3. metadata-eval 75.6%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      4. associate-/l* 75.6%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{\frac{0 \cdot \beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      5. metadata-eval 75.6%

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

      6. distribute-rgt1-in 75.6%

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

      7. associate-*r/ 75.6%

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

      8. div-sub 75.6%

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

   8. Simplified 75.6%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \frac{0 - \left(-\left(\left(-2 - \beta\right) - \beta\right)\right)}{\alpha}\right)}}{2} \]

   9. Taylor expanded in beta around 0 62.6%

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

       1. associate-*r/ 62.6%

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

       2. metadata-eval 62.6%

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

   11. Simplified 62.6%

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

   12. Taylor expanded in alpha around 0 62.6%

       \[\leadsto \frac{-1 \cdot \color{blue}{\frac{-4 \cdot i - 2}{\alpha}}}{2} \]

   if 2.45e117 < alpha < 8.2000000000000007e143

   1. Initial program 58.5%

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

   3. Simplified 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. Taylor expanded in beta around inf 77.6%

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

   6. Taylor expanded in beta around inf 77.2%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6.1 \cdot 10^{+54}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.45 \cdot 10^{+117} \lor \neg \left(\alpha \leq 8.2 \cdot 10^{+143}\right):\\ \;\;\;\;\frac{\frac{2 - i \cdot -4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \beta}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 \cdot i + \left(\beta + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.3 \cdot 10^{+120} \lor \neg \left(\alpha \leq 6.8 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{\frac{2 - i \cdot -4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.4e+54)
   (/ (+ 1.0 (/ beta (+ (* 2.0 i) (+ beta 2.0)))) 2.0)
   (if (or (<= alpha 1.3e+120) (not (<= alpha 6.8e+137)))
     (/ (/ (- 2.0 (* i -4.0)) alpha) 2.0)
     (/ (+ 1.0 (/ beta (+ alpha beta))) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.4e+54) {
		tmp = (1.0 + (beta / ((2.0 * i) + (beta + 2.0)))) / 2.0;
	} else if ((alpha <= 1.3e+120) || !(alpha <= 6.8e+137)) {
		tmp = ((2.0 - (i * -4.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (alpha + beta))) / 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 <= 4.4d+54) then
        tmp = (1.0d0 + (beta / ((2.0d0 * i) + (beta + 2.0d0)))) / 2.0d0
    else if ((alpha <= 1.3d+120) .or. (.not. (alpha <= 6.8d+137))) then
        tmp = ((2.0d0 - (i * (-4.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (alpha + beta))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.4e+54) {
		tmp = (1.0 + (beta / ((2.0 * i) + (beta + 2.0)))) / 2.0;
	} else if ((alpha <= 1.3e+120) || !(alpha <= 6.8e+137)) {
		tmp = ((2.0 - (i * -4.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.4e+54:
		tmp = (1.0 + (beta / ((2.0 * i) + (beta + 2.0)))) / 2.0
	elif (alpha <= 1.3e+120) or not (alpha <= 6.8e+137):
		tmp = ((2.0 - (i * -4.0)) / alpha) / 2.0
	else:
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4.4e+54)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(2.0 * i) + Float64(beta + 2.0)))) / 2.0);
	elseif ((alpha <= 1.3e+120) || !(alpha <= 6.8e+137))
		tmp = Float64(Float64(Float64(2.0 - Float64(i * -4.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(alpha + beta))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 4.4e+54)
		tmp = (1.0 + (beta / ((2.0 * i) + (beta + 2.0)))) / 2.0;
	elseif ((alpha <= 1.3e+120) || ~((alpha <= 6.8e+137)))
		tmp = ((2.0 - (i * -4.0)) / alpha) / 2.0;
	else
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4.4e+54], N[(N[(1.0 + N[(beta / N[(N[(2.0 * i), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 1.3e+120], N[Not[LessEqual[alpha, 6.8e+137]], $MachinePrecision]], N[(N[(N[(2.0 - N[(i * -4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 4.3999999999999998e54

   1. Initial program 81.1%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in beta around inf 96.3%

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

   6. Taylor expanded in alpha around 0 96.3%

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

      1. associate-+r+ 96.3%

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

      2. +-commutative 96.3%

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

   8. Simplified 96.3%

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

   if 4.3999999999999998e54 < alpha < 1.2999999999999999e120 or 6.79999999999999973e137 < alpha

   1. Initial program 7.5%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in alpha around -inf 75.5%

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

   6. Taylor expanded in i around 0 75.6%

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

      1. associate--l+ 75.6%

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

      2. distribute-lft1-in 75.6%

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

      3. metadata-eval 75.6%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      4. associate-/l* 75.6%

         \[\leadsto \frac{-1 \cdot \left(-4 \cdot \frac{i}{\alpha} + \left(\color{blue}{\frac{0 \cdot \beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)}{2} \]

      5. metadata-eval 75.6%

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

      6. distribute-rgt1-in 75.6%

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

      7. associate-*r/ 75.6%

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

      8. div-sub 75.6%

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

   8. Simplified 75.6%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \frac{0 - \left(-\left(\left(-2 - \beta\right) - \beta\right)\right)}{\alpha}\right)}}{2} \]

   9. Taylor expanded in beta around 0 62.6%

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

       1. associate-*r/ 62.6%

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

       2. metadata-eval 62.6%

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

   11. Simplified 62.6%

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

   12. Taylor expanded in alpha around 0 62.6%

       \[\leadsto \frac{-1 \cdot \color{blue}{\frac{-4 \cdot i - 2}{\alpha}}}{2} \]

   if 1.2999999999999999e120 < alpha < 6.79999999999999973e137

   1. Initial program 58.5%

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

   3. Simplified 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. Taylor expanded in beta around inf 77.6%

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

   6. Taylor expanded in beta around inf 77.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 \cdot i + \left(\beta + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.3 \cdot 10^{+120} \lor \neg \left(\alpha \leq 6.8 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{\frac{2 - i \cdot -4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \beta}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.2 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 9.5 \cdot 10^{+175}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \beta}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 8.5e+54)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (<= alpha 3.2e+113)
     (/ (/ (+ beta (- beta -2.0)) alpha) 2.0)
     (if (<= alpha 9.5e+175)
       (/ (+ 1.0 (/ beta (+ alpha beta))) 2.0)
       (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 8.5e+54) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 3.2e+113) {
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	} else if (alpha <= 9.5e+175) {
		tmp = (1.0 + (beta / (alpha + beta))) / 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 <= 8.5d+54) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if (alpha <= 3.2d+113) then
        tmp = ((beta + (beta - (-2.0d0))) / alpha) / 2.0d0
    else if (alpha <= 9.5d+175) then
        tmp = (1.0d0 + (beta / (alpha + beta))) / 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 <= 8.5e+54) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 3.2e+113) {
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	} else if (alpha <= 9.5e+175) {
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 8.5e+54:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif alpha <= 3.2e+113:
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0
	elif alpha <= 9.5e+175:
		tmp = (1.0 + (beta / (alpha + beta))) / 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 <= 8.5e+54)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif (alpha <= 3.2e+113)
		tmp = Float64(Float64(Float64(beta + Float64(beta - -2.0)) / alpha) / 2.0);
	elseif (alpha <= 9.5e+175)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(alpha + beta))) / 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 <= 8.5e+54)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif (alpha <= 3.2e+113)
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	elseif (alpha <= 9.5e+175)
		tmp = (1.0 + (beta / (alpha + beta))) / 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, 8.5e+54], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 3.2e+113], N[(N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 9.5e+175], N[(N[(1.0 + N[(beta / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $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 4 regimes

2. if alpha < 8.4999999999999995e54

   1. Initial program 81.1%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in i around 0 80.8%

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

      1. associate-+r+ 80.8%

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

      2. +-commutative 80.8%

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

   7. Simplified 80.8%

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

   8. Taylor expanded in alpha around 0 88.1%

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

      1. +-commutative 88.1%

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

   10. Simplified 88.1%

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

   if 8.4999999999999995e54 < alpha < 3.1999999999999998e113

   1. Initial program 23.5%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in i around 0 23.7%

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

      1. associate-+r+ 23.7%

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

      2. +-commutative 23.7%

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

   7. Simplified 23.7%

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

   8. Taylor expanded in alpha around -inf 61.0%

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

      1. sub-neg 61.0%

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

      2. mul-1-neg 61.0%

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

      3. associate-*r/ 61.0%

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

      4. mul-1-neg 61.0%

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

      5. mul-1-neg 61.0%

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

      6. +-commutative 61.0%

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

      7. mul-1-neg 61.0%

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

      8. unsub-neg 61.0%

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

      9. mul-1-neg 61.0%

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

      10. distribute-lft-in 61.0%

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

      11. metadata-eval 61.0%

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

      12. mul-1-neg 61.0%

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

      13. unsub-neg 61.0%

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

   10. Simplified 61.0%

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

   if 3.1999999999999998e113 < alpha < 9.5000000000000006e175

   1. Initial program 21.0%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in beta around inf 56.6%

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

   6. Taylor expanded in beta around inf 56.5%

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

   if 9.5000000000000006e175 < alpha

   1. Initial program 1.2%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in i around 0 8.9%

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

      1. associate-+r+ 8.9%

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

      2. +-commutative 8.9%

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

   7. Simplified 8.9%

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

   8. Taylor expanded in alpha around inf 53.4%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 53.4%

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

   10. Simplified 53.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.2 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 9.5 \cdot 10^{+175}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \beta}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 12600000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+132}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 12600000.0)
   0.5
   (if (<= beta 1.8e+42)
     (/ (- 2.0 (/ 2.0 beta)) 2.0)
     (if (<= beta 1.5e+132) 0.5 1.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 12600000.0) {
		tmp = 0.5;
	} else if (beta <= 1.8e+42) {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	} else if (beta <= 1.5e+132) {
		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 <= 12600000.0d0) then
        tmp = 0.5d0
    else if (beta <= 1.8d+42) then
        tmp = (2.0d0 - (2.0d0 / beta)) / 2.0d0
    else if (beta <= 1.5d+132) 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 <= 12600000.0) {
		tmp = 0.5;
	} else if (beta <= 1.8e+42) {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	} else if (beta <= 1.5e+132) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 12600000.0:
		tmp = 0.5
	elif beta <= 1.8e+42:
		tmp = (2.0 - (2.0 / beta)) / 2.0
	elif beta <= 1.5e+132:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 12600000.0)
		tmp = 0.5;
	elseif (beta <= 1.8e+42)
		tmp = Float64(Float64(2.0 - Float64(2.0 / beta)) / 2.0);
	elseif (beta <= 1.5e+132)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 12600000.0)
		tmp = 0.5;
	elseif (beta <= 1.8e+42)
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	elseif (beta <= 1.5e+132)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 12600000.0], 0.5, If[LessEqual[beta, 1.8e+42], N[(N[(2.0 - N[(2.0 / beta), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 1.5e+132], 0.5, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.26e7 or 1.8e42 < beta < 1.4999999999999999e132

   1. Initial program 73.7%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in i around inf 71.7%

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

   if 1.26e7 < beta < 1.8e42

   1. Initial program 80.3%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in i around 0 72.9%

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

      1. associate-+r+ 72.9%

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

      2. +-commutative 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in alpha around 0 72.8%

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

      1. +-commutative 72.8%

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

   10. Simplified 72.8%

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

   11. Taylor expanded in beta around inf 72.8%

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

       1. associate-*r/ 72.8%

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

       2. metadata-eval 72.8%

          \[\leadsto \frac{2 - \frac{\color{blue}{2}}{\beta}}{2} \]

   13. Simplified 72.8%

       \[\leadsto \frac{\color{blue}{2 - \frac{2}{\beta}}}{2} \]

   if 1.4999999999999999e132 < beta

   1. Initial program 12.0%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in beta around inf 76.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 12600000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+132}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 410000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.9 \cdot 10^{+42}:\\ \;\;\;\;1\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+132}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 410000000.0)
   0.5
   (if (<= beta 1.9e+42) 1.0 (if (<= beta 1.5e+132) 0.5 1.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 410000000.0) {
		tmp = 0.5;
	} else if (beta <= 1.9e+42) {
		tmp = 1.0;
	} else if (beta <= 1.5e+132) {
		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 <= 410000000.0d0) then
        tmp = 0.5d0
    else if (beta <= 1.9d+42) then
        tmp = 1.0d0
    else if (beta <= 1.5d+132) 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 <= 410000000.0) {
		tmp = 0.5;
	} else if (beta <= 1.9e+42) {
		tmp = 1.0;
	} else if (beta <= 1.5e+132) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 410000000.0:
		tmp = 0.5
	elif beta <= 1.9e+42:
		tmp = 1.0
	elif beta <= 1.5e+132:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 410000000.0)
		tmp = 0.5;
	elseif (beta <= 1.9e+42)
		tmp = 1.0;
	elseif (beta <= 1.5e+132)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 410000000.0)
		tmp = 0.5;
	elseif (beta <= 1.9e+42)
		tmp = 1.0;
	elseif (beta <= 1.5e+132)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 410000000.0], 0.5, If[LessEqual[beta, 1.9e+42], 1.0, If[LessEqual[beta, 1.5e+132], 0.5, 1.0]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.1e8 or 1.8999999999999999e42 < beta < 1.4999999999999999e132

   1. Initial program 73.7%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in i around inf 71.7%

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

   if 4.1e8 < beta < 1.8999999999999999e42 or 1.4999999999999999e132 < beta

   1. Initial program 22.8%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in beta around inf 74.7%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 410000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.9 \cdot 10^{+42}:\\ \;\;\;\;1\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+132}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 200000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 200000000.0) 0.5 (/ (+ 1.0 (/ beta (+ alpha beta))) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 200000000.0) {
		tmp = 0.5;
	} else {
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 200000000.0d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 + (beta / (alpha + beta))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 200000000.0) {
		tmp = 0.5;
	} else {
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 200000000.0:
		tmp = 0.5
	else:
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 200000000.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(alpha + beta))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 200000000.0)
		tmp = 0.5;
	else
		tmp = (1.0 + (beta / (alpha + beta))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 200000000.0], 0.5, N[(N[(1.0 + N[(beta / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e8

   1. Initial program 74.4%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in i around inf 74.1%

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

   if 2e8 < beta

   1. Initial program 35.1%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in beta around inf 83.0%

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

   6. Taylor expanded in beta around inf 66.0%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 200000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\alpha + \beta}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 1.5e+126) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.5e+126) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 1.5d+126) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.5e+126) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 1.5e+126:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 1.5e+126)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 1.5e+126)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 1.5e+126], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 1.5000000000000001e126

   1. Initial program 56.5%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in i around 0 71.0%

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

      1. associate-+r+ 71.0%

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

      2. +-commutative 71.0%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in alpha around 0 70.9%

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

      1. +-commutative 70.9%

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

   10. Simplified 70.9%

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

   if 1.5000000000000001e126 < i

   1. Initial program 70.8%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in i around inf 86.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.2% 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 61.2%

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

3. Simplified 80.0%

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

5. Taylor expanded in i around inf 61.2%

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

6. Final simplification 61.2%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(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))