Octave 3.8, jcobi/2

Percentage Accurate: 63.3% → 97.7%

Time: 25.9s

Alternatives: 12

Speedup: 4.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{2 + t\_1} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(t\_0 + \left(2 + t\_0\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1))
        -0.9999999)
     (/ (/ (+ (- beta beta) (+ t_0 (+ 2.0 t_0))) alpha) 2.0)
     (/
      (fma
       (+ alpha beta)
       (/
        (/ (- beta alpha) (+ alpha (+ beta (fma 2.0 i 2.0))))
        (+ alpha (fma 2.0 i beta)))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.9999999) {
		tmp = (((beta - beta) + (t_0 + (2.0 + t_0))) / alpha) / 2.0;
	} else {
		tmp = fma((alpha + beta), (((beta - alpha) / (alpha + (beta + fma(2.0, i, 2.0)))) / (alpha + fma(2.0, i, beta))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(2.0 + t_1)) <= -0.9999999)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(t_0 + Float64(2.0 + t_0))) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(alpha + beta), Float64(Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) / Float64(alpha + fma(2.0, i, beta))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], -0.9999999], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(alpha + beta), $MachinePrecision] * N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999900000000053

   1. Initial program 2.8%

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

   3. Simplified 16.0%

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

   5. Taylor expanded in alpha around -inf 90.4%

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

   if -0.999999900000000053 < (/.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 83.2%

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

   3. Simplified 99.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(\left(\beta + 2 \cdot i\right) + \left(2 + \left(\beta + 2 \cdot i\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{2 + t\_1} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(t\_0 + \left(2 + t\_0\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1))
        -0.9999999)
     (/ (/ (+ (- beta beta) (+ t_0 (+ 2.0 t_0))) alpha) 2.0)
     (/
      (+
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        (+ alpha (+ beta (fma 2.0 i 2.0))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.9999999) {
		tmp = (((beta - beta) + (t_0 + (2.0 + t_0))) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / (alpha + (beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(2.0 + t_1)) <= -0.9999999)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(t_0 + Float64(2.0 + t_0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(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)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(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$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], -0.9999999], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(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] + 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))) < -0.999999900000000053

   1. Initial program 2.8%

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

   3. Simplified 16.0%

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

   5. Taylor expanded in alpha around -inf 90.4%

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

   if -0.999999900000000053 < (/.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 83.2%

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

   3. Simplified 99.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(\left(\beta + 2 \cdot i\right) + \left(2 + \left(\beta + 2 \cdot i\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{2 + t\_1} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(t\_0 + \left(2 + t\_0\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\beta \cdot \left(\left(2 \cdot \frac{i}{\beta} + \left(2 \cdot \frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right) + 1\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1)) -0.999996)
     (/ (/ (+ (- beta beta) (+ t_0 (+ 2.0 t_0))) alpha) 2.0)
     (/
      (+
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        (*
         beta
         (+
          (+ (* 2.0 (/ i beta)) (+ (* 2.0 (/ 1.0 beta)) (/ alpha beta)))
          1.0)))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.999996) {
		tmp = (((beta - beta) + (t_0 + (2.0 + t_0))) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / (beta * (((2.0 * (i / beta)) + ((2.0 * (1.0 / beta)) + (alpha / beta))) + 1.0))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(2.0 + t_1)) <= -0.999996)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(t_0 + Float64(2.0 + t_0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / Float64(beta * Float64(Float64(Float64(2.0 * Float64(i / beta)) + Float64(Float64(2.0 * Float64(1.0 / beta)) + Float64(alpha / beta))) + 1.0))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(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$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], -0.999996], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta * N[(N[(N[(2.0 * N[(i / beta), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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))) < -0.999995999999999996

   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 16.9%

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

   5. Taylor expanded in alpha around -inf 89.8%

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

   if -0.999995999999999996 < (/.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 83.3%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in beta around inf 99.7%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(\left(\beta + 2 \cdot i\right) + \left(2 + \left(\beta + 2 \cdot i\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\beta \cdot \left(\left(2 \cdot \frac{i}{\beta} + \left(2 \cdot \frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right) + 1\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{2 + t\_1} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(t\_0 + \left(2 + t\_0\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{t\_0}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1)) -0.99999)
     (/ (/ (+ (- beta beta) (+ t_0 (+ 2.0 t_0))) alpha) 2.0)
     (/
      (+
       (/ (* (- beta alpha) (/ beta t_0)) (+ alpha (+ beta (fma 2.0 i 2.0))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.99999) {
		tmp = (((beta - beta) + (t_0 + (2.0 + t_0))) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) * (beta / t_0)) / (alpha + (beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(2.0 + t_1)) <= -0.99999)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(t_0 + Float64(2.0 + t_0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta / t_0)) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(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$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999990000000000046

   1. Initial program 5.2%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in alpha around -inf 89.0%

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

   if -0.999990000000000046 < (/.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 83.3%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 99.8%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(\left(\beta + 2 \cdot i\right) + \left(2 + \left(\beta + 2 \cdot i\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.5)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)
     (/
      (+
       (/
        beta
        (* (+ beta (* 2.0 i)) (+ (+ (* 2.0 (/ i beta)) (/ 2.0 beta)) 1.0)))
       1.0)
      2.0))))

C

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

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: 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 + ((beta * 2.0d0) + (i * 4.0d0)))) / alpha) / 2.0d0
    else
        tmp = ((beta / ((beta + (2.0d0 * i)) * (((2.0d0 * (i / beta)) + (2.0d0 / beta)) + 1.0d0))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5)
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	else
		tmp = ((beta / ((beta + (2.0 * i)) * (((2.0 * (i / beta)) + (2.0 / beta)) + 1.0))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

   1. Initial program 6.5%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in alpha around inf 88.2%

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

   if -0.5 < (/.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 83.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 100.0%

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

   6. Taylor expanded in alpha around 0 99.7%

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

      1. associate-*r/ 99.7%

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

      2. metadata-eval 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 97.1%

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

Alternative 6: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{2 + t\_1} \leq -0.5:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(t\_0 + \left(2 + t\_0\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0 \cdot \left(\left(2 \cdot \frac{i}{\beta} + \frac{2}{\beta}\right) + 1\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 6.5%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in alpha around -inf 88.2%

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

   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 83.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 100.0%

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

   6. Taylor expanded in alpha around 0 99.7%

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

      1. associate-*r/ 99.7%

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

      2. metadata-eval 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 97.1%

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

Alternative 7: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{\beta} + 1} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 9.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 1.3 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + 2 \cdot i\right) \cdot \left(2 \cdot \frac{i}{\beta} + 1\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.1e+35)
   (/ (+ (/ 1.0 (+ (/ 2.0 beta) 1.0)) 1.0) 2.0)
   (if (<= alpha 9.5e+63)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (if (<= alpha 1.3e+173)
       (/
        (+ (/ beta (* (+ beta (* 2.0 i)) (+ (* 2.0 (/ i beta)) 1.0))) 1.0)
        2.0)
       (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.1e+35) {
		tmp = ((1.0 / ((2.0 / beta) + 1.0)) + 1.0) / 2.0;
	} else if (alpha <= 9.5e+63) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else if (alpha <= 1.3e+173) {
		tmp = ((beta / ((beta + (2.0 * i)) * ((2.0 * (i / beta)) + 1.0))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.1d+35) then
        tmp = ((1.0d0 / ((2.0d0 / beta) + 1.0d0)) + 1.0d0) / 2.0d0
    else if (alpha <= 9.5d+63) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else if (alpha <= 1.3d+173) then
        tmp = ((beta / ((beta + (2.0d0 * i)) * ((2.0d0 * (i / beta)) + 1.0d0))) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.1e+35) {
		tmp = ((1.0 / ((2.0 / beta) + 1.0)) + 1.0) / 2.0;
	} else if (alpha <= 9.5e+63) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else if (alpha <= 1.3e+173) {
		tmp = ((beta / ((beta + (2.0 * i)) * ((2.0 * (i / beta)) + 1.0))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.1e+35:
		tmp = ((1.0 / ((2.0 / beta) + 1.0)) + 1.0) / 2.0
	elif alpha <= 9.5e+63:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	elif alpha <= 1.3e+173:
		tmp = ((beta / ((beta + (2.0 * i)) * ((2.0 * (i / beta)) + 1.0))) + 1.0) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.1e+35)
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(2.0 / beta) + 1.0)) + 1.0) / 2.0);
	elseif (alpha <= 9.5e+63)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	elseif (alpha <= 1.3e+173)
		tmp = Float64(Float64(Float64(beta / Float64(Float64(beta + Float64(2.0 * i)) * Float64(Float64(2.0 * Float64(i / beta)) + 1.0))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.1e+35)
		tmp = ((1.0 / ((2.0 / beta) + 1.0)) + 1.0) / 2.0;
	elseif (alpha <= 9.5e+63)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	elseif (alpha <= 1.3e+173)
		tmp = ((beta / ((beta + (2.0 * i)) * ((2.0 * (i / beta)) + 1.0))) + 1.0) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.1e+35], N[(N[(N[(1.0 / N[(N[(2.0 / beta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 9.5e+63], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.3e+173], N[(N[(N[(beta / N[(N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(i / beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 2.0999999999999999e35

   1. Initial program 85.6%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in beta around inf 99.0%

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

   6. Taylor expanded in alpha around 0 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in i around 0 91.2%

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

       1. associate-*r/ 91.2%

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

       2. metadata-eval 91.2%

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

   11. Simplified 91.2%

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

   if 2.0999999999999999e35 < alpha < 9.5000000000000003e63

   1. Initial program 3.4%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in alpha around inf 90.4%

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

   6. Taylor expanded in i around 0 90.5%

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

      1. distribute-rgt1-in 90.5%

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

      2. metadata-eval 90.5%

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

      3. mul0-lft 90.5%

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

      4. mul-1-neg 90.5%

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

   8. Simplified 90.5%

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

   if 9.5000000000000003e63 < alpha < 1.2999999999999999e173

   1. Initial program 37.4%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in beta around inf 65.4%

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

   6. Taylor expanded in alpha around 0 63.6%

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

      1. associate-*r/ 63.6%

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

      2. metadata-eval 63.6%

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

   8. Simplified 63.6%

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

   9. Taylor expanded in i around inf 63.6%

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

   if 1.2999999999999999e173 < alpha

   1. Initial program 1.1%

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

   3. Simplified 22.1%

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

   5. Taylor expanded in alpha around inf 84.9%

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

   6. Taylor expanded in beta around 0 70.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{\beta} + 1} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 9.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 1.3 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + 2 \cdot i\right) \cdot \left(2 \cdot \frac{i}{\beta} + 1\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{\beta} + 1} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 1.28 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.7 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + 2 \cdot i\right) \cdot \left(2 \cdot \frac{i}{\beta} + 1\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.7e+35)
   (/ (+ (/ 1.0 (+ (/ 2.0 beta) 1.0)) 1.0) 2.0)
   (if (<= alpha 1.28e+64)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (if (<= alpha 3.7e+171)
       (/
        (+ (/ beta (* (+ beta (* 2.0 i)) (+ (* 2.0 (/ i beta)) 1.0))) 1.0)
        2.0)
       (/
        (+
         (* 2.0 (/ beta alpha))
         (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))))
        2.0)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.7e+35) {
		tmp = ((1.0 / ((2.0 / beta) + 1.0)) + 1.0) / 2.0;
	} else if (alpha <= 1.28e+64) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else if (alpha <= 3.7e+171) {
		tmp = ((beta / ((beta + (2.0 * i)) * ((2.0 * (i / beta)) + 1.0))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.7d+35) then
        tmp = ((1.0d0 / ((2.0d0 / beta) + 1.0d0)) + 1.0d0) / 2.0d0
    else if (alpha <= 1.28d+64) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else if (alpha <= 3.7d+171) then
        tmp = ((beta / ((beta + (2.0d0 * i)) * ((2.0d0 * (i / beta)) + 1.0d0))) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 * (beta / alpha)) + ((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.7e+35) {
		tmp = ((1.0 / ((2.0 / beta) + 1.0)) + 1.0) / 2.0;
	} else if (alpha <= 1.28e+64) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else if (alpha <= 3.7e+171) {
		tmp = ((beta / ((beta + (2.0 * i)) * ((2.0 * (i / beta)) + 1.0))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.7e+35:
		tmp = ((1.0 / ((2.0 / beta) + 1.0)) + 1.0) / 2.0
	elif alpha <= 1.28e+64:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	elif alpha <= 3.7e+171:
		tmp = ((beta / ((beta + (2.0 * i)) * ((2.0 * (i / beta)) + 1.0))) + 1.0) / 2.0
	else:
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.7e+35)
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(2.0 / beta) + 1.0)) + 1.0) / 2.0);
	elseif (alpha <= 1.28e+64)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	elseif (alpha <= 3.7e+171)
		tmp = Float64(Float64(Float64(beta / Float64(Float64(beta + Float64(2.0 * i)) * Float64(Float64(2.0 * Float64(i / beta)) + 1.0))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.7e+35)
		tmp = ((1.0 / ((2.0 / beta) + 1.0)) + 1.0) / 2.0;
	elseif (alpha <= 1.28e+64)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	elseif (alpha <= 3.7e+171)
		tmp = ((beta / ((beta + (2.0 * i)) * ((2.0 * (i / beta)) + 1.0))) + 1.0) / 2.0;
	else
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.7e+35], N[(N[(N[(1.0 / N[(N[(2.0 / beta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.28e+64], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 3.7e+171], N[(N[(N[(beta / N[(N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(i / beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 1.7000000000000001e35

   1. Initial program 85.6%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in beta around inf 99.0%

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

   6. Taylor expanded in alpha around 0 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in i around 0 91.2%

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

       1. associate-*r/ 91.2%

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

       2. metadata-eval 91.2%

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

   11. Simplified 91.2%

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

   if 1.7000000000000001e35 < alpha < 1.28000000000000004e64

   1. Initial program 3.4%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in alpha around inf 90.4%

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

   6. Taylor expanded in i around 0 90.5%

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

      1. distribute-rgt1-in 90.5%

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

      2. metadata-eval 90.5%

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

      3. mul0-lft 90.5%

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

      4. mul-1-neg 90.5%

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

   8. Simplified 90.5%

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

   if 1.28000000000000004e64 < alpha < 3.69999999999999998e171

   1. Initial program 37.4%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in beta around inf 65.4%

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

   6. Taylor expanded in alpha around 0 63.6%

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

      1. associate-*r/ 63.6%

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

      2. metadata-eval 63.6%

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

   8. Simplified 63.6%

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

   9. Taylor expanded in i around inf 63.6%

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

   if 3.69999999999999998e171 < alpha

   1. Initial program 1.1%

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

   3. Simplified 22.1%

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

   5. Taylor expanded in alpha around inf 84.9%

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

   6. Taylor expanded in beta around 0 85.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{\beta} + 1} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 1.28 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.7 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + 2 \cdot i\right) \cdot \left(2 \cdot \frac{i}{\beta} + 1\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.85 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{\beta} + 1} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.85e+35)
   (/ (+ (/ 1.0 (+ (/ 2.0 beta) 1.0)) 1.0) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.85e+35) {
		tmp = ((1.0 / ((2.0 / beta) + 1.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.85d+35) then
        tmp = ((1.0d0 / ((2.0d0 / beta) + 1.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.85e+35) {
		tmp = ((1.0 / ((2.0 / beta) + 1.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.85e+35:
		tmp = ((1.0 / ((2.0 / beta) + 1.0)) + 1.0) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.85e+35)
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(2.0 / beta) + 1.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.85e+35)
		tmp = ((1.0 / ((2.0 / beta) + 1.0)) + 1.0) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.85e+35], N[(N[(N[(1.0 / N[(N[(2.0 / beta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.85e35

   1. Initial program 85.6%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in beta around inf 99.0%

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

   6. Taylor expanded in alpha around 0 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in i around 0 91.2%

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

       1. associate-*r/ 91.2%

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

       2. metadata-eval 91.2%

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

   11. Simplified 91.2%

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

   if 1.85e35 < alpha

   1. Initial program 11.6%

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

   3. Simplified 33.6%

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

   5. Taylor expanded in alpha around inf 72.7%

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

   6. Taylor expanded in beta around 0 60.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.85 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{\beta} + 1} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.8e+23) 0.5 (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.8e+23) {
		tmp = 0.5;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 4.8d+23) then
        tmp = 0.5d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.8e+23) {
		tmp = 0.5;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.8e+23:
		tmp = 0.5
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4.8e+23)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 4.8e+23)
		tmp = 0.5;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4.8e+23], 0.5, N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.8e23

   1. Initial program 85.9%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in i around inf 81.1%

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

   if 4.8e23 < alpha

   1. Initial program 12.8%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in alpha around inf 72.2%

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

   6. Taylor expanded in beta around 0 60.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.16 \cdot 10^{+47}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 1.16e+47) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.16e+47) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.16d+47) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.16e+47) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.16e+47:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.16e+47)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.16e+47)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.16e+47], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.1600000000000001e47

   1. Initial program 75.3%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in i around inf 74.1%

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

   if 1.1600000000000001e47 < beta

   1. Initial program 35.2%

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

   3. Simplified 92.0%

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

   5. Taylor expanded in beta around inf 73.4%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.16 \cdot 10^{+47}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 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 65.9%

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

3. Simplified 81.6%

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

5. Taylor expanded in i around inf 64.7%

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

6. Final simplification 64.7%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024068 
(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))