Octave 3.8, jcobi/2

Percentage Accurate: 63.6% → 97.6%

Time: 18.6s

Alternatives: 9

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: 63.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-2 - \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{2 + t\_1} \leq -0.99999:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-{t\_0}^{2}\right) - \mathsf{log1p}\left(t\_0\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (- -2.0 (fma 2.0 beta (* i 4.0))) alpha))
        (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1)) -0.99999)
     (/ (- (log1p (- (pow t_0 2.0))) (log1p t_0)) 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 = (-2.0 - fma(2.0, beta, (i * 4.0))) / alpha;
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.99999) {
		tmp = (log1p(-pow(t_0, 2.0)) - log1p(t_0)) / 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(-2.0 - fma(2.0, beta, Float64(i * 4.0))) / alpha)
	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(log1p(Float64(-(t_0 ^ 2.0))) - log1p(t_0)) / 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[(-2.0 - N[(2.0 * beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $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[Log[1 + (-N[Power[t$95$0, 2.0], $MachinePrecision])], $MachinePrecision] - N[Log[1 + t$95$0], $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))) < -0.999990000000000046

   1. Initial program 2.9%

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

   3. Simplified 17.5%

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

      1. add-log-exp 17.6%

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

      2. associate-/l* 17.6%

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

      3. fma-define 17.2%

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

      4. +-commutative 17.2%

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

      5. +-commutative 17.2%

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

   6. Applied egg-rr 17.2%

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

   7. Taylor expanded in alpha around inf 9.4%

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

      1. flip-- 9.4%

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

      2. log-div 8.0%

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

   9. Applied egg-rr 8.0%

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

   11. Simplified 89.4%

       \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-{\left(\frac{-2 - \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)}^{2}\right) - \mathsf{log1p}\left(\frac{-2 - \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)}}{2} \]

   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 78.9%

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

   3. Simplified 99.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-{\left(\frac{-2 - \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)}^{2}\right) - \mathsf{log1p}\left(\frac{-2 - \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(N[(beta + N[(N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * -2.0), $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))) < -0.999990000000000046

   1. Initial program 2.9%

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

   3. Simplified 17.5%

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

   5. Taylor expanded in beta around 0 17.5%

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

   6. Taylor expanded in alpha around -inf 88.8%

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

      1. mul-1-neg 88.8%

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

      2. unsub-neg 88.8%

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

      3. *-commutative 88.8%

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

      4. +-commutative 88.8%

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

   8. Simplified 88.8%

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

   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 78.9%

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

   3. Simplified 99.9%

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

4. Final simplification 97.2%

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

Alternative 3: 97.7% accurate, 0.1× speedup

Math

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

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)) -0.99999)
     (/ (/ (+ beta (- (+ 2.0 (+ beta (* 2.0 i))) (* i -2.0))) 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)) <= -0.99999) {
		tmp = ((beta + ((2.0 + (beta + (2.0 * i))) - (i * -2.0))) / 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)) <= -0.99999)
		tmp = Float64(Float64(Float64(beta + Float64(Float64(2.0 + Float64(beta + Float64(2.0 * i))) - Float64(i * -2.0))) / 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], -0.99999], N[(N[(N[(beta + N[(N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(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))) < -0.999990000000000046

   1. Initial program 2.9%

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

   3. Simplified 17.5%

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

   5. Taylor expanded in beta around 0 17.5%

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

   6. Taylor expanded in alpha around -inf 88.8%

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

      1. mul-1-neg 88.8%

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

      2. unsub-neg 88.8%

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

      3. *-commutative 88.8%

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

      4. +-commutative 88.8%

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

   8. Simplified 88.8%

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

   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 78.9%

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

   3. Simplified 99.9%

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

4. Final simplification 97.2%

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

Alternative 4: 94.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta + \left(\left(2 + \left(\beta + 2 \cdot i\right)\right) - i \cdot -2\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_1 \leq 10^{-8}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha}{\alpha + 2 \cdot i}}{i \cdot \left(2 + \left(\frac{\alpha}{i} + \frac{2}{i}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.99999)
     (/ (/ (+ beta (- (+ 2.0 (+ beta (* 2.0 i))) (* i -2.0))) alpha) 2.0)
     (if (<= t_1 1e-8)
       (/
        (+
         1.0
         (/
          (* (- beta alpha) (/ alpha (+ alpha (* 2.0 i))))
          (* i (+ 2.0 (+ (/ alpha i) (/ 2.0 i))))))
        2.0)
       (/ (+ 1.0 (/ (- beta alpha) (+ alpha beta))) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.99999) {
		tmp = ((beta + ((2.0 + (beta + (2.0 * i))) - (i * -2.0))) / alpha) / 2.0;
	} else if (t_1 <= 1e-8) {
		tmp = (1.0 + (((beta - alpha) * (alpha / (alpha + (2.0 * i)))) / (i * (2.0 + ((alpha / i) + (2.0 / i)))))) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)
    if (t_1 <= (-0.99999d0)) then
        tmp = ((beta + ((2.0d0 + (beta + (2.0d0 * i))) - (i * (-2.0d0)))) / alpha) / 2.0d0
    else if (t_1 <= 1d-8) then
        tmp = (1.0d0 + (((beta - alpha) * (alpha / (alpha + (2.0d0 * i)))) / (i * (2.0d0 + ((alpha / i) + (2.0d0 / i)))))) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / (alpha + beta))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.99999) {
		tmp = ((beta + ((2.0 + (beta + (2.0 * i))) - (i * -2.0))) / alpha) / 2.0;
	} else if (t_1 <= 1e-8) {
		tmp = (1.0 + (((beta - alpha) * (alpha / (alpha + (2.0 * i)))) / (i * (2.0 + ((alpha / i) + (2.0 / i)))))) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + beta))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)
	tmp = 0
	if t_1 <= -0.99999:
		tmp = ((beta + ((2.0 + (beta + (2.0 * i))) - (i * -2.0))) / alpha) / 2.0
	elif t_1 <= 1e-8:
		tmp = (1.0 + (((beta - alpha) * (alpha / (alpha + (2.0 * i)))) / (i * (2.0 + ((alpha / i) + (2.0 / i)))))) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / (alpha + beta))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -0.99999)
		tmp = Float64(Float64(Float64(beta + Float64(Float64(2.0 + Float64(beta + Float64(2.0 * i))) - Float64(i * -2.0))) / alpha) / 2.0);
	elseif (t_1 <= 1e-8)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(alpha / Float64(alpha + Float64(2.0 * i)))) / Float64(i * Float64(2.0 + Float64(Float64(alpha / i) + Float64(2.0 / i)))))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(alpha + beta))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	tmp = 0.0;
	if (t_1 <= -0.99999)
		tmp = ((beta + ((2.0 + (beta + (2.0 * i))) - (i * -2.0))) / alpha) / 2.0;
	elseif (t_1 <= 1e-8)
		tmp = (1.0 + (((beta - alpha) * (alpha / (alpha + (2.0 * i)))) / (i * (2.0 + ((alpha / i) + (2.0 / i)))))) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / (alpha + beta))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.99999], N[(N[(N[(beta + N[(N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e-8], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(alpha / N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * N[(2.0 + N[(N[(alpha / i), $MachinePrecision] + N[(2.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 2.9%

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

   3. Simplified 17.5%

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

   5. Taylor expanded in beta around 0 17.5%

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

   6. Taylor expanded in alpha around -inf 88.8%

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

      1. mul-1-neg 88.8%

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

      2. unsub-neg 88.8%

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

      3. *-commutative 88.8%

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

      4. +-commutative 88.8%

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

   8. Simplified 88.8%

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

   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))) < 1e-8

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in beta around 0 99.8%

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

   6. Taylor expanded in i around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

      3. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in beta around 0 99.8%

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

       1. associate-*r/ 99.8%

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

       2. metadata-eval 99.8%

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

       3. +-commutative 99.8%

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

   11. Simplified 99.8%

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

   if 1e-8 < (/.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 27.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 97.1%

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

      1. associate-+r+ 97.1%

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

   7. Simplified 97.1%

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

   8. Taylor expanded in alpha around inf 97.1%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta + \left(\left(2 + \left(\beta + 2 \cdot i\right)\right) - i \cdot -2\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 10^{-8}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha}{\alpha + 2 \cdot i}}{i \cdot \left(2 + \left(\frac{\alpha}{i} + \frac{2}{i}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \beta}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.6 \cdot 10^{+33}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5.6e+33)
   (/ (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 2.0)
   (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.6e+33) {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 5.6d+33) then
        tmp = (1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) / 2.0d0
    else
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.6e+33) {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5.6e+33:
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0
	else:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5.6e+33)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 5.6e+33)
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	else
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5.6e+33], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.6000000000000002e33

   1. Initial program 82.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in i around 0 92.5%

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

      1. associate-+r+ 92.5%

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

   7. Simplified 92.5%

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

   if 5.6000000000000002e33 < alpha

   1. Initial program 16.7%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in beta around 0 40.1%

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

   6. Taylor expanded in alpha around inf 65.7%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.6 \cdot 10^{+33}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.65 \cdot 10^{+35}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + i \cdot 4\right) - \beta}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.65e+35)
   (/ (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 2.0)
   (/ (/ (- (+ 2.0 (* i 4.0)) beta) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.65e+35) {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	} else {
		tmp = (((2.0 + (i * 4.0)) - beta) / 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 <= 2.65d+35) then
        tmp = (1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) / 2.0d0
    else
        tmp = (((2.0d0 + (i * 4.0d0)) - beta) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.65e+35) {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	} else {
		tmp = (((2.0 + (i * 4.0)) - beta) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.65e+35:
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0
	else:
		tmp = (((2.0 + (i * 4.0)) - beta) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.65e+35)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(i * 4.0)) - beta) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.65e+35)
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	else
		tmp = (((2.0 + (i * 4.0)) - beta) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.65e+35], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.65000000000000005e35

   1. Initial program 82.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in i around 0 92.5%

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

      1. associate-+r+ 92.5%

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

   7. Simplified 92.5%

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

   if 2.65000000000000005e35 < alpha

   1. Initial program 16.7%

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

   3. Simplified 26.7%

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

   5. Taylor expanded in beta around 0 24.5%

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

      1. associate-*r/ 24.5%

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

      2. neg-mul-1 24.5%

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

      3. *-commutative 24.5%

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

      4. associate-+r+ 24.5%

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

   7. Simplified 24.5%

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

   8. Taylor expanded in alpha around inf 51.4%

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

      1. mul-1-neg 51.4%

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

   10. Simplified 51.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.65 \cdot 10^{+35}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + i \cdot 4\right) - \beta}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+117}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.8e+117) 0.5 (/ (+ 1.0 (/ (- beta alpha) (+ alpha beta))) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.8e+117) {
		tmp = 0.5;
	} else {
		tmp = (1.0 + ((beta - alpha) / (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 <= 3.8d+117) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 + ((beta - alpha) / (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 <= 3.8e+117) {
		tmp = 0.5;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + beta))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.8e+117:
		tmp = 0.5
	else:
		tmp = (1.0 + ((beta - alpha) / (alpha + beta))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.8e+117)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(alpha + beta))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.8e+117)
		tmp = 0.5;
	else
		tmp = (1.0 + ((beta - alpha) / (alpha + beta))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 3.8e+117], 0.5, N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.8000000000000002e117

   1. Initial program 72.7%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in i around inf 72.1%

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

   if 3.8000000000000002e117 < beta

   1. Initial program 20.6%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in i around 0 84.7%

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

      1. associate-+r+ 84.7%

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

   7. Simplified 84.7%

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

   8. Taylor expanded in alpha around inf 84.7%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+117}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \beta}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+118}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 1.05e+118) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.05e+118) {
		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 <= 1.05d+118) 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 <= 1.05e+118) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.05e+118:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.05e+118)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.05e+118)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.05e+118], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.05e118

   1. Initial program 72.7%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in i around inf 72.1%

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

   if 1.05e118 < beta

   1. Initial program 20.6%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in beta around inf 81.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+118}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.9% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 60.5%

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

3. Simplified 66.3%

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

5. Taylor expanded in i around inf 60.9%

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

Reproduce

herbie shell --seed 2024135 
(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))