Octave 3.8, jcobi/2

Percentage Accurate: 63.5% → 97.8%

Time: 37.2s

Alternatives: 10

Speedup: 9.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.999998:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 \cdot \left(\beta + 1\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{t_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.999998)
     (/ (+ (* 4.0 (/ i alpha)) (/ (* 2.0 (+ beta 1.0)) alpha)) 2.0)
     (/
      (+
       1.0
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        t_1))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.999998) {
		tmp = ((4.0 * (i / alpha)) + ((2.0 * (beta + 1.0)) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / t_1)) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.999998)
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(Float64(2.0 * Float64(beta + 1.0)) / 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)))) / t_1)) / 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]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.999998], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999998000000000054

   1. Initial program 3.7%

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

      1. associate-/l/ 2.9%

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

      2. *-commutative 2.9%

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

      3. times-frac 17.5%

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

      4. associate-+l+ 17.5%

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

      5. fma-def 17.5%

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

      6. +-commutative 17.5%

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

      7. fma-def 17.5%

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

   3. Simplified 17.5%

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

   4. Taylor expanded in alpha around inf 8.3%

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

   5. Taylor expanded in i around 0 88.9%

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

      1. sub-neg 88.9%

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

      2. associate-+r+ 88.9%

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

      3. +-commutative 88.9%

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

      4. distribute-lft1-in 88.9%

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

      5. metadata-eval 88.9%

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

      6. mul0-lft 88.9%

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

      7. +-lft-identity 88.9%

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

      8. mul-1-neg 88.9%

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

      9. remove-double-neg 88.9%

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

      10. *-commutative 88.9%

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

      11. distribute-rgt1-in 88.9%

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

   7. Simplified 88.9%

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

   if -0.999998000000000054 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 83.9%

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

      1. expm1-log1p-u 79.8%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 79.8%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 94.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 94.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\beta + \alpha}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. +-commutative 94.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. fma-udef 94.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. +-commutative 94.4%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \color{blue}{\beta + \alpha}\right)}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 94.4%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 94.4%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 99.9%

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

      3. associate-/r/ 99.9%

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

      4. *-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999998:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 \cdot \left(\beta + 1\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 2: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 \cdot \left(\beta + 1\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta}{\beta + 2 \cdot i}}{2}\\ \end{array} \end{array} \]

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)) -0.9)
     (/ (+ (* 4.0 (/ i alpha)) (/ (* 2.0 (+ beta 1.0)) alpha)) 2.0)
     (/
      (+
       1.0
       (*
        (/ (- beta alpha) (+ (+ alpha beta) (fma 2.0 i 2.0)))
        (/ beta (+ beta (* 2.0 i)))))
      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)) <= -0.9) {
		tmp = ((4.0 * (i / alpha)) + ((2.0 * (beta + 1.0)) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) / ((alpha + beta) + fma(2.0, i, 2.0))) * (beta / (beta + (2.0 * i))))) / 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)) <= -0.9)
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(Float64(2.0 * Float64(beta + 1.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))) * Float64(beta / Float64(beta + Float64(2.0 * i))))) / 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], -0.9], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta / N[(beta + N[(2.0 * i), $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 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.900000000000000022

   1. Initial program 5.1%

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

      1. associate-/l/ 4.4%

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

      2. *-commutative 4.4%

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

      3. times-frac 18.7%

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

      4. associate-+l+ 18.7%

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

      5. fma-def 18.7%

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

      6. +-commutative 18.7%

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

      7. fma-def 18.7%

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

   3. Simplified 18.7%

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

   4. Taylor expanded in alpha around inf 9.1%

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

   5. Taylor expanded in i around 0 88.1%

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

      1. sub-neg 88.1%

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

      2. associate-+r+ 88.1%

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

      3. +-commutative 88.1%

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

      4. distribute-lft1-in 88.1%

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

      5. metadata-eval 88.1%

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

      6. mul0-lft 88.1%

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

      7. +-lft-identity 88.1%

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

      8. mul-1-neg 88.1%

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

      9. remove-double-neg 88.1%

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

      10. *-commutative 88.1%

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

      11. distribute-rgt1-in 88.1%

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

   7. Simplified 88.1%

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

   if -0.900000000000000022 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 83.9%

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

      1. associate-/l/ 83.4%

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

      2. *-commutative 83.4%

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

      3. times-frac 100.0%

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

      4. associate-+l+ 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 99.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 \cdot \left(\beta + 1\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta}{\beta + 2 \cdot i}}{2}\\ \end{array} \]

Alternative 3: 97.0% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.5)
     (/ (+ (* 4.0 (/ i alpha)) (/ (* 2.0 (+ beta 1.0)) alpha)) 2.0)
     (/
      (+
       1.0
       (* (/ beta (+ beta (* 2.0 i))) (/ beta (+ beta (+ 2.0 (* 2.0 i))))))
      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 = ((4.0 * (i / alpha)) + ((2.0 * (beta + 1.0)) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + ((beta / (beta + (2.0 * i))) * (beta / (beta + (2.0 + (2.0 * i)))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.5d0)) then
        tmp = ((4.0d0 * (i / alpha)) + ((2.0d0 * (beta + 1.0d0)) / alpha)) / 2.0d0
    else
        tmp = (1.0d0 + ((beta / (beta + (2.0d0 * i))) * (beta / (beta + (2.0d0 + (2.0d0 * i)))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5) {
		tmp = ((4.0 * (i / alpha)) + ((2.0 * (beta + 1.0)) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + ((beta / (beta + (2.0 * i))) * (beta / (beta + (2.0 + (2.0 * i)))))) / 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 = ((4.0 * (i / alpha)) + ((2.0 * (beta + 1.0)) / alpha)) / 2.0
	else:
		tmp = (1.0 + ((beta / (beta + (2.0 * i))) * (beta / (beta + (2.0 + (2.0 * i)))))) / 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(4.0 * Float64(i / alpha)) + Float64(Float64(2.0 * Float64(beta + 1.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta / Float64(beta + Float64(2.0 * i))) * Float64(beta / Float64(beta + Float64(2.0 + Float64(2.0 * i)))))) / 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 = ((4.0 * (i / alpha)) + ((2.0 * (beta + 1.0)) / alpha)) / 2.0;
	else
		tmp = (1.0 + ((beta / (beta + (2.0 * i))) * (beta / (beta + (2.0 + (2.0 * i)))))) / 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[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta / N[(beta + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $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 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.5

   1. Initial program 6.9%

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

      1. associate-/l/ 6.2%

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

      2. *-commutative 6.2%

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

      3. times-frac 20.2%

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

      4. associate-+l+ 20.2%

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

      5. fma-def 20.2%

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

      6. +-commutative 20.2%

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

      7. fma-def 20.2%

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

   3. Simplified 20.2%

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

   4. Taylor expanded in alpha around inf 9.3%

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

   5. Taylor expanded in i around 0 86.9%

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

      1. sub-neg 86.9%

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

      2. associate-+r+ 86.9%

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

      3. +-commutative 86.9%

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

      4. distribute-lft1-in 86.9%

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

      5. metadata-eval 86.9%

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

      6. mul0-lft 86.9%

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

      7. +-lft-identity 86.9%

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

      8. mul-1-neg 86.9%

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

      9. remove-double-neg 86.9%

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

      10. *-commutative 86.9%

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

      11. distribute-rgt1-in 86.9%

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

   7. Simplified 86.9%

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

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 83.9%

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

      1. expm1-log1p-u 80.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 80.6%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 95.3%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 95.3%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\beta + \alpha}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. +-commutative 95.3%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. fma-udef 95.3%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. +-commutative 95.3%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \color{blue}{\beta + \alpha}\right)}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 95.3%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 95.3%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 100.0%

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

      3. associate-/r/ 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in alpha around 0 82.7%

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

      1. unpow2 82.7%

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

      2. times-frac 99.4%

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

      3. +-commutative 99.4%

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

   8. Simplified 99.4%

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

4. Final simplification 96.8%

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

Alternative 4: 83.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 \cdot \left(\beta + 1\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.8e+51)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ (* 4.0 (/ i alpha)) (/ (* 2.0 (+ beta 1.0)) alpha)) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.8e+51) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + ((2.0 * (beta + 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 <= 2.8d+51) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((4.0d0 * (i / alpha)) + ((2.0d0 * (beta + 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 <= 2.8e+51) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + ((2.0 * (beta + 1.0)) / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.8e+51:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((4.0 * (i / alpha)) + ((2.0 * (beta + 1.0)) / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.8e+51)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(Float64(2.0 * Float64(beta + 1.0)) / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.8e+51)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((4.0 * (i / alpha)) + ((2.0 * (beta + 1.0)) / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.8e+51], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.80000000000000005e51

   1. Initial program 84.3%

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

      1. expm1-log1p-u 82.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 82.6%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\beta + \alpha}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. +-commutative 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. fma-udef 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. +-commutative 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \color{blue}{\beta + \alpha}\right)}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 95.8%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 95.8%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 98.9%

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

      3. associate-/r/ 98.9%

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

      4. *-commutative 98.9%

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

      5. +-commutative 98.9%

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

      6. +-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in alpha around 0 82.5%

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

      1. unpow2 82.5%

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

      2. times-frac 97.5%

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

      3. +-commutative 97.5%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in i around 0 88.5%

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

   if 2.80000000000000005e51 < alpha

   1. Initial program 18.9%

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

      1. associate-/l/ 18.1%

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

      2. *-commutative 18.1%

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

      3. times-frac 37.5%

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

      4. associate-+l+ 37.5%

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

      5. fma-def 37.5%

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

      6. +-commutative 37.5%

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

      7. fma-def 37.5%

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

   3. Simplified 37.5%

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

   4. Taylor expanded in alpha around inf 8.2%

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

   5. Taylor expanded in i around 0 69.1%

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

      1. sub-neg 69.1%

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

      2. associate-+r+ 69.1%

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

      3. +-commutative 69.1%

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

      4. distribute-lft1-in 69.1%

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

      5. metadata-eval 69.1%

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

      6. mul0-lft 69.1%

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

      7. +-lft-identity 69.1%

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

      8. mul-1-neg 69.1%

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

      9. remove-double-neg 69.1%

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

      10. *-commutative 69.1%

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

      11. distribute-rgt1-in 69.1%

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

   7. Simplified 69.1%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 \cdot \left(\beta + 1\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 89.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+150}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 \cdot \left(\beta + 1\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.4e+150)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (+ (* 4.0 (/ i alpha)) (/ (* 2.0 (+ beta 1.0)) alpha)) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.4e+150) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + ((2.0 * (beta + 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 <= 2.4d+150) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((4.0d0 * (i / alpha)) + ((2.0d0 * (beta + 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 <= 2.4e+150) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + ((2.0 * (beta + 1.0)) / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.4e+150:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((4.0 * (i / alpha)) + ((2.0 * (beta + 1.0)) / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.4e+150)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(Float64(2.0 * Float64(beta + 1.0)) / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.4e+150)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((4.0 * (i / alpha)) + ((2.0 * (beta + 1.0)) / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.4e+150], N[(N[(1.0 + N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.40000000000000003e150

   1. Initial program 80.3%

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

   2. Taylor expanded in beta around inf 92.1%

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

   if 2.40000000000000003e150 < alpha

   1. Initial program 1.2%

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

      1. associate-/l/ 0.0%

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

      2. *-commutative 0.0%

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

      3. times-frac 26.3%

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

      4. associate-+l+ 26.3%

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

      5. fma-def 26.3%

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

      6. +-commutative 26.3%

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

      7. fma-def 26.3%

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

   3. Simplified 26.3%

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

   4. Taylor expanded in alpha around inf 7.5%

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

   5. Taylor expanded in i around 0 80.2%

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

      1. sub-neg 80.2%

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

      2. associate-+r+ 80.2%

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

      3. +-commutative 80.2%

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

      4. distribute-lft1-in 80.2%

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

      5. metadata-eval 80.2%

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

      6. mul0-lft 80.2%

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

      7. +-lft-identity 80.2%

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

      8. mul-1-neg 80.2%

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

      9. remove-double-neg 80.2%

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

      10. *-commutative 80.2%

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

      11. distribute-rgt1-in 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+150}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 \cdot \left(\beta + 1\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 83.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.55 \cdot 10^{+51}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.55e+51)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.55e+51) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.55d+51) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (((i * 4.0d0) + (2.0d0 + (beta * 2.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.55e+51) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.55e+51:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.55e+51)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.55e+51)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.55e+51], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.55000000000000005e51

   1. Initial program 84.3%

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

      1. expm1-log1p-u 82.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 82.6%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\beta + \alpha}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. +-commutative 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. fma-udef 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. +-commutative 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \color{blue}{\beta + \alpha}\right)}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 95.8%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 95.8%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 98.9%

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

      3. associate-/r/ 98.9%

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

      4. *-commutative 98.9%

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

      5. +-commutative 98.9%

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

      6. +-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in alpha around 0 82.5%

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

      1. unpow2 82.5%

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

      2. times-frac 97.5%

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

      3. +-commutative 97.5%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in i around 0 88.5%

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

   if 2.55000000000000005e51 < alpha

   1. Initial program 18.9%

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

      1. expm1-log1p-u 7.9%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 7.9%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 15.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 15.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\beta + \alpha}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. +-commutative 15.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. fma-udef 15.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. +-commutative 15.0%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \color{blue}{\beta + \alpha}\right)}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 15.0%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 15.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 37.6%

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

      3. associate-/r/ 37.6%

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

      4. *-commutative 37.6%

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

      5. +-commutative 37.6%

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

      6. +-commutative 37.6%

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

   5. Simplified 37.6%

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

   6. Taylor expanded in beta around 0 34.1%

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

   7. Taylor expanded in alpha around inf 69.1%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.55 \cdot 10^{+51}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 76.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.8 \cdot 10^{+187}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.8e+187)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta 2.0) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.8e+187) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.8d+187) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + 2.0d0) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.8e+187) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.8e+187:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + 2.0) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.8e+187)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + 2.0) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.8e+187)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + 2.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.8e+187], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.80000000000000018e187

   1. Initial program 77.1%

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

      1. expm1-log1p-u 72.7%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 72.7%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 85.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 85.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\beta + \alpha}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. +-commutative 85.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. fma-udef 85.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. +-commutative 85.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \color{blue}{\beta + \alpha}\right)}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 85.2%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 85.2%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 92.1%

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

      3. associate-/r/ 92.1%

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

      4. *-commutative 92.1%

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

      5. +-commutative 92.1%

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

      6. +-commutative 92.1%

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

   5. Simplified 92.1%

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

   6. Taylor expanded in alpha around 0 76.3%

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

      1. unpow2 76.3%

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

      2. times-frac 90.4%

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

      3. +-commutative 90.4%

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

   8. Simplified 90.4%

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

   9. Taylor expanded in i around 0 81.6%

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

   if 1.80000000000000018e187 < alpha

   1. Initial program 1.3%

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

   2. Taylor expanded in alpha around inf 14.1%

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

      1. mul-1-neg 14.1%

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

   4. Simplified 14.1%

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

   5. Taylor expanded in i around 0 7.1%

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

      1. mul-1-neg 7.1%

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

      2. unsub-neg 7.1%

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

      3. +-commutative 7.1%

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

   7. Simplified 7.1%

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

   8. Taylor expanded in alpha around inf 50.2%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.8 \cdot 10^{+187}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 80.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 6.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 6.5e+50)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 6.5e+50) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 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 <= 6.5d+50) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 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 <= 6.5e+50) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 6.5e+50:
		tmp = (1.0 + (beta / (beta + 2.0))) / 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 <= 6.5e+50)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 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 <= 6.5e+50)
		tmp = (1.0 + (beta / (beta + 2.0))) / 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, 6.5e+50], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.5000000000000003e50

   1. Initial program 84.3%

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

      1. expm1-log1p-u 82.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 82.6%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\beta + \alpha}}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. +-commutative 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. fma-udef 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\color{blue}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. +-commutative 95.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \color{blue}{\beta + \alpha}\right)}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 95.8%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 95.8%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + \alpha}{\frac{\mathsf{fma}\left(2, i, \beta + \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 98.9%

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

      3. associate-/r/ 98.9%

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

      4. *-commutative 98.9%

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

      5. +-commutative 98.9%

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

      6. +-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in alpha around 0 82.5%

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

      1. unpow2 82.5%

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

      2. times-frac 97.5%

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

      3. +-commutative 97.5%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in i around 0 88.5%

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

   if 6.5000000000000003e50 < alpha

   1. Initial program 18.9%

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

      1. associate-/l/ 18.1%

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

      2. *-commutative 18.1%

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

      3. times-frac 37.5%

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

      4. associate-+l+ 37.5%

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

      5. fma-def 37.5%

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

      6. +-commutative 37.5%

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

      7. fma-def 37.5%

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

   3. Simplified 37.5%

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

   4. Taylor expanded in alpha around inf 8.2%

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

   5. Taylor expanded in beta around 0 59.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 73.3% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+71}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 3.5e+71) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.5e+71) {
		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 <= 3.5d+71) 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 <= 3.5e+71) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.5e+71:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.5e+71)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.5e+71)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 3.5e+71], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 3.4999999999999999e71

   1. Initial program 78.2%

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

      1. associate-/l/ 78.0%

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

      2. *-commutative 78.0%

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

      3. times-frac 81.2%

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

      4. associate-+l+ 81.2%

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

      5. fma-def 81.2%

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

      6. +-commutative 81.2%

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

      7. fma-def 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in i around inf 76.4%

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

   if 3.4999999999999999e71 < beta

   1. Initial program 39.6%

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

      1. associate-/l/ 37.9%

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

      2. *-commutative 37.9%

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

      3. times-frac 91.2%

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

      4. associate-+l+ 91.2%

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

      5. fma-def 91.2%

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

      6. +-commutative 91.2%

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

      7. fma-def 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in beta around inf 79.0%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+71}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 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 68.2%

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

   1. associate-/l/ 67.7%

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

   2. *-commutative 67.7%

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

   3. times-frac 83.8%

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

   4. associate-+l+ 83.8%

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

   5. fma-def 83.8%

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

   6. +-commutative 83.8%

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

   7. fma-def 83.8%

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

3. Simplified 83.8%

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

4. Taylor expanded in i around inf 63.9%

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

5. Final simplification 63.9%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023187 
(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))