Octave 3.8, jcobi/2

Percentage Accurate: 64.2% → 97.8%

Time: 30.8s

Alternatives: 8

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: 64.2% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := 2 + t_1\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{t_2} \leq -0.999998:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{t_2} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (+ 2.0 t_1)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) t_2) -0.999998)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0)
     (/
      (+
       (/ (* (- beta alpha) (/ (+ alpha beta) (+ beta (fma 2.0 i alpha)))) t_2)
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.999998) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / (beta + fma(2.0, i, alpha)))) / t_2) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(2.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / t_2) <= -0.999998)
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / Float64(beta + fma(2.0, i, alpha)))) / t_2) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], -0.999998], N[(N[(N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.6%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. associate-/l* 0.0%

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

      4. +-commutative 0.0%

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

      5. associate-+r+ 0.0%

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

      6. +-commutative 0.0%

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

      7. fma-def 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. expm1-def 0.0%

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

      2. expm1-log1p 14.1%

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

      3. associate-/r/ 14.1%

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

      4. +-commutative 14.1%

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

   5. Simplified 14.1%

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

   6. Taylor expanded in alpha around -inf 92.3%

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

      1. associate-*r/ 92.3%

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

   8. Simplified 92.3%

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

   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 82.8%

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

      1. expm1-log1p-u 77.2%

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

      2. expm1-udef 77.2%

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

      3. associate-/l* 92.8%

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

      4. +-commutative 92.8%

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

      5. associate-+r+ 92.8%

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

      6. +-commutative 92.8%

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

      7. fma-def 92.8%

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

   3. Applied egg-rr 92.8%

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

      1. expm1-def 92.8%

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

      2. expm1-log1p 99.8%

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

      3. associate-/r/ 99.8%

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

      4. +-commutative 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 98.3%

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

Alternative 2: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := 2 + t_1\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{t_2} \leq -0.999998:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 + t_0\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{t_2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (+ 2.0 t_1)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) t_2) -0.999998)
     (/ (/ (+ t_0 (+ 2.0 t_0)) alpha) 2.0)
     (/ (+ 1.0 (/ (- beta alpha) t_2)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.999998) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / t_2)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = beta + (2.0d0 * i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = 2.0d0 + t_1
    if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= (-0.999998d0)) then
        tmp = ((t_0 + (2.0d0 + t_0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / t_2)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.999998) {
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / t_2)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = 2.0 + t_1
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.999998:
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / t_2)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(2.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / t_2) <= -0.999998)
		tmp = Float64(Float64(Float64(t_0 + Float64(2.0 + t_0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / t_2)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = 2.0 + t_1;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.999998)
		tmp = ((t_0 + (2.0 + t_0)) / alpha) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / t_2)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], -0.999998], N[(N[(N[(t$95$0 + N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / t$95$2), $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 3.6%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. associate-/l* 0.0%

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

      4. +-commutative 0.0%

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

      5. associate-+r+ 0.0%

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

      6. +-commutative 0.0%

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

      7. fma-def 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. expm1-def 0.0%

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

      2. expm1-log1p 14.1%

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

      3. associate-/r/ 14.1%

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

      4. +-commutative 14.1%

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

   5. Simplified 14.1%

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

   6. Taylor expanded in alpha around -inf 92.3%

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

      1. associate-*r/ 92.3%

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

   8. Simplified 92.3%

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

   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 82.8%

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

   2. Taylor expanded in i around 0 98.6%

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

4. Final simplification 97.3%

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

Alternative 3: 85.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{+132}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.9e+132)
   (/ (+ 1.0 (/ (- beta alpha) (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (+ 2.0 (* 2.0 (+ beta i))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.9e+132) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + (2.0 * (beta + i))) / 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 <= 2.9d+132) then
        tmp = (1.0d0 + ((beta - alpha) / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((2.0d0 + (2.0d0 * (beta + i))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.9e+132) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.9e+132:
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.9e+132)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(2.0 * Float64(beta + i))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.9e+132)
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.9e+132], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(2.0 * N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.8999999999999999e132

   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 i around 0 94.3%

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

   if 2.8999999999999999e132 < alpha

   1. Initial program 1.5%

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

   2. Taylor expanded in i around 0 14.5%

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

   3. Taylor expanded in alpha around inf 70.3%

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

      1. distribute-lft-out 70.3%

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

   5. Simplified 70.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{+132}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 4: 78.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.65 \cdot 10^{+132}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.65e+132)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* 2.0 (+ beta i))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.65e+132) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.65d+132) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (2.0d0 * (beta + i))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.65e+132) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.65e+132:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.65e+132)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(2.0 * Float64(beta + i))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.65e+132)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.65e+132], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(2.0 * N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.65e132

   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 i around 0 94.3%

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

   3. Taylor expanded in i around 0 80.3%

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

      1. +-commutative 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in alpha around 0 84.8%

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

   if 2.65e132 < alpha

   1. Initial program 1.5%

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

   2. Taylor expanded in i around 0 14.5%

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

   3. Taylor expanded in alpha around inf 70.3%

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

      1. distribute-lft-out 70.3%

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

   5. Simplified 70.3%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.65 \cdot 10^{+132}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 75.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2.9 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 2.9e+85) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.9e+85) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 2.9d+85) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.9e+85) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 2.9e+85:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 2.9e+85)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 2.9e+85)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 2.9e+85], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 2.89999999999999997e85

   1. Initial program 61.6%

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

   2. Taylor expanded in i around 0 74.2%

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

   3. Taylor expanded in i around 0 73.4%

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

      1. +-commutative 73.4%

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

   5. Simplified 73.4%

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

   6. Taylor expanded in alpha around 0 70.3%

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

   if 2.89999999999999997e85 < i

   1. Initial program 75.6%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in i around inf 82.3%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.9 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 6: 78.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+131}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.75e+131)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.75e+131) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.75d+131) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.75e+131) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.75e+131:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.75e+131)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.75e+131)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.75e+131], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.7499999999999999e131

   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 i around 0 94.3%

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

   3. Taylor expanded in i around 0 80.3%

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

      1. +-commutative 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in alpha around 0 84.8%

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

   if 1.7499999999999999e131 < alpha

   1. Initial program 1.5%

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

   2. Taylor expanded in i around 0 14.5%

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

   3. Taylor expanded in i around 0 10.0%

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

      1. +-commutative 10.0%

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

   5. Simplified 10.0%

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

   6. Taylor expanded in alpha around inf 67.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+131}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 72.3% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+70}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 2.7e+70) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.7e+70) {
		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 <= 2.7d+70) 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 <= 2.7e+70) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.7e+70:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.7e+70)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.7e+70)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2.7e+70], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7e70

   1. Initial program 80.3%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in i around inf 75.4%

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

   if 2.7e70 < beta

   1. Initial program 19.6%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in beta around inf 71.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+70}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 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 67.0%

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

3. Simplified 82.7%

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

4. Taylor expanded in i around inf 63.7%

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

5. Final simplification 63.7%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023283 
(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))