Octave 3.8, jcobi/2

Percentage Accurate: 63.1% → 97.5%

Time: 13.6s

Alternatives: 7

Speedup: 4.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.1% 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: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.6:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}}\right)}^{3}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.6)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)
     (/
      (fma
       (+ alpha beta)
       (/
        (pow (cbrt (/ (- beta alpha) (+ (+ alpha beta) (fma 2.0 i 2.0)))) 3.0)
        (+ alpha (fma 2.0 i beta)))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.6) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = fma((alpha + beta), (pow(cbrt(((beta - alpha) / ((alpha + beta) + fma(2.0, i, 2.0)))), 3.0) / (alpha + fma(2.0, i, beta))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.6)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0)))) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(alpha + beta), Float64((cbrt(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0)))) ^ 3.0) / Float64(alpha + fma(2.0, i, beta))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.6], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(alpha + beta), $MachinePrecision] * N[(N[Power[N[Power[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.599999999999999978

   1. Initial program 2.1%

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

   3. Simplified 10.9%

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

   5. Taylor expanded in alpha around inf 93.9%

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

   if -0.599999999999999978 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 79.2%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \frac{\color{blue}{\left(\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}\right) \cdot \sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]

      2. pow3 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \frac{\color{blue}{{\left(\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}\right)}^{3}}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]

      3. fma-define 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \frac{{\left(\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + \color{blue}{\left(2 \cdot i + 2\right)}\right)}}\right)}^{3}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]

      4. associate-+r+ 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \frac{{\left(\sqrt[3]{\frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}}\right)}^{3}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \frac{{\left(\sqrt[3]{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + \left(2 \cdot i + 2\right)}}\right)}^{3}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]

      6. fma-define 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \frac{{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}}}\right)}^{3}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]

   6. Applied egg-rr 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.6:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}}\right)}^{3}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.6:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.6)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        (+ alpha (+ beta (fma 2.0 i 2.0)))))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.6) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / (alpha + (beta + fma(2.0, i, 2.0))))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.6)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0))))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.6], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.599999999999999978

   1. Initial program 2.1%

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

   3. Simplified 10.9%

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

   5. Taylor expanded in alpha around inf 93.9%

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

   if -0.599999999999999978 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 79.2%

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

   3. Simplified 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.6:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.6:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\frac{1 - \frac{\left(\alpha + \beta\right) \cdot \left(\alpha - \beta\right)}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(-2 \cdot \frac{\alpha}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.6)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)
     (if (<= t_1 1.0)
       (/
        (-
         1.0
         (/
          (* (+ alpha beta) (- alpha beta))
          (*
           (+ (+ alpha beta) (+ 2.0 (* 2.0 i)))
           (+ beta (+ alpha (* 2.0 i))))))
        2.0)
       (+ 1.0 (* 0.5 (* -2.0 (/ alpha beta))))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.6) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else if (t_1 <= 1.0) {
		tmp = (1.0 - (((alpha + beta) * (alpha - beta)) / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i)))))) / 2.0;
	} else {
		tmp = 1.0 + (0.5 * (-2.0 * (alpha / beta)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)
    if (t_1 <= (-0.6d0)) then
        tmp = (((beta - beta) + (2.0d0 + ((beta * 2.0d0) + (i * 4.0d0)))) / alpha) / 2.0d0
    else if (t_1 <= 1.0d0) then
        tmp = (1.0d0 - (((alpha + beta) * (alpha - beta)) / (((alpha + beta) + (2.0d0 + (2.0d0 * i))) * (beta + (alpha + (2.0d0 * i)))))) / 2.0d0
    else
        tmp = 1.0d0 + (0.5d0 * ((-2.0d0) * (alpha / beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.6) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else if (t_1 <= 1.0) {
		tmp = (1.0 - (((alpha + beta) * (alpha - beta)) / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i)))))) / 2.0;
	} else {
		tmp = 1.0 + (0.5 * (-2.0 * (alpha / beta)));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)
	tmp = 0
	if t_1 <= -0.6:
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0
	elif t_1 <= 1.0:
		tmp = (1.0 - (((alpha + beta) * (alpha - beta)) / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i)))))) / 2.0
	else:
		tmp = 1.0 + (0.5 * (-2.0 * (alpha / beta)))
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -0.6)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0)))) / alpha) / 2.0);
	elseif (t_1 <= 1.0)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(alpha + beta) * Float64(alpha - beta)) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))) * Float64(beta + Float64(alpha + Float64(2.0 * i)))))) / 2.0);
	else
		tmp = Float64(1.0 + Float64(0.5 * Float64(-2.0 * Float64(alpha / beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	tmp = 0.0;
	if (t_1 <= -0.6)
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	elseif (t_1 <= 1.0)
		tmp = (1.0 - (((alpha + beta) * (alpha - beta)) / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i)))))) / 2.0;
	else
		tmp = 1.0 + (0.5 * (-2.0 * (alpha / beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.6], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[(1.0 - N[(N[(N[(alpha + beta), $MachinePrecision] * N[(alpha - beta), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 + N[(0.5 * N[(-2.0 * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.599999999999999978

   1. Initial program 2.1%

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

   3. Simplified 10.9%

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

   5. Taylor expanded in alpha around inf 93.9%

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

   if -0.599999999999999978 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 1

   1. Initial program 100.0%

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

      1. associate-/l/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   if 1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 9.2%

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

   3. Simplified 39.0%

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

   5. Taylor expanded in beta around inf 81.5%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \frac{\left(\alpha + -1 \cdot \alpha\right) - \left(2 + \left(2 \cdot \alpha + 4 \cdot i\right)\right)}{\beta}} \]

   6. Taylor expanded in alpha around inf 84.8%

      \[\leadsto 1 + 0.5 \cdot \color{blue}{\left(-2 \cdot \frac{\alpha}{\beta}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.6:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 1:\\ \;\;\;\;\frac{1 - \frac{\left(\alpha + \beta\right) \cdot \left(\alpha - \beta\right)}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(-2 \cdot \frac{\alpha}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.6e+67)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.6e+67) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.6d+67) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (((beta - beta) + (2.0d0 + ((beta * 2.0d0) + (i * 4.0d0)))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.6e+67) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.6e+67:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.6e+67)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0)))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.6e+67)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.6e+67], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.59999999999999991e67

   1. Initial program 82.3%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in i around 0 84.1%

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

   6. Taylor expanded in alpha around 0 87.2%

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

   if 1.59999999999999991e67 < alpha

   1. Initial program 8.8%

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

   3. Simplified 18.6%

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

   5. Taylor expanded in alpha around inf 71.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 1.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= (* 2.0 i) 1.5e+142) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((2.0 * i) <= 1.5e+142) {
		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 ((2.0d0 * i) <= 1.5d+142) 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 ((2.0 * i) <= 1.5e+142) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (2.0 * i) <= 1.5e+142:
		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 (Float64(2.0 * i) <= 1.5e+142)
		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 ((2.0 * i) <= 1.5e+142)
		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[N[(2.0 * i), $MachinePrecision], 1.5e+142], 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 (*.f64 #s(literal 2 binary64) i) < 1.49999999999999987e142

   1. Initial program 60.8%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in i around 0 74.4%

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

   6. Taylor expanded in alpha around 0 76.7%

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

   if 1.49999999999999987e142 < (*.f64 #s(literal 2 binary64) i)

   1. Initial program 78.1%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in i around inf 96.0%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 1.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+61}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 8.5e+61) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.5e+61) {
		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 <= 8.5d+61) 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 <= 8.5e+61) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 8.5e+61:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8.5e+61)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8.5e+61)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 8.5e+61], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 8.50000000000000035e61

   1. Initial program 78.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in i around inf 79.8%

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

   if 8.50000000000000035e61 < beta

   1. Initial program 37.1%

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

   3. Simplified 90.9%

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

      1. add-cube-cbrt 90.9%

         \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \frac{\color{blue}{\left(\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}\right) \cdot \sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]

      2. pow3 90.9%

         \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \frac{\color{blue}{{\left(\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}\right)}^{3}}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]

      3. fma-define 90.9%

         \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \frac{{\left(\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + \color{blue}{\left(2 \cdot i + 2\right)}\right)}}\right)}^{3}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]

      4. associate-+r+ 90.9%

         \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \frac{{\left(\sqrt[3]{\frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}}\right)}^{3}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]

      5. +-commutative 90.9%

         \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \frac{{\left(\sqrt[3]{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + \left(2 \cdot i + 2\right)}}\right)}^{3}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]

      6. fma-define 90.9%

         \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \frac{{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}}}\right)}^{3}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]

   6. Applied egg-rr 90.9%

      \[\leadsto \frac{\mathsf{fma}\left(\alpha + \beta, \frac{\color{blue}{{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + \mathsf{fma}\left(2, i, 2\right)}}\right)}^{3}}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2} \]

   7. Taylor expanded in beta around inf 71.5%

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

Alternative 7: 62.0% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 65.4%

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

3. Simplified 72.5%

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

5. Taylor expanded in i around inf 65.9%

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

Reproduce

herbie shell --seed 2024165 
(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))