Octave 3.8, jcobi/3

Percentage Accurate: 99.8% → 99.8%

Time: 42.1s

Alternatives: 14

Speedup: 1.2×

• could not determine a ground truth

  1. alpha = 4.1334891586105655e+249
  2. beta = 6.759880953206116e+186

Specification

\[\alpha > -1 \land \beta > -1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 2: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \frac{\frac{\left(\alpha + \beta\right) + \left(1 + \alpha \cdot \beta\right)}{t_0}}{t_0 \cdot \left(\left(\alpha + \beta\right) + 3\right)} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (/
    (/ (+ (+ alpha beta) (+ 1.0 (* alpha beta))) t_0)
    (* t_0 (+ (+ alpha beta) 3.0)))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return (((alpha + beta) + (1.0 + (alpha * beta))) / t_0) / (t_0 * ((alpha + beta) + 3.0));
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = alpha + (beta + 2.0d0)
    code = (((alpha + beta) + (1.0d0 + (alpha * beta))) / t_0) / (t_0 * ((alpha + beta) + 3.0d0))
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return (((alpha + beta) + (1.0 + (alpha * beta))) / t_0) / (t_0 * ((alpha + beta) + 3.0));
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	return (((alpha + beta) + (1.0 + (alpha * beta))) / t_0) / (t_0 * ((alpha + beta) + 3.0))

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	return Float64(Float64(Float64(Float64(alpha + beta) + Float64(1.0 + Float64(alpha * beta))) / t_0) / Float64(t_0 * Float64(Float64(alpha + beta) + 3.0)))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = (((alpha + beta) + (1.0 + (alpha * beta))) / t_0) / (t_0 * ((alpha + beta) + 3.0));
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(1.0 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(N[(alpha + beta), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. associate-/l/ 99.8%

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

   2. associate-+l+ 99.8%

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

   3. *-commutative 99.8%

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

   4. metadata-eval 99.8%

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

   5. associate-+l+ 99.8%

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

   6. metadata-eval 99.8%

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

   7. associate-+l+ 99.8%

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

   8. metadata-eval 99.8%

      \[\leadsto \frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + \color{blue}{3}\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]

   9. metadata-eval 99.8%

      \[\leadsto \frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)} \]

   10. associate-+l+ 99.8%

       \[\leadsto \frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2\right)\right)}} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) + \left(\alpha \cdot \beta + 1\right)}{\alpha + \left(\beta + 2\right)}}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

4. Final simplification 99.8%

   \[\leadsto \frac{\frac{\left(\alpha + \beta\right) + \left(1 + \alpha \cdot \beta\right)}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)} \]

Alternative 3: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{t_0 \cdot t_0} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (* (/ (+ beta 1.0) (+ alpha (+ beta 3.0))) (/ (+ alpha 1.0) (* t_0 t_0)))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return ((beta + 1.0) / (alpha + (beta + 3.0))) * ((alpha + 1.0) / (t_0 * t_0));
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = alpha + (beta + 2.0d0)
    code = ((beta + 1.0d0) / (alpha + (beta + 3.0d0))) * ((alpha + 1.0d0) / (t_0 * t_0))
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return ((beta + 1.0) / (alpha + (beta + 3.0))) * ((alpha + 1.0) / (t_0 * t_0));
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	return ((beta + 1.0) / (alpha + (beta + 3.0))) * ((alpha + 1.0) / (t_0 * t_0))

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	return Float64(Float64(Float64(beta + 1.0) / Float64(alpha + Float64(beta + 3.0))) * Float64(Float64(alpha + 1.0) / Float64(t_0 * t_0)))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((alpha + 1.0) / (t_0 * t_0));
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. associate-/l/ 99.8%

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

   2. associate-/l/ 92.9%

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

   3. associate-+l+ 92.9%

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

   4. +-commutative 92.9%

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

   5. associate-+r+ 92.9%

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

   6. associate-+l+ 92.9%

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

   7. distribute-rgt1-in 92.9%

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

   8. *-rgt-identity 92.9%

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

   9. distribute-lft-out 92.9%

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

   10. +-commutative 92.9%

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

   11. times-frac 99.7%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

4. Final simplification 99.7%

   \[\leadsto \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

Alternative 4: 86.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \frac{\alpha + 1}{t_0 \cdot t_0} \cdot \frac{\beta + 1}{\beta + 3} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (* (/ (+ alpha 1.0) (* t_0 t_0)) (/ (+ beta 1.0) (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return ((alpha + 1.0) / (t_0 * t_0)) * ((beta + 1.0) / (beta + 3.0));
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = alpha + (beta + 2.0d0)
    code = ((alpha + 1.0d0) / (t_0 * t_0)) * ((beta + 1.0d0) / (beta + 3.0d0))
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return ((alpha + 1.0) / (t_0 * t_0)) * ((beta + 1.0) / (beta + 3.0));
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	return ((alpha + 1.0) / (t_0 * t_0)) * ((beta + 1.0) / (beta + 3.0))

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	return Float64(Float64(Float64(alpha + 1.0) / Float64(t_0 * t_0)) * Float64(Float64(beta + 1.0) / Float64(beta + 3.0)))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = ((alpha + 1.0) / (t_0 * t_0)) * ((beta + 1.0) / (beta + 3.0));
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. associate-/l/ 99.8%

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

   2. associate-/l/ 92.9%

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

   3. associate-+l+ 92.9%

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

   4. +-commutative 92.9%

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

   5. associate-+r+ 92.9%

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

   6. associate-+l+ 92.9%

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

   7. distribute-rgt1-in 92.9%

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

   8. *-rgt-identity 92.9%

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

   9. distribute-lft-out 92.9%

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

   10. +-commutative 92.9%

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

   11. times-frac 99.7%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

4. Taylor expanded in alpha around 0 85.5%

   \[\leadsto \color{blue}{\frac{\beta + 1}{\beta + 3}} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

5. Final simplification 85.5%

   \[\leadsto \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \cdot \frac{\beta + 1}{\beta + 3} \]

Alternative 5: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+19}:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{1}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \left(\beta - -5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta} + \frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+19)
   (* (+ beta 1.0) (/ 1.0 (* (+ beta 2.0) (+ 6.0 (* beta (- beta -5.0))))))
   (+ (/ (/ 1.0 beta) beta) (/ alpha (* beta beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+19) {
		tmp = (beta + 1.0) * (1.0 / ((beta + 2.0) * (6.0 + (beta * (beta - -5.0)))));
	} else {
		tmp = ((1.0 / beta) / beta) + (alpha / (beta * beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1d+19) then
        tmp = (beta + 1.0d0) * (1.0d0 / ((beta + 2.0d0) * (6.0d0 + (beta * (beta - (-5.0d0))))))
    else
        tmp = ((1.0d0 / beta) / beta) + (alpha / (beta * beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+19) {
		tmp = (beta + 1.0) * (1.0 / ((beta + 2.0) * (6.0 + (beta * (beta - -5.0)))));
	} else {
		tmp = ((1.0 / beta) / beta) + (alpha / (beta * beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1e+19:
		tmp = (beta + 1.0) * (1.0 / ((beta + 2.0) * (6.0 + (beta * (beta - -5.0)))))
	else:
		tmp = ((1.0 / beta) / beta) + (alpha / (beta * beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+19)
		tmp = Float64(Float64(beta + 1.0) * Float64(1.0 / Float64(Float64(beta + 2.0) * Float64(6.0 + Float64(beta * Float64(beta - -5.0))))));
	else
		tmp = Float64(Float64(Float64(1.0 / beta) / beta) + Float64(alpha / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1e+19)
		tmp = (beta + 1.0) * (1.0 / ((beta + 2.0) * (6.0 + (beta * (beta - -5.0)))));
	else
		tmp = ((1.0 / beta) / beta) + (alpha / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1e+19], N[(N[(beta + 1.0), $MachinePrecision] * N[(1.0 / N[(N[(beta + 2.0), $MachinePrecision] * N[(6.0 + N[(beta * N[(beta - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision] + N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e19

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. distribute-lft-out 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-*r/ 99.7%

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

      11. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      2. associate-+r+ 99.7%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      3. distribute-lft-in 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)}} \]

      4. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \left(\beta + 2\right)} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\beta + 2\right)}} \]

   6. Taylor expanded in beta around 0 99.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \color{blue}{\left(\beta \cdot \left(5 + \alpha\right) + \left({\beta}^{2} + 2 \cdot \left(3 + \alpha\right)\right)\right)}} \]

   7. Taylor expanded in alpha around 0 64.4%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\frac{1}{\left(6 + \left(5 \cdot \beta + {\beta}^{2}\right)\right) \cdot \left(\beta + 2\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 64.4%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1}{\left(6 + \color{blue}{\left({\beta}^{2} + 5 \cdot \beta\right)}\right) \cdot \left(\beta + 2\right)} \]

      2. metadata-eval 64.4%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1}{\left(6 + \left({\beta}^{2} + \color{blue}{\left(--5\right)} \cdot \beta\right)\right) \cdot \left(\beta + 2\right)} \]

      3. cancel-sign-sub-inv 64.4%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1}{\left(6 + \color{blue}{\left({\beta}^{2} - -5 \cdot \beta\right)}\right) \cdot \left(\beta + 2\right)} \]

      4. unpow2 64.4%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1}{\left(6 + \left(\color{blue}{\beta \cdot \beta} - -5 \cdot \beta\right)\right) \cdot \left(\beta + 2\right)} \]

      5. distribute-rgt-out-- 64.4%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1}{\left(6 + \color{blue}{\beta \cdot \left(\beta - -5\right)}\right) \cdot \left(\beta + 2\right)} \]

   9. Simplified 64.4%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\frac{1}{\left(6 + \beta \cdot \left(\beta - -5\right)\right) \cdot \left(\beta + 2\right)}} \]

   if 1e19 < beta

   1. Initial program 99.8%

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

   2. Taylor expanded in beta around inf 99.8%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 99.8%

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

   4. Simplified 99.8%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in alpha around 0 99.8%

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

       1. unpow2 99.8%

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

       2. associate-/r* 99.8%

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

       3. unpow2 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+19}:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{1}{\left(\beta + 2\right) \cdot \left(6 + \beta \cdot \left(\beta - -5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta} + \frac{\alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 6: 73.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.3:\\ \;\;\;\;\left(\beta + 1\right) \cdot \left(0.08333333333333333 + \alpha \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.3)
   (* (+ beta 1.0) (+ 0.08333333333333333 (* alpha -0.027777777777777776)))
   (/ (/ (+ alpha 1.0) beta) (+ 1.0 (+ (+ alpha beta) 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3) {
		tmp = (beta + 1.0) * (0.08333333333333333 + (alpha * -0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + ((alpha + beta) + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.3d0) then
        tmp = (beta + 1.0d0) * (0.08333333333333333d0 + (alpha * (-0.027777777777777776d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (1.0d0 + ((alpha + beta) + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.3) {
		tmp = (beta + 1.0) * (0.08333333333333333 + (alpha * -0.027777777777777776));
	} else {
		tmp = ((alpha + 1.0) / beta) / (1.0 + ((alpha + beta) + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.3:
		tmp = (beta + 1.0) * (0.08333333333333333 + (alpha * -0.027777777777777776))
	else:
		tmp = ((alpha + 1.0) / beta) / (1.0 + ((alpha + beta) + 2.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.3)
		tmp = Float64(Float64(beta + 1.0) * Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(1.0 + Float64(Float64(alpha + beta) + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.3)
		tmp = (beta + 1.0) * (0.08333333333333333 + (alpha * -0.027777777777777776));
	else
		tmp = ((alpha + 1.0) / beta) / (1.0 + ((alpha + beta) + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.3], N[(N[(beta + 1.0), $MachinePrecision] * N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.30000000000000004

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      2. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      3. distribute-lft-in 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)}} \]

      4. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \left(\beta + 2\right)} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\beta + 2\right)}} \]

   6. Taylor expanded in beta around 0 93.0%

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

      1. distribute-rgt-out 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)\right)}} \]

      2. *-commutative 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   8. Simplified 93.0%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   9. Taylor expanded in alpha around 0 62.8%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\left(0.08333333333333333 + -0.027777777777777776 \cdot \alpha\right)} \]
   10. Step-by-step derivation

       1. *-commutative 62.8%

          \[\leadsto \left(\beta + 1\right) \cdot \left(0.08333333333333333 + \color{blue}{\alpha \cdot -0.027777777777777776}\right) \]

   11. Simplified 62.8%

       \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\left(0.08333333333333333 + \alpha \cdot -0.027777777777777776\right)} \]

   if 1.30000000000000004 < beta

   1. Initial program 99.8%

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

   2. Taylor expanded in beta around inf 97.8%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3:\\ \;\;\;\;\left(\beta + 1\right) \cdot \left(0.08333333333333333 + \alpha \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \]

Alternative 7: 73.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.95:\\ \;\;\;\;\left(\beta + 1\right) \cdot \left(0.08333333333333333 + \alpha \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta} + \frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.95)
   (* (+ beta 1.0) (+ 0.08333333333333333 (* alpha -0.027777777777777776)))
   (+ (/ (/ 1.0 beta) beta) (/ alpha (* beta beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.95) {
		tmp = (beta + 1.0) * (0.08333333333333333 + (alpha * -0.027777777777777776));
	} else {
		tmp = ((1.0 / beta) / beta) + (alpha / (beta * beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.95d0) then
        tmp = (beta + 1.0d0) * (0.08333333333333333d0 + (alpha * (-0.027777777777777776d0)))
    else
        tmp = ((1.0d0 / beta) / beta) + (alpha / (beta * beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.95) {
		tmp = (beta + 1.0) * (0.08333333333333333 + (alpha * -0.027777777777777776));
	} else {
		tmp = ((1.0 / beta) / beta) + (alpha / (beta * beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.95:
		tmp = (beta + 1.0) * (0.08333333333333333 + (alpha * -0.027777777777777776))
	else:
		tmp = ((1.0 / beta) / beta) + (alpha / (beta * beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.95)
		tmp = Float64(Float64(beta + 1.0) * Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 / beta) / beta) + Float64(alpha / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.95)
		tmp = (beta + 1.0) * (0.08333333333333333 + (alpha * -0.027777777777777776));
	else
		tmp = ((1.0 / beta) / beta) + (alpha / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.95], N[(N[(beta + 1.0), $MachinePrecision] * N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision] + N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.94999999999999996

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      2. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      3. distribute-lft-in 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)}} \]

      4. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \left(\beta + 2\right)} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\beta + 2\right)}} \]

   6. Taylor expanded in beta around 0 93.0%

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

      1. distribute-rgt-out 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)\right)}} \]

      2. *-commutative 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   8. Simplified 93.0%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   9. Taylor expanded in alpha around 0 62.8%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\left(0.08333333333333333 + -0.027777777777777776 \cdot \alpha\right)} \]
   10. Step-by-step derivation

       1. *-commutative 62.8%

          \[\leadsto \left(\beta + 1\right) \cdot \left(0.08333333333333333 + \color{blue}{\alpha \cdot -0.027777777777777776}\right) \]

   11. Simplified 62.8%

       \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\left(0.08333333333333333 + \alpha \cdot -0.027777777777777776\right)} \]

   if 1.94999999999999996 < beta

   1. Initial program 99.8%

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

   2. Taylor expanded in beta around inf 97.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 97.7%

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

   4. Simplified 97.7%

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

      1. clear-num 97.6%

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

      2. inv-pow 97.6%

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

   6. Applied egg-rr 97.6%

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

      1. unpow-1 97.6%

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

      2. associate-/l* 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in alpha around 0 97.7%

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

       1. unpow2 97.7%

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

       2. associate-/r* 97.7%

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

       3. unpow2 97.7%

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

   11. Simplified 97.7%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.95:\\ \;\;\;\;\left(\beta + 1\right) \cdot \left(0.08333333333333333 + \alpha \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta} + \frac{\alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 8: 73.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\beta + 1}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.8)
   (* 0.16666666666666666 (/ (+ beta 1.0) (+ beta 2.0)))
   (/ (+ alpha 1.0) (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.16666666666666666 * ((beta + 1.0) / (beta + 2.0));
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.8d0) then
        tmp = 0.16666666666666666d0 * ((beta + 1.0d0) / (beta + 2.0d0))
    else
        tmp = (alpha + 1.0d0) / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.8) {
		tmp = 0.16666666666666666 * ((beta + 1.0) / (beta + 2.0));
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.8:
		tmp = 0.16666666666666666 * ((beta + 1.0) / (beta + 2.0))
	else:
		tmp = (alpha + 1.0) / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.8)
		tmp = Float64(0.16666666666666666 * Float64(Float64(beta + 1.0) / Float64(beta + 2.0)));
	else
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.8)
		tmp = 0.16666666666666666 * ((beta + 1.0) / (beta + 2.0));
	else
		tmp = (alpha + 1.0) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.8], N[(0.16666666666666666 * N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.7999999999999998

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]

   4. Taylor expanded in beta around 0 98.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}} \]

   5. Taylor expanded in alpha around 0 63.3%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{\beta + 1}{\beta + 2}} \]

   if 2.7999999999999998 < beta

   1. Initial program 99.8%

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

   2. Taylor expanded in beta around inf 97.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 97.7%

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

   4. Simplified 97.7%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\beta + 1}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 9: 72.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\left(\beta + 1\right) \cdot \left(0.08333333333333333 + \alpha \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0)
   (* (+ beta 1.0) (+ 0.08333333333333333 (* alpha -0.027777777777777776)))
   (/ (+ alpha 1.0) (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = (beta + 1.0) * (0.08333333333333333 + (alpha * -0.027777777777777776));
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = (beta + 1.0d0) * (0.08333333333333333d0 + (alpha * (-0.027777777777777776d0)))
    else
        tmp = (alpha + 1.0d0) / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = (beta + 1.0) * (0.08333333333333333 + (alpha * -0.027777777777777776));
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = (beta + 1.0) * (0.08333333333333333 + (alpha * -0.027777777777777776))
	else:
		tmp = (alpha + 1.0) / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(Float64(beta + 1.0) * Float64(0.08333333333333333 + Float64(alpha * -0.027777777777777776)));
	else
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = (beta + 1.0) * (0.08333333333333333 + (alpha * -0.027777777777777776));
	else
		tmp = (alpha + 1.0) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(N[(beta + 1.0), $MachinePrecision] * N[(0.08333333333333333 + N[(alpha * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      2. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      3. distribute-lft-in 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)}} \]

      4. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \left(\beta + 2\right)} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\beta + 2\right)}} \]

   6. Taylor expanded in beta around 0 93.0%

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

      1. distribute-rgt-out 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)\right)}} \]

      2. *-commutative 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   8. Simplified 93.0%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   9. Taylor expanded in alpha around 0 62.8%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\left(0.08333333333333333 + -0.027777777777777776 \cdot \alpha\right)} \]
   10. Step-by-step derivation

       1. *-commutative 62.8%

          \[\leadsto \left(\beta + 1\right) \cdot \left(0.08333333333333333 + \color{blue}{\alpha \cdot -0.027777777777777776}\right) \]

   11. Simplified 62.8%

       \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\left(0.08333333333333333 + \alpha \cdot -0.027777777777777776\right)} \]

   if 2 < beta

   1. Initial program 99.8%

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

   2. Taylor expanded in beta around inf 97.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 97.7%

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

   4. Simplified 97.7%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\left(\beta + 1\right) \cdot \left(0.08333333333333333 + \alpha \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 10: 72.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\left(\beta + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0)
   (* (+ beta 1.0) 0.08333333333333333)
   (/ (+ alpha 1.0) (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = (beta + 1.0) * 0.08333333333333333;
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = (beta + 1.0d0) * 0.08333333333333333d0
    else
        tmp = (alpha + 1.0d0) / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = (beta + 1.0) * 0.08333333333333333;
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = (beta + 1.0) * 0.08333333333333333
	else:
		tmp = (alpha + 1.0) / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(Float64(beta + 1.0) * 0.08333333333333333);
	else
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = (beta + 1.0) * 0.08333333333333333;
	else
		tmp = (alpha + 1.0) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(N[(beta + 1.0), $MachinePrecision] * 0.08333333333333333), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      2. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      3. distribute-lft-in 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)}} \]

      4. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \left(\beta + 2\right)} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\beta + 2\right)}} \]

   6. Taylor expanded in beta around 0 93.0%

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

      1. distribute-rgt-out 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)\right)}} \]

      2. *-commutative 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   8. Simplified 93.0%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   9. Taylor expanded in alpha around 0 63.3%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{0.08333333333333333} \]

   if 2 < beta

   1. Initial program 99.8%

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

   2. Taylor expanded in beta around inf 97.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 97.7%

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

   4. Simplified 97.7%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\left(\beta + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 11: 63.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;\left(\beta + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.55)
   (* (+ beta 1.0) 0.08333333333333333)
   (/ 0.5 (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = (beta + 1.0) * 0.08333333333333333;
	} else {
		tmp = 0.5 / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.55d0) then
        tmp = (beta + 1.0d0) * 0.08333333333333333d0
    else
        tmp = 0.5d0 / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.55) {
		tmp = (beta + 1.0) * 0.08333333333333333;
	} else {
		tmp = 0.5 / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.55:
		tmp = (beta + 1.0) * 0.08333333333333333
	else:
		tmp = 0.5 / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.55)
		tmp = Float64(Float64(beta + 1.0) * 0.08333333333333333);
	else
		tmp = Float64(0.5 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.55)
		tmp = (beta + 1.0) * 0.08333333333333333;
	else
		tmp = 0.5 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.55], N[(N[(beta + 1.0), $MachinePrecision] * 0.08333333333333333), $MachinePrecision], N[(0.5 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55000000000000004

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      2. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      3. distribute-lft-in 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)}} \]

      4. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \left(\beta + 2\right)} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\beta + 2\right)}} \]

   6. Taylor expanded in beta around 0 93.0%

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

      1. distribute-rgt-out 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)\right)}} \]

      2. *-commutative 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   8. Simplified 93.0%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   9. Taylor expanded in alpha around 0 63.3%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{0.08333333333333333} \]

   if 1.55000000000000004 < beta

   1. Initial program 99.8%

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

      1. associate-/l/ 99.7%

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

      2. associate-/r* 88.3%

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

      3. associate-+l+ 88.3%

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

      4. +-commutative 88.3%

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

      5. associate-+r+ 88.3%

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

      6. associate-+l+ 88.3%

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

      7. distribute-rgt1-in 88.3%

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

      8. *-rgt-identity 88.3%

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

      9. distribute-lft-out 88.3%

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

      10. *-commutative 88.3%

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

      11. metadata-eval 88.3%

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

      12. associate-+l+ 88.3%

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

      13. +-commutative 88.3%

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

   3. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}} \]

   4. Taylor expanded in beta around inf 86.3%

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

      1. unpow2 86.3%

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

   6. Simplified 86.3%

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

   7. Taylor expanded in beta around 0 64.0%

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

      1. unpow2 64.0%

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

      2. +-commutative 64.0%

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

   9. Simplified 64.0%

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

   10. Taylor expanded in alpha around 0 64.0%

       \[\leadsto \color{blue}{\frac{0.5}{{\beta}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 64.0%

          \[\leadsto \frac{0.5}{\color{blue}{\beta \cdot \beta}} \]

   12. Simplified 64.0%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;\left(\beta + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\beta \cdot \beta}\\ \end{array} \]

Alternative 12: 72.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\left(\beta + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (* (+ beta 1.0) 0.08333333333333333) (/ 1.0 (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = (beta + 1.0) * 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = (beta + 1.0d0) * 0.08333333333333333d0
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = (beta + 1.0) * 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = (beta + 1.0) * 0.08333333333333333
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(Float64(beta + 1.0) * 0.08333333333333333);
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = (beta + 1.0) * 0.08333333333333333;
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(N[(beta + 1.0), $MachinePrecision] * 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      2. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      3. distribute-lft-in 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)}} \]

      4. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \left(\beta + 2\right)} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\beta + 2\right)}} \]

   6. Taylor expanded in beta around 0 93.0%

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

      1. distribute-rgt-out 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)\right)}} \]

      2. *-commutative 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   8. Simplified 93.0%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   9. Taylor expanded in alpha around 0 63.3%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{0.08333333333333333} \]

   if 2 < beta

   1. Initial program 99.8%

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

   2. Taylor expanded in beta around inf 97.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in alpha around 0 93.1%

      \[\leadsto \color{blue}{\frac{1}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 93.1%

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

   7. Simplified 93.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\left(\beta + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 13: 72.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\left(\beta + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (* (+ beta 1.0) 0.08333333333333333) (/ (/ 1.0 beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = (beta + 1.0) * 0.08333333333333333;
	} else {
		tmp = (1.0 / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = (beta + 1.0d0) * 0.08333333333333333d0
    else
        tmp = (1.0d0 / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = (beta + 1.0) * 0.08333333333333333;
	} else {
		tmp = (1.0 / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = (beta + 1.0) * 0.08333333333333333
	else:
		tmp = (1.0 / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(Float64(beta + 1.0) * 0.08333333333333333);
	else
		tmp = Float64(Float64(1.0 / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = (beta + 1.0) * 0.08333333333333333;
	else
		tmp = (1.0 / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(N[(beta + 1.0), $MachinePrecision] * 0.08333333333333333), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 99.9%

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

      1. associate-/l/ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. associate-+l+ 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-*r/ 99.8%

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

      11. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

      2. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

      3. distribute-lft-in 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)}} \]

      4. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \left(\beta + 2\right)} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\beta + 2\right)}} \]

   6. Taylor expanded in beta around 0 93.0%

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

      1. distribute-rgt-out 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)\right)}} \]

      2. *-commutative 93.0%

         \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   8. Simplified 93.0%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

   9. Taylor expanded in alpha around 0 63.3%

      \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{0.08333333333333333} \]

   if 2 < beta

   1. Initial program 99.8%

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

   2. Taylor expanded in beta around inf 97.7%

      \[\leadsto \color{blue}{\frac{1 + \alpha}{{\beta}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 97.7%

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

   4. Simplified 97.7%

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

      1. clear-num 97.6%

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

      2. inv-pow 97.6%

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

   6. Applied egg-rr 97.6%

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

      1. unpow-1 97.6%

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

      2. associate-/l* 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in alpha around 0 93.1%

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

       1. unpow2 93.1%

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

       2. associate-/r* 93.1%

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

   11. Simplified 93.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\left(\beta + 1\right) \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 14: 49.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \left(\beta + 1\right) \cdot 0.08333333333333333 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (* (+ beta 1.0) 0.08333333333333333))

C

double code(double alpha, double beta) {
	return (beta + 1.0) * 0.08333333333333333;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (beta + 1.0d0) * 0.08333333333333333d0
end function

Java

public static double code(double alpha, double beta) {
	return (beta + 1.0) * 0.08333333333333333;
}

Python

def code(alpha, beta):
	return (beta + 1.0) * 0.08333333333333333

Julia

function code(alpha, beta)
	return Float64(Float64(beta + 1.0) * 0.08333333333333333)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (beta + 1.0) * 0.08333333333333333;
end

Wolfram

code[alpha_, beta_] := N[(N[(beta + 1.0), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-/l/ 99.8%

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

   2. associate-+l+ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. associate-+l+ 99.8%

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

   6. distribute-rgt1-in 99.7%

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

   7. *-rgt-identity 99.7%

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

   8. distribute-lft-out 99.7%

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

   9. +-commutative 99.7%

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

   10. associate-*r/ 99.7%

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

   11. associate-*r/ 97.8%

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

3. Simplified 97.8%

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

   1. *-commutative 97.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]

   2. associate-+r+ 97.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right)} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]

   3. distribute-lft-in 97.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 3\right) \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)}} \]

   4. associate-+r+ 97.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \alpha + \left(\left(\alpha + \beta\right) + 3\right) \cdot \left(\beta + 2\right)} \]

   5. associate-+r+ 97.8%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)} \cdot \left(\beta + 2\right)} \]

5. Applied egg-rr 97.8%

   \[\leadsto \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \alpha + \left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\beta + 2\right)}} \]

6. Taylor expanded in beta around 0 76.5%

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

   1. distribute-rgt-out 76.4%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(3 + \alpha\right) \cdot \left(2 + \alpha\right)\right)}} \]

   2. *-commutative 76.4%

      \[\leadsto \left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \color{blue}{\left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

8. Simplified 76.4%

   \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{\frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)\right)}} \]

9. Taylor expanded in alpha around 0 52.0%

   \[\leadsto \left(\beta + 1\right) \cdot \color{blue}{0.08333333333333333} \]

10. Final simplification 52.0%

    \[\leadsto \left(\beta + 1\right) \cdot 0.08333333333333333 \]

Reproduce

herbie shell --seed 2023278 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))