Octave 3.8, jcobi/3

Percentage Accurate: 94.1% → 99.8%

Time: 22.9s

Alternatives: 16

Speedup: 1.4×

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: 94.1% 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, 1.4× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return (((1.0 + beta) / t_0) * ((1.0 + alpha) / 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 = (((1.0d0 + beta) / t_0) * ((1.0d0 + alpha) / 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 (((1.0 + beta) / t_0) * ((1.0 + alpha) / t_0)) / (alpha + (beta + 3.0));
}

Python

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

Julia

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

MATLAB

function tmp = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = (((1.0 + beta) / t_0) * ((1.0 + alpha) / 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[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 93.7%

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

3. Simplified 96.4%

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

   1. clear-num 96.4%

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

   2. associate-+r+ 96.4%

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

   3. *-commutative 96.4%

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

   4. frac-times 91.4%

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

   5. *-un-lft-identity 91.4%

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

   6. +-commutative 91.4%

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

   7. *-commutative 91.4%

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

   8. associate-+r+ 91.4%

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

6. Applied egg-rr 91.4%

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

   1. associate-/r* 96.5%

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

   2. associate-/l* 91.6%

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

   3. associate-*l/ 96.4%

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

   4. *-commutative 96.4%

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

   5. times-frac 99.8%

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

   6. associate-/r* 96.4%

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

   7. *-commutative 96.4%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

8. Simplified 99.8%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	return (((1.0 + beta) / t_0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / 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 = (((1.0d0 + beta) / t_0) / (alpha + (beta + 3.0d0))) * ((1.0d0 + alpha) / t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

3. Simplified 96.4%

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

   1. clear-num 96.4%

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

   2. associate-+r+ 96.4%

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

   3. *-commutative 96.4%

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

   4. frac-times 91.4%

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

   5. *-un-lft-identity 91.4%

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

   6. +-commutative 91.4%

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

   7. *-commutative 91.4%

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

   8. associate-+r+ 91.4%

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

6. Applied egg-rr 91.4%

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

   1. associate-/r* 96.5%

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

   2. associate-/l* 91.6%

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

   3. associate-*l/ 96.4%

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

   4. *-commutative 96.4%

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

   5. times-frac 99.8%

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

   6. associate-/r* 96.4%

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

   7. *-commutative 96.4%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 72.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.55:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \frac{-1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.55)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ beta alpha))))
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
    (/ (+ 1.0 (/ -1.0 beta)) (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.55) {
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 + (-1.0 / beta)) / (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.55d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (beta + alpha)))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * ((1.0d0 + ((-1.0d0) / beta)) / (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.55) {
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 + (-1.0 / beta)) / (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.55:
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 + (-1.0 / beta)) / (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.55)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 + Float64(-1.0 / beta)) / Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.55)
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 + (-1.0 / beta)) / (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.55], N[(0.25 / N[(1.0 + N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5499999999999998

   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. Add Preprocessing

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 69.0%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 2.5499999999999998 < beta

   1. Initial program 84.4%

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

   3. Simplified 91.6%

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

      1. clear-num 91.6%

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

      2. associate-+r+ 91.6%

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

      3. *-commutative 91.6%

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

      4. frac-times 79.1%

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

      5. *-un-lft-identity 79.1%

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

      6. +-commutative 79.1%

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

      7. *-commutative 79.1%

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

      8. associate-+r+ 79.1%

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

   6. Applied egg-rr 79.1%

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

      1. associate-/r* 91.7%

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

      2. associate-/l* 79.7%

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

      3. associate-*l/ 91.7%

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

      4. *-commutative 91.7%

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

      5. times-frac 99.7%

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

      6. associate-/r* 91.6%

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

      7. *-commutative 91.6%

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

      8. associate-/r* 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around inf 83.4%

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

       1. associate-*r/ 83.4%

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

       2. distribute-lft-in 83.4%

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

       3. metadata-eval 83.4%

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

       4. neg-mul-1 83.4%

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

       5. unsub-neg 83.4%

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

   11. Simplified 83.4%

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

   12. Taylor expanded in alpha around 0 83.8%

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

       1. +-commutative 83.8%

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

   14. Simplified 83.8%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.55:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \frac{-1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 88000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{t_0 \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{t_0} \cdot \frac{1 + \frac{-1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 88000000.0)
     (/ (/ (+ 1.0 beta) (+ beta 2.0)) (* t_0 (+ 3.0 (+ beta alpha))))
     (* (/ (+ 1.0 alpha) t_0) (/ (+ 1.0 (/ -1.0 beta)) (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 88000000.0) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / (t_0 * (3.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + (-1.0 / beta)) / (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 88000000.0d0) then
        tmp = ((1.0d0 + beta) / (beta + 2.0d0)) / (t_0 * (3.0d0 + (beta + alpha)))
    else
        tmp = ((1.0d0 + alpha) / t_0) * ((1.0d0 + ((-1.0d0) / beta)) / (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 88000000.0) {
		tmp = ((1.0 + beta) / (beta + 2.0)) / (t_0 * (3.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + (-1.0 / beta)) / (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 88000000.0:
		tmp = ((1.0 + beta) / (beta + 2.0)) / (t_0 * (3.0 + (beta + alpha)))
	else:
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + (-1.0 / beta)) / (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 88000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(beta + 2.0)) / Float64(t_0 * Float64(3.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 + Float64(-1.0 / beta)) / Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 88000000.0)
		tmp = ((1.0 + beta) / (beta + 2.0)) / (t_0 * (3.0 + (beta + alpha)));
	else
		tmp = ((1.0 + alpha) / t_0) * ((1.0 + (-1.0 / beta)) / (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 88000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.8e7

   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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in alpha around 0 84.9%

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

   if 8.8e7 < beta

   1. Initial program 84.1%

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

   3. Simplified 91.5%

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

      1. clear-num 91.5%

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

      2. associate-+r+ 91.5%

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

      3. *-commutative 91.5%

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

      4. frac-times 78.7%

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

      5. *-un-lft-identity 78.7%

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

      6. +-commutative 78.7%

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

      7. *-commutative 78.7%

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

      8. associate-+r+ 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. associate-/r* 91.6%

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

      2. associate-/l* 79.2%

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

      3. associate-*l/ 91.6%

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

      4. *-commutative 91.6%

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

      5. times-frac 99.7%

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

      6. associate-/r* 91.5%

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

      7. *-commutative 91.5%

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

      8. associate-/r* 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in beta around inf 84.4%

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

       1. associate-*r/ 84.4%

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

       2. distribute-lft-in 84.4%

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

       3. metadata-eval 84.4%

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

       4. neg-mul-1 84.4%

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

       5. unsub-neg 84.4%

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

   11. Simplified 84.4%

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

   12. Taylor expanded in alpha around 0 84.7%

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

       1. +-commutative 84.7%

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

   14. Simplified 84.7%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 88000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \frac{-1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ beta alpha))))
   (/
    (* (+ 1.0 alpha) (/ 1.0 (+ alpha (+ beta 2.0))))
    (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (alpha + (beta + 2.0)))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (beta + alpha)))
    else
        tmp = ((1.0d0 + alpha) * (1.0d0 / (alpha + (beta + 2.0d0)))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) * (1.0 / (alpha + (beta + 2.0)))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)))
	else:
		tmp = ((1.0 + alpha) * (1.0 / (alpha + (beta + 2.0)))) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 / Float64(alpha + Float64(beta + 2.0)))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	else
		tmp = ((1.0 + alpha) * (1.0 / (alpha + (beta + 2.0)))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.25 / N[(1.0 + N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   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. Add Preprocessing

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 69.0%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 2 < beta

   1. Initial program 84.4%

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

      1. clear-num 84.4%

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

      2. inv-pow 84.4%

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

      3. metadata-eval 84.4%

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

      4. associate-+r+ 84.4%

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

      5. +-commutative 84.4%

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

      6. *-commutative 84.4%

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

      7. associate-+r+ 84.4%

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

      8. +-commutative 84.4%

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

      9. distribute-rgt1-in 84.4%

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

      10. fma-def 84.4%

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

      11. +-commutative 84.4%

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

      12. metadata-eval 84.4%

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

      13. associate-+r+ 84.4%

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

   4. Applied egg-rr 84.4%

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

      1. unpow-1 84.4%

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

      2. +-commutative 84.4%

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

      3. fma-udef 84.4%

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

      4. *-commutative 84.4%

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

      5. +-commutative 84.4%

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

      6. associate-+r+ 84.4%

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

      7. +-commutative 84.4%

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

      8. distribute-lft1-in 84.4%

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

      9. +-commutative 84.4%

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

      10. *-commutative 84.4%

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

      11. +-commutative 84.4%

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

      12. associate-*l/ 99.8%

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

      13. +-commutative 99.8%

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

      14. *-commutative 99.8%

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

      15. +-commutative 99.8%

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

   6. Simplified 99.8%

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

      1. div-inv 99.7%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. +-commutative 99.6%

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

      8. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r/ 84.4%

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

      5. associate-/l* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in beta around inf 84.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \alpha\right) \cdot \frac{1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

3. Simplified 96.4%

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

   1. clear-num 96.4%

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

   2. associate-+r+ 96.4%

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

   3. *-commutative 96.4%

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

   4. frac-times 91.4%

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

   5. *-un-lft-identity 91.4%

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

   6. +-commutative 91.4%

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

   7. *-commutative 91.4%

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

   8. associate-+r+ 91.4%

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

6. Applied egg-rr 91.4%

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

   1. associate-/r* 96.5%

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

   2. associate-/l* 91.6%

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

   3. associate-*l/ 96.4%

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

   4. *-commutative 96.4%

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

   5. times-frac 99.8%

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

   6. associate-/r* 96.4%

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

   7. *-commutative 96.4%

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

   8. associate-/r* 99.8%

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

   9. +-commutative 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in alpha around 0 73.8%

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

    1. associate-/r* 74.5%

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

    2. +-commutative 74.5%

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

11. Simplified 74.5%

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

12. Final simplification 74.5%

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

Alternative 7: 72.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ beta alpha))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (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 <= 4.8d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (beta + alpha)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.8:
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.8)
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.8], N[(0.25 / N[(1.0 + N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999982

   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. Add Preprocessing

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 69.0%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 4.79999999999999982 < beta

   1. Initial program 84.4%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in beta around inf 83.5%

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

      1. associate-*l/ 83.6%

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

      2. +-commutative 83.6%

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

   7. Applied egg-rr 83.6%

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

      1. associate-*r/ 83.6%

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

      2. *-rgt-identity 83.6%

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

      3. +-commutative 83.6%

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

   9. Simplified 83.6%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.4)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ beta alpha))))
   (/ (/ (+ 1.0 alpha) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.4) {
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.4d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (beta + alpha)))
    else
        tmp = ((1.0d0 + alpha) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.4) {
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	} else {
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.4:
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)))
	else:
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.4)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.4)
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	else
		tmp = ((1.0 + alpha) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.4], N[(0.25 / N[(1.0 + N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.4000000000000004

   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. Add Preprocessing

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 69.0%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 4.4000000000000004 < beta

   1. Initial program 84.4%

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

      1. clear-num 84.4%

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

      2. inv-pow 84.4%

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

      3. metadata-eval 84.4%

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

      4. associate-+r+ 84.4%

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

      5. +-commutative 84.4%

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

      6. *-commutative 84.4%

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

      7. associate-+r+ 84.4%

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

      8. +-commutative 84.4%

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

      9. distribute-rgt1-in 84.4%

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

      10. fma-def 84.4%

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

      11. +-commutative 84.4%

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

      12. metadata-eval 84.4%

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

      13. associate-+r+ 84.4%

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

   4. Applied egg-rr 84.4%

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

      1. unpow-1 84.4%

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

      2. +-commutative 84.4%

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

      3. fma-udef 84.4%

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

      4. *-commutative 84.4%

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

      5. +-commutative 84.4%

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

      6. associate-+r+ 84.4%

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

      7. +-commutative 84.4%

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

      8. distribute-lft1-in 84.4%

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

      9. +-commutative 84.4%

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

      10. *-commutative 84.4%

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

      11. +-commutative 84.4%

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

      12. associate-*l/ 99.8%

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

      13. +-commutative 99.8%

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

      14. *-commutative 99.8%

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

      15. +-commutative 99.8%

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

   6. Simplified 99.8%

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

      1. div-inv 99.7%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. +-commutative 99.6%

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

      8. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r/ 84.4%

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

      5. associate-/l* 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+r+ 99.8%

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

   10. Simplified 99.8%

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

       1. inv-pow 99.8%

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

       2. add-sqr-sqrt 99.5%

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

       3. unpow-prod-down 99.4%

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

       4. +-commutative 99.4%

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

       5. +-commutative 99.4%

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

   12. Applied egg-rr 99.4%

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

       1. pow-sqr 99.6%

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

       2. associate-+r+ 99.6%

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

       3. metadata-eval 99.6%

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

   14. Simplified 99.6%

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

   15. Taylor expanded in beta around inf 83.7%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.4:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1 + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2)
   (/ 0.25 (+ beta 3.0))
   (* (/ 1.0 beta) (/ (+ 1.0 alpha) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = (1.0 / beta) * ((1.0 + alpha) / 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 <= 6.2d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = (1.0d0 / beta) * ((1.0d0 + alpha) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.2:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(Float64(1.0 + alpha) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.2)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.2], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   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. Add Preprocessing

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 67.5%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   5. Step-by-step derivation

      1. +-commutative 67.5%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   6. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   if 6.20000000000000018 < beta

   1. Initial program 84.4%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in beta around inf 83.5%

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

   6. Taylor expanded in beta around inf 83.3%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1 + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1 + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.2)
   (/ 0.25 (+ 1.0 (+ 2.0 (+ beta alpha))))
   (* (/ 1.0 beta) (/ (+ 1.0 alpha) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	} else {
		tmp = (1.0 / beta) * ((1.0 + alpha) / 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 <= 6.2d0) then
        tmp = 0.25d0 / (1.0d0 + (2.0d0 + (beta + alpha)))
    else
        tmp = (1.0d0 / beta) * ((1.0d0 + alpha) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.2) {
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	} else {
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.2:
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)))
	else:
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.2)
		tmp = Float64(0.25 / Float64(1.0 + Float64(2.0 + Float64(beta + alpha))));
	else
		tmp = Float64(Float64(1.0 / beta) * Float64(Float64(1.0 + alpha) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.2)
		tmp = 0.25 / (1.0 + (2.0 + (beta + alpha)));
	else
		tmp = (1.0 / beta) * ((1.0 + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.2], N[(0.25 / N[(1.0 + N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.20000000000000018

   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. Add Preprocessing

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 69.0%

      \[\leadsto \frac{\color{blue}{0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   if 6.20000000000000018 < beta

   1. Initial program 84.4%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in beta around inf 83.5%

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

   6. Taylor expanded in beta around inf 83.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1 + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.0) (/ 0.25 (+ beta 3.0)) (/ 1.0 (* beta (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.0d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = 1.0 / (beta * (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.0:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = 1.0 / (beta * (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.0)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.0)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = 1.0 / (beta * (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.0], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4

   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. Add Preprocessing

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 67.5%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   5. Step-by-step derivation

      1. +-commutative 67.5%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   6. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   if 4 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around -inf 83.7%

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

   4. Taylor expanded in alpha around 0 76.5%

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

      1. +-commutative 76.5%

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

   6. Simplified 76.5%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.6) (/ 0.25 (+ beta 3.0)) (/ (/ 1.0 beta) (+ beta 2.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = (1.0 / beta) / (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 <= 4.6d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = (1.0d0 / beta) / (beta + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.6) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = (1.0 / beta) / (beta + 2.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.6:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = (1.0 / beta) / (beta + 2.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.6)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.6)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = (1.0 / beta) / (beta + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.6], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5999999999999996

   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. Add Preprocessing

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 67.5%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   5. Step-by-step derivation

      1. +-commutative 67.5%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   6. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   if 4.5999999999999996 < beta

   1. Initial program 84.4%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in beta around inf 83.5%

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

   6. Taylor expanded in alpha around 0 76.5%

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

      1. associate-/r* 76.6%

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

   8. Simplified 76.6%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.0) (/ 0.25 (+ beta 3.0)) (/ (/ 1.0 beta) (+ beta 3.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 4.0d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = (1.0 / beta) / (beta + 3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.0:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = (1.0 / beta) / (beta + 3.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.0)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.0)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = (1.0 / beta) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.0], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4

   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. Add Preprocessing

   3. Taylor expanded in beta around 0 99.0%

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

   4. Taylor expanded in alpha around 0 67.5%

      \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
   5. Step-by-step derivation

      1. +-commutative 67.5%

         \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

   6. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

   if 4 < beta

   1. Initial program 84.4%

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

      1. associate-/l/ 79.7%

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

      2. +-commutative 79.7%

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

      3. associate-+l+ 79.7%

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

      4. *-commutative 79.7%

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

      5. metadata-eval 79.7%

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

      6. associate-+l+ 79.7%

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

      7. metadata-eval 79.7%

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

      8. +-commutative 79.7%

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

      9. metadata-eval 79.7%

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

      10. metadata-eval 79.7%

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

      11. associate-+l+ 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in beta around -inf 87.7%

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

   6. Taylor expanded in alpha around 0 76.6%

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

      1. associate-/r* 76.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   8. Simplified 76.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   9. Taylor expanded in beta around inf 76.6%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 11.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0) 0.16666666666666666 (/ 1.0 beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.16666666666666666;
	} else {
		tmp = 1.0 / 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 <= 6.0d0) then
        tmp = 0.16666666666666666d0
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.16666666666666666;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.16666666666666666
	else:
		tmp = 1.0 / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = 0.16666666666666666;
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.16666666666666666;
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.0], 0.16666666666666666, N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-+l+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in beta around -inf 29.2%

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

   6. Taylor expanded in alpha around 0 14.0%

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

      1. associate-/r* 14.0%

         \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   8. Simplified 14.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

   9. Taylor expanded in beta around 0 14.0%

      \[\leadsto \color{blue}{0.16666666666666666} \]

   if 6 < beta

   1. Initial program 84.4%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in beta around inf 83.5%

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

   6. Taylor expanded in alpha around inf 6.9%

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

4. Final simplification 11.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\beta + 3} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 0.25 (+ beta 3.0)))

C

double code(double alpha, double beta) {
	return 0.25 / (beta + 3.0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.25d0 / (beta + 3.0d0)
end function

Java

public static double code(double alpha, double beta) {
	return 0.25 / (beta + 3.0);
}

Python

def code(alpha, beta):
	return 0.25 / (beta + 3.0)

Julia

function code(alpha, beta)
	return Float64(0.25 / Float64(beta + 3.0))
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.25 / (beta + 3.0);
end

Wolfram

code[alpha_, beta_] := N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

3. Taylor expanded in beta around 0 67.2%

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

4. Taylor expanded in alpha around 0 43.4%

   \[\leadsto \color{blue}{\frac{0.25}{3 + \beta}} \]
5. Step-by-step derivation

   1. +-commutative 43.4%

      \[\leadsto \frac{0.25}{\color{blue}{\beta + 3}} \]

6. Simplified 43.4%

   \[\leadsto \color{blue}{\frac{0.25}{\beta + 3}} \]

7. Final simplification 43.4%

   \[\leadsto \frac{0.25}{\beta + 3} \]
8. Add Preprocessing

Alternative 16: 10.6% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 0.16666666666666666 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.16666666666666666)

C

double code(double alpha, double beta) {
	return 0.16666666666666666;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.16666666666666666d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.16666666666666666;
}

Python

def code(alpha, beta):
	return 0.16666666666666666

Julia

function code(alpha, beta)
	return 0.16666666666666666
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.16666666666666666;
end

Wolfram

code[alpha_, beta_] := 0.16666666666666666

Derivation

1. Initial program 93.7%

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

   1. associate-/l/ 91.6%

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

   2. +-commutative 91.6%

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

   3. associate-+l+ 91.6%

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

   4. *-commutative 91.6%

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

   5. metadata-eval 91.6%

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

   6. associate-+l+ 91.6%

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

   7. metadata-eval 91.6%

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

   8. +-commutative 91.6%

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

   9. metadata-eval 91.6%

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

   10. metadata-eval 91.6%

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

   11. associate-+l+ 91.6%

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

3. Simplified 91.6%

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

5. Taylor expanded in beta around -inf 52.5%

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

6. Taylor expanded in alpha around 0 39.0%

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

   1. associate-/r* 39.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

8. Simplified 39.0%

   \[\leadsto \color{blue}{\frac{\frac{1}{2 + \beta}}{3 + \beta}} \]

9. Taylor expanded in beta around 0 10.1%

   \[\leadsto \color{blue}{0.16666666666666666} \]

10. Final simplification 10.1%

    \[\leadsto 0.16666666666666666 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(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)))