Octave 3.8, jcobi/3

Percentage Accurate: 94.1% → 99.8%

Time: 17.5s

Alternatives: 23

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, 1.3× speedup

Math

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

FPCore

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

C

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

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 + alpha) * ((1.0d0 + beta) / t_0)) / t_0) / (1.0d0 + (2.0d0 + (alpha + beta)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. div-inv 94.0%

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

   2. +-commutative 94.0%

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

   3. associate-+l+ 94.0%

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

   4. *-commutative 94.0%

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

   5. metadata-eval 94.0%

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

   6. associate-+r+ 94.0%

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

   7. metadata-eval 94.0%

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

   8. associate-+r+ 94.0%

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

3. Applied egg-rr 94.0%

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

   1. associate-*l/ 94.0%

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

   2. associate-*r/ 94.0%

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

   3. *-rgt-identity 94.0%

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

   4. associate-+r+ 94.0%

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

   5. *-rgt-identity 94.0%

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

   6. +-commutative 94.0%

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

   7. distribute-rgt1-in 94.0%

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

   8. distribute-lft-in 94.0%

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

   9. associate-*r/ 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. associate-/l/ 92.9%

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

   2. associate-+l+ 92.9%

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

   3. +-commutative 92.9%

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

   4. associate-+r+ 92.9%

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

   5. associate-+l+ 92.9%

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

   6. distribute-rgt1-in 92.9%

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

   7. *-rgt-identity 92.9%

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

   8. distribute-lft-out 92.9%

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

   9. +-commutative 92.9%

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

   10. associate-*l/ 96.9%

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

   11. *-commutative 96.9%

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

   12. associate-*r/ 92.6%

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

3. Simplified 92.6%

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

   1. associate-*r/ 97.0%

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

   2. +-commutative 97.0%

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

5. Applied egg-rr 97.0%

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

   1. +-commutative 97.0%

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

   2. *-commutative 97.0%

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

   3. +-commutative 97.0%

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

   4. associate-*r/ 96.9%

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

   5. +-commutative 96.9%

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

   6. associate-/r* 99.7%

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

   7. +-commutative 99.7%

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

   8. +-commutative 99.7%

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

   9. +-commutative 99.7%

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

   10. associate-+r+ 99.7%

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

   11. +-commutative 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 71.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+42}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+42)
   (*
    (+ 1.0 alpha)
    (/ (/ (+ 1.0 beta) (+ alpha (+ beta 2.0))) (* (+ beta 2.0) (+ beta 3.0))))
   (/ (/ (- alpha -1.0) beta) (+ 3.0 (+ alpha beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+42) {
		tmp = (1.0 + alpha) * (((1.0 + beta) / (alpha + (beta + 2.0))) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / beta) / (3.0 + (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 <= 1d+42) then
        tmp = (1.0d0 + alpha) * (((1.0d0 + beta) / (alpha + (beta + 2.0d0))) / ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+42) {
		tmp = (1.0 + alpha) * (((1.0 + beta) / (alpha + (beta + 2.0))) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((alpha - -1.0) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1e+42:
		tmp = (1.0 + alpha) * (((1.0 + beta) / (alpha + (beta + 2.0))) / ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((alpha - -1.0) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+42)
		tmp = Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(1.0 + beta) / Float64(alpha + Float64(beta + 2.0))) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1e+42)
		tmp = (1.0 + alpha) * (((1.0 + beta) / (alpha + (beta + 2.0))) / ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((alpha - -1.0) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1e+42], N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000004e42

   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/ 99.0%

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

      2. associate-+l+ 99.0%

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

      3. +-commutative 99.0%

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

      4. associate-+r+ 99.0%

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

      5. associate-+l+ 99.0%

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

      6. distribute-rgt1-in 98.9%

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

      7. *-rgt-identity 98.9%

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

      8. distribute-lft-out 98.9%

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

      9. +-commutative 98.9%

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

      10. associate-*l/ 98.9%

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

      11. *-commutative 98.9%

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

      12. associate-*r/ 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in alpha around 0 64.9%

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

   if 1.00000000000000004e42 < beta

   1. Initial program 79.6%

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

   2. Taylor expanded in beta around -inf 90.9%

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

      1. *-un-lft-identity 90.9%

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

      2. mul-1-neg 90.9%

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

      3. fma-neg 90.9%

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

      4. metadata-eval 90.9%

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

      5. metadata-eval 90.9%

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

      6. associate-+l+ 90.9%

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

      7. metadata-eval 90.9%

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

   4. Applied egg-rr 90.9%

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

      1. *-lft-identity 90.9%

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

      2. metadata-eval 90.9%

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

      3. fma-neg 90.9%

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

      4. distribute-neg-frac 90.9%

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

      5. sub-neg 90.9%

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

      6. neg-mul-1 90.9%

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

      7. distribute-neg-in 90.9%

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

      8. +-commutative 90.9%

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

      9. mul-1-neg 90.9%

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

      10. distribute-lft-in 90.9%

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

      11. metadata-eval 90.9%

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

      12. neg-mul-1 90.9%

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

      13. unsub-neg 90.9%

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

      14. +-commutative 90.9%

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

      15. +-commutative 90.9%

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

   6. Simplified 90.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+42}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 4: 71.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 10^{+42}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{t_0}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 1e+42)
     (* (+ 1.0 alpha) (/ (/ (+ 1.0 beta) t_0) (* (+ beta 2.0) (+ beta 3.0))))
     (/ (/ (+ 1.0 alpha) t_0) (+ 1.0 (+ 2.0 (+ alpha beta)))))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 1e+42) {
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_0) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / (1.0 + (2.0 + (alpha + beta)));
	}
	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 <= 1d+42) then
        tmp = (1.0d0 + alpha) * (((1.0d0 + beta) / t_0) / ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = ((1.0d0 + alpha) / t_0) / (1.0d0 + (2.0d0 + (alpha + beta)))
    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 <= 1e+42) {
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_0) / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = ((1.0 + alpha) / t_0) / (1.0 + (2.0 + (alpha + beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 1e+42:
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_0) / ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = ((1.0 + alpha) / t_0) / (1.0 + (2.0 + (alpha + beta)))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 1e+42)
		tmp = Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(1.0 + beta) / t_0) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(1.0 + Float64(2.0 + Float64(alpha + beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 1e+42)
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_0) / ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = ((1.0 + alpha) / t_0) / (1.0 + (2.0 + (alpha + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+42], N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000004e42

   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/ 99.0%

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

      2. associate-+l+ 99.0%

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

      3. +-commutative 99.0%

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

      4. associate-+r+ 99.0%

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

      5. associate-+l+ 99.0%

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

      6. distribute-rgt1-in 98.9%

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

      7. *-rgt-identity 98.9%

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

      8. distribute-lft-out 98.9%

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

      9. +-commutative 98.9%

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

      10. associate-*l/ 98.9%

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

      11. *-commutative 98.9%

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

      12. associate-*r/ 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in alpha around 0 64.9%

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

   if 1.00000000000000004e42 < beta

   1. Initial program 79.6%

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

      1. div-inv 79.6%

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

      2. +-commutative 79.6%

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

      3. associate-+l+ 79.6%

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

      4. *-commutative 79.6%

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

      5. metadata-eval 79.6%

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

      6. associate-+r+ 79.6%

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

      7. metadata-eval 79.6%

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

      8. associate-+r+ 79.6%

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

   3. Applied egg-rr 79.6%

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

      1. associate-*l/ 79.6%

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

      2. associate-*r/ 79.6%

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

      3. *-rgt-identity 79.6%

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

      4. associate-+r+ 79.6%

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

      5. *-rgt-identity 79.6%

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

      6. +-commutative 79.6%

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

      7. distribute-rgt1-in 79.6%

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

      8. distribute-lft-in 79.6%

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

      9. associate-*r/ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in beta around inf 91.1%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+42}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]

Alternative 5: 71.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \frac{1 + \beta}{t_0}\\ \mathbf{if}\;\beta \leq 10^{+42}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{t_1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot t_1}{t_0}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))) (t_1 (/ (+ 1.0 beta) t_0)))
   (if (<= beta 1e+42)
     (* (+ 1.0 alpha) (/ t_1 (* (+ beta 2.0) (+ beta 3.0))))
     (/ (/ (* (+ 1.0 alpha) t_1) t_0) beta))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (1.0 + beta) / t_0;
	double tmp;
	if (beta <= 1e+42) {
		tmp = (1.0 + alpha) * (t_1 / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = (((1.0 + alpha) * t_1) / t_0) / beta;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    t_1 = (1.0d0 + beta) / t_0
    if (beta <= 1d+42) then
        tmp = (1.0d0 + alpha) * (t_1 / ((beta + 2.0d0) * (beta + 3.0d0)))
    else
        tmp = (((1.0d0 + alpha) * t_1) / t_0) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double t_1 = (1.0 + beta) / t_0;
	double tmp;
	if (beta <= 1e+42) {
		tmp = (1.0 + alpha) * (t_1 / ((beta + 2.0) * (beta + 3.0)));
	} else {
		tmp = (((1.0 + alpha) * t_1) / t_0) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	t_1 = (1.0 + beta) / t_0
	tmp = 0
	if beta <= 1e+42:
		tmp = (1.0 + alpha) * (t_1 / ((beta + 2.0) * (beta + 3.0)))
	else:
		tmp = (((1.0 + alpha) * t_1) / t_0) / beta
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	t_1 = Float64(Float64(1.0 + beta) / t_0)
	tmp = 0.0
	if (beta <= 1e+42)
		tmp = Float64(Float64(1.0 + alpha) * Float64(t_1 / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) * t_1) / t_0) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	t_1 = (1.0 + beta) / t_0;
	tmp = 0.0;
	if (beta <= 1e+42)
		tmp = (1.0 + alpha) * (t_1 / ((beta + 2.0) * (beta + 3.0)));
	else
		tmp = (((1.0 + alpha) * t_1) / t_0) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[beta, 1e+42], N[(N[(1.0 + alpha), $MachinePrecision] * N[(t$95$1 / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision] / beta), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.00000000000000004e42

   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/ 99.0%

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

      2. associate-+l+ 99.0%

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

      3. +-commutative 99.0%

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

      4. associate-+r+ 99.0%

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

      5. associate-+l+ 99.0%

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

      6. distribute-rgt1-in 98.9%

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

      7. *-rgt-identity 98.9%

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

      8. distribute-lft-out 98.9%

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

      9. +-commutative 98.9%

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

      10. associate-*l/ 98.9%

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

      11. *-commutative 98.9%

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

      12. associate-*r/ 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in alpha around 0 64.9%

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

   if 1.00000000000000004e42 < beta

   1. Initial program 79.6%

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

      1. div-inv 79.6%

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

      2. +-commutative 79.6%

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

      3. associate-+l+ 79.6%

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

      4. *-commutative 79.6%

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

      5. metadata-eval 79.6%

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

      6. associate-+r+ 79.6%

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

      7. metadata-eval 79.6%

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

      8. associate-+r+ 79.6%

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

   3. Applied egg-rr 79.6%

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

      1. associate-*l/ 79.6%

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

      2. associate-*r/ 79.6%

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

      3. *-rgt-identity 79.6%

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

      4. associate-+r+ 79.6%

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

      5. *-rgt-identity 79.6%

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

      6. +-commutative 79.6%

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

      7. distribute-rgt1-in 79.6%

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

      8. distribute-lft-in 79.6%

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

      9. associate-*r/ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in beta around inf 91.1%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+42}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}}{\beta}\\ \end{array} \]

Alternative 6: 72.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 94.0%

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

   1. div-inv 94.0%

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

   2. +-commutative 94.0%

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

   3. associate-+l+ 94.0%

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

   4. *-commutative 94.0%

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

   5. metadata-eval 94.0%

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

   6. associate-+r+ 94.0%

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

   7. metadata-eval 94.0%

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

   8. associate-+r+ 94.0%

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

3. Applied egg-rr 94.0%

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

   1. associate-*l/ 94.0%

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

   2. associate-*r/ 94.0%

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

   3. *-rgt-identity 94.0%

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

   4. associate-+r+ 94.0%

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

   5. *-rgt-identity 94.0%

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

   6. +-commutative 94.0%

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

   7. distribute-rgt1-in 94.0%

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

   8. distribute-lft-in 94.0%

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

   9. associate-*r/ 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in alpha around 0 73.0%

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

7. Final simplification 73.0%

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

Alternative 7: 70.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{0.5 \cdot \frac{1 + \alpha}{\alpha + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \left(1 + \frac{-1}{\beta}\right)}{\alpha + \left(\beta + 2\right)}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.5)
   (/ (* 0.5 (/ (+ 1.0 alpha) (+ alpha 2.0))) (+ beta 3.0))
   (/
    (/ (* (+ 1.0 alpha) (+ 1.0 (/ -1.0 beta))) (+ alpha (+ beta 2.0)))
    (+ beta 3.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = (0.5 * ((1.0 + alpha) / (alpha + 2.0))) / (beta + 3.0);
	} else {
		tmp = (((1.0 + alpha) * (1.0 + (-1.0 / beta))) / (alpha + (beta + 2.0))) / (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 <= 2.5d0) then
        tmp = (0.5d0 * ((1.0d0 + alpha) / (alpha + 2.0d0))) / (beta + 3.0d0)
    else
        tmp = (((1.0d0 + alpha) * (1.0d0 + ((-1.0d0) / beta))) / (alpha + (beta + 2.0d0))) / (beta + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.5) {
		tmp = (0.5 * ((1.0 + alpha) / (alpha + 2.0))) / (beta + 3.0);
	} else {
		tmp = (((1.0 + alpha) * (1.0 + (-1.0 / beta))) / (alpha + (beta + 2.0))) / (beta + 3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.5:
		tmp = (0.5 * ((1.0 + alpha) / (alpha + 2.0))) / (beta + 3.0)
	else:
		tmp = (((1.0 + alpha) * (1.0 + (-1.0 / beta))) / (alpha + (beta + 2.0))) / (beta + 3.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.5)
		tmp = Float64(Float64(0.5 * Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0))) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) * Float64(1.0 + Float64(-1.0 / beta))) / Float64(alpha + Float64(beta + 2.0))) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.5)
		tmp = (0.5 * ((1.0 + alpha) / (alpha + 2.0))) / (beta + 3.0);
	else
		tmp = (((1.0 + alpha) * (1.0 + (-1.0 / beta))) / (alpha + (beta + 2.0))) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.5], N[(N[(0.5 * N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.5

   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. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+r+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-+r+ 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. *-rgt-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-rgt1-in 99.8%

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

      8. distribute-lft-in 99.8%

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

      9. associate-*r/ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in alpha around 0 64.9%

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

   7. Taylor expanded in alpha around 0 64.0%

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

   8. Taylor expanded in beta around 0 63.3%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{1 + \alpha}{2 + \alpha}}}{\beta + 3} \]
   9. Step-by-step derivation

      1. +-commutative 63.3%

         \[\leadsto \frac{0.5 \cdot \frac{1 + \alpha}{\color{blue}{\alpha + 2}}}{\beta + 3} \]

   10. Simplified 63.3%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{1 + \alpha}{\alpha + 2}}}{\beta + 3} \]

   if 2.5 < beta

   1. Initial program 82.6%

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

      1. div-inv 82.6%

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

      2. +-commutative 82.6%

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

      3. associate-+l+ 82.6%

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

      4. *-commutative 82.6%

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

      5. metadata-eval 82.6%

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

      6. associate-+r+ 82.6%

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

      7. metadata-eval 82.6%

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

      8. associate-+r+ 82.6%

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

   3. Applied egg-rr 82.6%

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

      1. associate-*l/ 82.6%

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

      2. associate-*r/ 82.6%

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

      3. *-rgt-identity 82.6%

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

      4. associate-+r+ 82.6%

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

      5. *-rgt-identity 82.6%

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

      6. +-commutative 82.6%

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

      7. distribute-rgt1-in 82.6%

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

      8. distribute-lft-in 82.6%

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

      9. associate-*r/ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in alpha around 0 89.1%

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

   7. Taylor expanded in alpha around 0 88.8%

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

   8. Taylor expanded in beta around inf 88.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5:\\ \;\;\;\;\frac{0.5 \cdot \frac{1 + \alpha}{\alpha + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \left(1 + \frac{-1}{\beta}\right)}{\alpha + \left(\beta + 2\right)}}{\beta + 3}\\ \end{array} \]

Alternative 8: 71.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. div-inv 94.0%

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

   2. +-commutative 94.0%

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

   3. associate-+l+ 94.0%

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

   4. *-commutative 94.0%

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

   5. metadata-eval 94.0%

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

   6. associate-+r+ 94.0%

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

   7. metadata-eval 94.0%

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

   8. associate-+r+ 94.0%

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

3. Applied egg-rr 94.0%

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

   1. associate-*l/ 94.0%

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

   2. associate-*r/ 94.0%

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

   3. *-rgt-identity 94.0%

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

   4. associate-+r+ 94.0%

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

   5. *-rgt-identity 94.0%

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

   6. +-commutative 94.0%

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

   7. distribute-rgt1-in 94.0%

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

   8. distribute-lft-in 94.0%

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

   9. associate-*r/ 99.8%

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

   10. +-commutative 99.8%

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

   11. +-commutative 99.8%

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

   12. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in alpha around 0 73.0%

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

7. Taylor expanded in alpha around 0 72.3%

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

8. Final simplification 72.3%

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

Alternative 9: 70.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.5 \cdot \frac{1 + \alpha}{\alpha + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.0)
   (/ (* 0.5 (/ (+ 1.0 alpha) (+ alpha 2.0))) (+ beta 3.0))
   (/ (/ (- alpha -1.0) beta) (+ 3.0 (+ alpha beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = (0.5 * ((1.0 + alpha) / (alpha + 2.0))) / (beta + 3.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (3.0 + (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 <= 4.0d0) then
        tmp = (0.5d0 * ((1.0d0 + alpha) / (alpha + 2.0d0))) / (beta + 3.0d0)
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.0) {
		tmp = (0.5 * ((1.0 + alpha) / (alpha + 2.0))) / (beta + 3.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.0:
		tmp = (0.5 * ((1.0 + alpha) / (alpha + 2.0))) / (beta + 3.0)
	else:
		tmp = ((alpha - -1.0) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.0)
		tmp = Float64(Float64(0.5 * Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0))) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.0)
		tmp = (0.5 * ((1.0 + alpha) / (alpha + 2.0))) / (beta + 3.0);
	else
		tmp = ((alpha - -1.0) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.0], N[(N[(0.5 * N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $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. Step-by-step derivation

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+r+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-+r+ 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. *-rgt-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-rgt1-in 99.8%

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

      8. distribute-lft-in 99.8%

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

      9. associate-*r/ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in alpha around 0 64.9%

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

   7. Taylor expanded in alpha around 0 64.0%

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

   8. Taylor expanded in beta around 0 63.3%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{1 + \alpha}{2 + \alpha}}}{\beta + 3} \]
   9. Step-by-step derivation

      1. +-commutative 63.3%

         \[\leadsto \frac{0.5 \cdot \frac{1 + \alpha}{\color{blue}{\alpha + 2}}}{\beta + 3} \]

   10. Simplified 63.3%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{1 + \alpha}{\alpha + 2}}}{\beta + 3} \]

   if 4 < beta

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around -inf 87.2%

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

      1. *-un-lft-identity 87.2%

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

      2. mul-1-neg 87.2%

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

      3. fma-neg 87.2%

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

      4. metadata-eval 87.2%

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

      5. metadata-eval 87.2%

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

      6. associate-+l+ 87.2%

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

      7. metadata-eval 87.2%

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

   4. Applied egg-rr 87.2%

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

      1. *-lft-identity 87.2%

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

      2. metadata-eval 87.2%

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

      3. fma-neg 87.2%

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

      4. distribute-neg-frac 87.2%

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

      5. sub-neg 87.2%

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

      6. neg-mul-1 87.2%

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

      7. distribute-neg-in 87.2%

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

      8. +-commutative 87.2%

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

      9. mul-1-neg 87.2%

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

      10. distribute-lft-in 87.2%

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

      11. metadata-eval 87.2%

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

      12. neg-mul-1 87.2%

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

      13. unsub-neg 87.2%

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

      14. +-commutative 87.2%

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

      15. +-commutative 87.2%

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

   6. Simplified 87.2%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.5 \cdot \frac{1 + \alpha}{\alpha + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 10: 70.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{0.5 \cdot \frac{1 + \alpha}{\alpha + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0)
   (/ (* 0.5 (/ (+ 1.0 alpha) (+ alpha 2.0))) (+ beta 3.0))
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) (+ beta 3.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = (0.5 * ((1.0 + alpha) / (alpha + 2.0))) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (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 = (0.5d0 * ((1.0d0 + alpha) / (alpha + 2.0d0))) / (beta + 3.0d0)
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / (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 = (0.5 * ((1.0 + alpha) / (alpha + 2.0))) / (beta + 3.0);
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + 3.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = (0.5 * ((1.0 + alpha) / (alpha + 2.0))) / (beta + 3.0)
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + 3.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(Float64(0.5 * Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0))) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / Float64(beta + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = (0.5 * ((1.0 + alpha) / (alpha + 2.0))) / (beta + 3.0);
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / (beta + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(N[(0.5 * N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $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. Step-by-step derivation

      1. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+r+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-+r+ 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. *-rgt-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-rgt1-in 99.8%

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

      8. distribute-lft-in 99.8%

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

      9. associate-*r/ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in alpha around 0 64.9%

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

   7. Taylor expanded in alpha around 0 64.0%

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

   8. Taylor expanded in beta around 0 63.3%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{1 + \alpha}{2 + \alpha}}}{\beta + 3} \]
   9. Step-by-step derivation

      1. +-commutative 63.3%

         \[\leadsto \frac{0.5 \cdot \frac{1 + \alpha}{\color{blue}{\alpha + 2}}}{\beta + 3} \]

   10. Simplified 63.3%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{1 + \alpha}{\alpha + 2}}}{\beta + 3} \]

   if 2 < beta

   1. Initial program 82.6%

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

      1. div-inv 82.6%

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

      2. +-commutative 82.6%

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

      3. associate-+l+ 82.6%

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

      4. *-commutative 82.6%

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

      5. metadata-eval 82.6%

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

      6. associate-+r+ 82.6%

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

      7. metadata-eval 82.6%

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

      8. associate-+r+ 82.6%

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

   3. Applied egg-rr 82.6%

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

      1. associate-*l/ 82.6%

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

      2. associate-*r/ 82.6%

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

      3. *-rgt-identity 82.6%

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

      4. associate-+r+ 82.6%

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

      5. *-rgt-identity 82.6%

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

      6. +-commutative 82.6%

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

      7. distribute-rgt1-in 82.6%

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

      8. distribute-lft-in 82.6%

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

      9. associate-*r/ 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in alpha around 0 89.1%

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

   7. Taylor expanded in beta around inf 87.3%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{0.5 \cdot \frac{1 + \alpha}{\alpha + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta + 3}\\ \end{array} \]

Alternative 11: 70.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.6:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + \left(\beta + 2\right)}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.6)
   (/ (/ 0.5 (+ alpha (+ beta 2.0))) (+ beta 3.0))
   (/ (/ (- alpha -1.0) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.6) {
		tmp = (0.5 / (alpha + (beta + 2.0))) / (beta + 3.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 8.6d0) then
        tmp = (0.5d0 / (alpha + (beta + 2.0d0))) / (beta + 3.0d0)
    else
        tmp = ((alpha - (-1.0d0)) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.6) {
		tmp = (0.5 / (alpha + (beta + 2.0))) / (beta + 3.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 8.6:
		tmp = (0.5 / (alpha + (beta + 2.0))) / (beta + 3.0)
	else:
		tmp = ((alpha - -1.0) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.6)
		tmp = Float64(Float64(0.5 / Float64(alpha + Float64(beta + 2.0))) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.6)
		tmp = (0.5 / (alpha + (beta + 2.0))) / (beta + 3.0);
	else
		tmp = ((alpha - -1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.6], N[(N[(0.5 / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.59999999999999964

   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. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+r+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-+r+ 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. *-rgt-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-rgt1-in 99.8%

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

      8. distribute-lft-in 99.8%

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

      9. associate-*r/ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in alpha around 0 64.9%

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

   7. Taylor expanded in beta around 0 63.5%

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

      1. +-commutative 63.5%

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

   9. Simplified 63.5%

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

   10. Taylor expanded in alpha around 0 63.5%

       \[\leadsto \frac{\frac{\color{blue}{0.5}}{\left(\beta + 2\right) + \alpha}}{\beta + 3} \]

   if 8.59999999999999964 < beta

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around -inf 87.2%

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

   3. Taylor expanded in beta around inf 87.0%

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

      1. expm1-log1p-u 87.0%

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

      2. expm1-udef 48.2%

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

      3. mul-1-neg 48.2%

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

      4. *-commutative 48.2%

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

      5. fma-neg 48.2%

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

      6. metadata-eval 48.2%

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

   5. Applied egg-rr 48.2%

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

      1. expm1-def 87.0%

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

      2. expm1-log1p 87.0%

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

      3. distribute-neg-frac 87.0%

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

      4. fma-udef 87.0%

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

      5. distribute-lft1-in 87.0%

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

      6. +-commutative 87.0%

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

      7. *-commutative 87.0%

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

      8. distribute-lft-in 87.0%

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

      9. metadata-eval 87.0%

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

      10. neg-mul-1 87.0%

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

      11. unsub-neg 87.0%

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

   7. Simplified 87.0%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.6:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + \left(\beta + 2\right)}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]

Alternative 12: 70.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + \left(\beta + 2\right)}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.5)
   (/ (/ 0.5 (+ alpha (+ beta 2.0))) (+ beta 3.0))
   (/ (/ (- alpha -1.0) beta) (+ 3.0 (+ alpha beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = (0.5 / (alpha + (beta + 2.0))) / (beta + 3.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (3.0 + (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 <= 4.5d0) then
        tmp = (0.5d0 / (alpha + (beta + 2.0d0))) / (beta + 3.0d0)
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (3.0d0 + (alpha + beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.5) {
		tmp = (0.5 / (alpha + (beta + 2.0))) / (beta + 3.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (3.0 + (alpha + beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 4.5:
		tmp = (0.5 / (alpha + (beta + 2.0))) / (beta + 3.0)
	else:
		tmp = ((alpha - -1.0) / beta) / (3.0 + (alpha + beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.5)
		tmp = Float64(Float64(0.5 / Float64(alpha + Float64(beta + 2.0))) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(3.0 + Float64(alpha + beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 4.5)
		tmp = (0.5 / (alpha + (beta + 2.0))) / (beta + 3.0);
	else
		tmp = ((alpha - -1.0) / beta) / (3.0 + (alpha + beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.5], N[(N[(0.5 / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5

   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. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+r+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-+r+ 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. *-rgt-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-rgt1-in 99.8%

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

      8. distribute-lft-in 99.8%

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

      9. associate-*r/ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in alpha around 0 64.9%

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

   7. Taylor expanded in beta around 0 63.5%

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

      1. +-commutative 63.5%

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

   9. Simplified 63.5%

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

   10. Taylor expanded in alpha around 0 63.5%

       \[\leadsto \frac{\frac{\color{blue}{0.5}}{\left(\beta + 2\right) + \alpha}}{\beta + 3} \]

   if 4.5 < beta

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around -inf 87.2%

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

      1. *-un-lft-identity 87.2%

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

      2. mul-1-neg 87.2%

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

      3. fma-neg 87.2%

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

      4. metadata-eval 87.2%

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

      5. metadata-eval 87.2%

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

      6. associate-+l+ 87.2%

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

      7. metadata-eval 87.2%

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

   4. Applied egg-rr 87.2%

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

      1. *-lft-identity 87.2%

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

      2. metadata-eval 87.2%

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

      3. fma-neg 87.2%

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

      4. distribute-neg-frac 87.2%

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

      5. sub-neg 87.2%

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

      6. neg-mul-1 87.2%

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

      7. distribute-neg-in 87.2%

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

      8. +-commutative 87.2%

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

      9. mul-1-neg 87.2%

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

      10. distribute-lft-in 87.2%

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

      11. metadata-eval 87.2%

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

      12. neg-mul-1 87.2%

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

      13. unsub-neg 87.2%

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

      14. +-commutative 87.2%

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

      15. +-commutative 87.2%

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

   6. Simplified 87.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + \left(\beta + 2\right)}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{3 + \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 13: 70.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + 2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6)
   (* (/ (+ 1.0 alpha) (+ alpha 2.0)) 0.16666666666666666)
   (/ (/ (- alpha -1.0) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) * 0.16666666666666666;
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.6d0) then
        tmp = ((1.0d0 + alpha) / (alpha + 2.0d0)) * 0.16666666666666666d0
    else
        tmp = ((alpha - (-1.0d0)) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = ((1.0 + alpha) / (alpha + 2.0)) * 0.16666666666666666;
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.6:
		tmp = ((1.0 + alpha) / (alpha + 2.0)) * 0.16666666666666666
	else:
		tmp = ((alpha - -1.0) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6)
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + 2.0)) * 0.16666666666666666);
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.6)
		tmp = ((1.0 + alpha) / (alpha + 2.0)) * 0.16666666666666666;
	else
		tmp = ((alpha - -1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.6], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.60000000000000009

   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. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+r+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-+r+ 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. *-rgt-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-rgt1-in 99.8%

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

      8. distribute-lft-in 99.8%

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

      9. associate-*r/ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in alpha around 0 64.9%

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

   7. Taylor expanded in alpha around 0 64.0%

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

   8. Taylor expanded in beta around 0 62.6%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{1 + \alpha}{2 + \alpha}} \]
   9. Step-by-step derivation

      1. +-commutative 62.6%

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

   10. Simplified 62.6%

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

   if 3.60000000000000009 < beta

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around -inf 87.2%

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

   3. Taylor expanded in beta around inf 87.0%

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

      1. expm1-log1p-u 87.0%

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

      2. expm1-udef 48.2%

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

      3. mul-1-neg 48.2%

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

      4. *-commutative 48.2%

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

      5. fma-neg 48.2%

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

      6. metadata-eval 48.2%

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

   5. Applied egg-rr 48.2%

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

      1. expm1-def 87.0%

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

      2. expm1-log1p 87.0%

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

      3. distribute-neg-frac 87.0%

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

      4. fma-udef 87.0%

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

      5. distribute-lft1-in 87.0%

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

      6. +-commutative 87.0%

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

      7. *-commutative 87.0%

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

      8. distribute-lft-in 87.0%

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

      9. metadata-eval 87.0%

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

      10. neg-mul-1 87.0%

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

      11. unsub-neg 87.0%

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

   7. Simplified 87.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + 2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]

Alternative 14: 70.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 9.5)
   (/ (/ 0.5 (+ beta 2.0)) (+ beta 3.0))
   (/ (/ (- alpha -1.0) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.5d0) then
        tmp = (0.5d0 / (beta + 2.0d0)) / (beta + 3.0d0)
    else
        tmp = ((alpha - (-1.0d0)) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5) {
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 9.5:
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0)
	else:
		tmp = ((alpha - -1.0) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.5)
		tmp = Float64(Float64(0.5 / Float64(beta + 2.0)) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.5)
		tmp = (0.5 / (beta + 2.0)) / (beta + 3.0);
	else
		tmp = ((alpha - -1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 9.5], N[(N[(0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5

   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. div-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. associate-+r+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-+r+ 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. *-rgt-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-rgt1-in 99.8%

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

      8. distribute-lft-in 99.8%

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

      9. associate-*r/ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in alpha around 0 64.9%

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

   7. Taylor expanded in beta around 0 63.5%

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

      1. +-commutative 63.5%

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

   9. Simplified 63.5%

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

   10. Taylor expanded in alpha around 0 62.6%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{\beta + 2}}}{\beta + 3} \]

   if 9.5 < beta

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around -inf 87.2%

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

   3. Taylor expanded in beta around inf 87.0%

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

      1. expm1-log1p-u 87.0%

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

      2. expm1-udef 48.2%

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

      3. mul-1-neg 48.2%

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

      4. *-commutative 48.2%

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

      5. fma-neg 48.2%

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

      6. metadata-eval 48.2%

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

   5. Applied egg-rr 48.2%

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

      1. expm1-def 87.0%

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

      2. expm1-log1p 87.0%

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

      3. distribute-neg-frac 87.0%

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

      4. fma-udef 87.0%

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

      5. distribute-lft1-in 87.0%

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

      6. +-commutative 87.0%

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

      7. *-commutative 87.0%

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

      8. distribute-lft-in 87.0%

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

      9. metadata-eval 87.0%

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

      10. neg-mul-1 87.0%

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

      11. unsub-neg 87.0%

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

   7. Simplified 87.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]

Alternative 15: 28.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.9e-18) (/ 1.0 (* beta (+ beta 3.0))) (/ (/ alpha beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.9e-18) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.9d-18) then
        tmp = 1.0d0 / (beta * (beta + 3.0d0))
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.9e-18) {
		tmp = 1.0 / (beta * (beta + 3.0));
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.9e-18:
		tmp = 1.0 / (beta * (beta + 3.0))
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.9e-18)
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.9e-18)
		tmp = 1.0 / (beta * (beta + 3.0));
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.9e-18], N[(1.0 / N[(beta * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.9e-18

   1. Initial program 99.9%

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

   2. Taylor expanded in beta around -inf 38.1%

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

   3. Taylor expanded in alpha around 0 37.4%

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

   if 2.9e-18 < alpha

   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. Taylor expanded in beta around -inf 19.9%

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

   3. Taylor expanded in beta around inf 19.8%

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

   4. Taylor expanded in alpha around inf 19.8%

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

      1. mul-1-neg 19.8%

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

      2. distribute-frac-neg 19.8%

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

   6. Simplified 19.8%

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

      1. expm1-log1p-u 19.8%

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

      2. expm1-udef 16.0%

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

      3. mul-1-neg 16.0%

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

   8. Applied egg-rr 16.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{-\alpha}{\beta}}{\beta}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 19.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\frac{-\alpha}{\beta}}{\beta}\right)\right)} \]

      2. expm1-log1p 19.8%

         \[\leadsto \color{blue}{\frac{-\frac{-\alpha}{\beta}}{\beta}} \]

      3. distribute-frac-neg 19.8%

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

      4. remove-double-neg 19.8%

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

   10. Simplified 19.8%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 16: 29.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.5e+154)
   (/ (+ 1.0 alpha) (* beta beta))
   (/ (/ alpha beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.5d+154) then
        tmp = (1.0d0 + alpha) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.5e+154) {
		tmp = (1.0 + alpha) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.5e+154:
		tmp = (1.0 + alpha) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.5e+154)
		tmp = Float64(Float64(1.0 + alpha) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.5e+154)
		tmp = (1.0 + alpha) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.5e+154], N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.50000000000000013e154

   1. Initial program 99.3%

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

      1. div-inv 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-+l+ 99.3%

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

      4. *-commutative 99.3%

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

      5. metadata-eval 99.3%

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

      6. associate-+r+ 99.3%

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

      7. metadata-eval 99.3%

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

      8. associate-+r+ 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.3%

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

      2. associate-*r/ 99.3%

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

      3. *-rgt-identity 99.3%

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

      4. associate-+r+ 99.3%

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

      5. *-rgt-identity 99.3%

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

      6. +-commutative 99.3%

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

      7. distribute-rgt1-in 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. associate-*r/ 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in beta around inf 19.1%

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

      1. unpow2 19.1%

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

   8. Simplified 19.1%

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

   if 1.50000000000000013e154 < beta

   1. Initial program 68.7%

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

   2. Taylor expanded in beta around -inf 91.6%

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

   3. Taylor expanded in beta around inf 91.5%

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

   4. Taylor expanded in alpha around inf 90.8%

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

      1. mul-1-neg 90.8%

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

      2. distribute-frac-neg 90.8%

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

   6. Simplified 90.8%

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

      1. expm1-log1p-u 90.8%

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

      2. expm1-udef 88.9%

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

      3. mul-1-neg 88.9%

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

   8. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{-\alpha}{\beta}}{\beta}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 90.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\frac{-\alpha}{\beta}}{\beta}\right)\right)} \]

      2. expm1-log1p 90.8%

         \[\leadsto \color{blue}{\frac{-\frac{-\alpha}{\beta}}{\beta}} \]

      3. distribute-frac-neg 90.8%

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

      4. remove-double-neg 90.8%

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

   10. Simplified 90.8%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 17: 28.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.9e-18) (/ (/ 1.0 beta) (+ beta 3.0)) (/ (/ alpha beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.9e-18) {
		tmp = (1.0 / beta) / (beta + 3.0);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.9d-18) then
        tmp = (1.0d0 / beta) / (beta + 3.0d0)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.9e-18) {
		tmp = (1.0 / beta) / (beta + 3.0);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.9e-18:
		tmp = (1.0 / beta) / (beta + 3.0)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.9e-18)
		tmp = Float64(Float64(1.0 / beta) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.9e-18)
		tmp = (1.0 / beta) / (beta + 3.0);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.9e-18], N[(N[(1.0 / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.9e-18

   1. Initial program 99.9%

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

   2. Taylor expanded in beta around -inf 38.1%

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

   3. Taylor expanded in alpha around 0 38.1%

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

      1. mul-1-neg 38.1%

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

      2. distribute-frac-neg 38.1%

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

   5. Simplified 38.1%

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

   6. Taylor expanded in alpha around 0 37.4%

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

      1. associate-/r* 37.6%

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

      2. +-commutative 37.6%

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

   8. Simplified 37.6%

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

   if 2.9e-18 < alpha

   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. Taylor expanded in beta around -inf 19.9%

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

   3. Taylor expanded in beta around inf 19.8%

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

   4. Taylor expanded in alpha around inf 19.8%

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

      1. mul-1-neg 19.8%

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

      2. distribute-frac-neg 19.8%

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

   6. Simplified 19.8%

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

      1. expm1-log1p-u 19.8%

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

      2. expm1-udef 16.0%

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

      3. mul-1-neg 16.0%

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

   8. Applied egg-rr 16.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{-\alpha}{\beta}}{\beta}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 19.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\frac{-\alpha}{\beta}}{\beta}\right)\right)} \]

      2. expm1-log1p 19.8%

         \[\leadsto \color{blue}{\frac{-\frac{-\alpha}{\beta}}{\beta}} \]

      3. distribute-frac-neg 19.8%

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

      4. remove-double-neg 19.8%

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

   10. Simplified 19.8%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 18: 20.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.15 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha \cdot \alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.15e+124) (/ 1.0 beta) (/ 1.0 (* alpha alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.15e+124) {
		tmp = 1.0 / beta;
	} else {
		tmp = 1.0 / (alpha * alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.15d+124) then
        tmp = 1.0d0 / beta
    else
        tmp = 1.0d0 / (alpha * alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.15e+124) {
		tmp = 1.0 / beta;
	} else {
		tmp = 1.0 / (alpha * alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.15e+124:
		tmp = 1.0 / beta
	else:
		tmp = 1.0 / (alpha * alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.15e+124)
		tmp = Float64(1.0 / beta);
	else
		tmp = Float64(1.0 / Float64(alpha * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.15e+124)
		tmp = 1.0 / beta;
	else
		tmp = 1.0 / (alpha * alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.15e+124], N[(1.0 / beta), $MachinePrecision], N[(1.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.14999999999999992e124

   1. Initial program 97.9%

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

   2. Taylor expanded in beta around -inf 34.9%

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

   3. Taylor expanded in alpha around inf 4.6%

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

   if 1.14999999999999992e124 < alpha

   1. Initial program 76.9%

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

      1. associate-/l/ 71.3%

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

      2. associate-/l/ 57.7%

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

      3. associate-+l+ 57.7%

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

      4. +-commutative 57.7%

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

      5. associate-+r+ 57.7%

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

      6. associate-+l+ 57.7%

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

      7. distribute-rgt1-in 57.7%

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

      8. *-rgt-identity 57.7%

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

      9. distribute-lft-out 57.7%

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

      10. +-commutative 57.7%

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

      11. times-frac 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in alpha around inf 84.9%

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

      1. +-commutative 84.9%

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

      2. unpow2 84.9%

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

   6. Simplified 84.9%

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

   7. Taylor expanded in beta around 0 83.0%

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

      1. unpow2 83.0%

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

   9. Simplified 83.0%

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

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.15 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha \cdot \alpha}\\ \end{array} \]

Alternative 19: 27.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.9e-18) (/ 1.0 (* beta beta)) (/ alpha (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.9e-18) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.9d-18) then
        tmp = 1.0d0 / (beta * beta)
    else
        tmp = alpha / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.9e-18) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.9e-18:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = alpha / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.9e-18)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.9e-18)
		tmp = 1.0 / (beta * beta);
	else
		tmp = alpha / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.9e-18], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.9e-18

   1. Initial program 99.9%

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

   2. Taylor expanded in beta around -inf 38.1%

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

   3. Taylor expanded in beta around inf 38.5%

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

   4. Taylor expanded in alpha around 0 37.9%

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

      1. unpow2 37.9%

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

   6. Simplified 37.9%

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

   if 2.9e-18 < alpha

   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. Taylor expanded in beta around -inf 19.9%

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

   3. Taylor expanded in beta around inf 19.8%

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

   4. Taylor expanded in alpha around inf 19.0%

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

      1. unpow2 19.0%

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

   6. Simplified 19.0%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 20: 28.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.9e-18) (/ 1.0 (* beta beta)) (/ (/ alpha beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.9e-18) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.9d-18) then
        tmp = 1.0d0 / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.9e-18) {
		tmp = 1.0 / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.9e-18:
		tmp = 1.0 / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.9e-18)
		tmp = Float64(1.0 / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.9e-18)
		tmp = 1.0 / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.9e-18], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.9e-18

   1. Initial program 99.9%

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

   2. Taylor expanded in beta around -inf 38.1%

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

   3. Taylor expanded in beta around inf 38.5%

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

   4. Taylor expanded in alpha around 0 37.9%

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

      1. unpow2 37.9%

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

   6. Simplified 37.9%

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

   if 2.9e-18 < alpha

   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. Taylor expanded in beta around -inf 19.9%

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

   3. Taylor expanded in beta around inf 19.8%

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

   4. Taylor expanded in alpha around inf 19.8%

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

      1. mul-1-neg 19.8%

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

      2. distribute-frac-neg 19.8%

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

   6. Simplified 19.8%

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

      1. expm1-log1p-u 19.8%

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

      2. expm1-udef 16.0%

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

      3. mul-1-neg 16.0%

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

   8. Applied egg-rr 16.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{-\alpha}{\beta}}{\beta}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 19.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\frac{-\alpha}{\beta}}{\beta}\right)\right)} \]

      2. expm1-log1p 19.8%

         \[\leadsto \color{blue}{\frac{-\frac{-\alpha}{\beta}}{\beta}} \]

      3. distribute-frac-neg 19.8%

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

      4. remove-double-neg 19.8%

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

   10. Simplified 19.8%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 21: 29.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\alpha - -1}{\beta}}{\beta} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ (/ (- alpha -1.0) beta) beta))

C

double code(double alpha, double beta) {
	return ((alpha - -1.0) / beta) / beta;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = ((alpha - (-1.0d0)) / beta) / beta
end function

Java

public static double code(double alpha, double beta) {
	return ((alpha - -1.0) / beta) / beta;
}

Python

def code(alpha, beta):
	return ((alpha - -1.0) / beta) / beta

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(alpha - -1.0) / beta) / beta)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = ((alpha - -1.0) / beta) / beta;
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]

Derivation

1. Initial program 94.0%

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

2. Taylor expanded in beta around -inf 31.3%

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

3. Taylor expanded in beta around inf 31.5%

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

   1. expm1-log1p-u 31.5%

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

   2. expm1-udef 18.5%

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

   3. mul-1-neg 18.5%

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

   4. *-commutative 18.5%

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

   5. fma-neg 18.5%

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

   6. metadata-eval 18.5%

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

5. Applied egg-rr 18.5%

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

   1. expm1-def 31.5%

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

   2. expm1-log1p 31.5%

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

   3. distribute-neg-frac 31.5%

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

   4. fma-udef 31.5%

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

   5. distribute-lft1-in 31.5%

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

   6. +-commutative 31.5%

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

   7. *-commutative 31.5%

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

   8. distribute-lft-in 31.5%

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

   9. metadata-eval 31.5%

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

   10. neg-mul-1 31.5%

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

   11. unsub-neg 31.5%

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

7. Simplified 31.5%

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

8. Final simplification 31.5%

   \[\leadsto \frac{\frac{\alpha - -1}{\beta}}{\beta} \]

Alternative 22: 26.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\beta \cdot \beta} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 1.0 (* beta beta)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

2. Taylor expanded in beta around -inf 31.3%

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

3. Taylor expanded in beta around inf 31.5%

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

4. Taylor expanded in alpha around 0 29.8%

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

   1. unpow2 29.8%

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

6. Simplified 29.8%

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

7. Final simplification 29.8%

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

Alternative 23: 4.2% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\beta} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 1.0 beta))

C

double code(double alpha, double beta) {
	return 1.0 / beta;
}

Fortran

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

Java

public static double code(double alpha, double beta) {
	return 1.0 / beta;
}

Python

def code(alpha, beta):
	return 1.0 / beta

Julia

function code(alpha, beta)
	return Float64(1.0 / beta)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 1.0 / beta;
end

Wolfram

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

Derivation

1. Initial program 94.0%

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

2. Taylor expanded in beta around -inf 31.3%

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

3. Taylor expanded in alpha around inf 4.5%

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

4. Final simplification 4.5%

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

Reproduce

herbie shell --seed 2023188 
(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)))