Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.8%

Time: 16.7s

Alternatives: 15

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: 94.2% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \frac{\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{t\_0}}{t\_0 + -1}}{\beta + \left(\alpha + 2\right)} \end{array} \end{array} \]

FPCore

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

C

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

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 + 3.0d0)
    code = (((1.0d0 + beta) * ((1.0d0 + alpha) / t_0)) / (t_0 + (-1.0d0))) / (beta + (alpha + 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Simplified 85.1%

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

   1. times-frac 96.7%

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

   2. +-commutative 96.7%

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

6. Applied egg-rr 96.7%

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

   1. expm1-log1p-u 94.1%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\alpha + \left(\beta + 2\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   2. log1p-define 95.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\alpha + \left(\beta + 2\right)\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   3. +-commutative 95.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \left(\beta + 2\right)\right) + 1\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   4. associate-+r+ 95.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} + 1\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. associate-+l+ 95.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   6. metadata-eval 95.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   7. associate-+r+ 95.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

8. Applied egg-rr 95.0%

   \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
9. Step-by-step derivation

   1. associate-*l/ 95.0%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)}} \]

   2. +-commutative 95.0%

      \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)} \]

   3. expm1-undefine 94.4%

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

   4. add-exp-log 96.7%

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

10. Applied egg-rr 96.7%

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

    1. associate-*r/ 92.2%

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

    2. *-commutative 92.2%

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

    3. times-frac 99.8%

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

    4. associate--l+ 99.8%

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

    5. +-commutative 99.8%

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

    6. +-commutative 99.8%

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

    7. associate-+r+ 99.8%

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

    8. +-commutative 99.8%

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

    9. associate-+r+ 99.8%

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

12. Simplified 99.8%

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

    1. associate-*l/ 99.8%

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

    2. +-commutative 99.8%

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

    3. associate-+r- 99.8%

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

    4. +-commutative 99.8%

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

14. Applied egg-rr 99.8%

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

15. Final simplification 99.8%

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

Alternative 2: 91.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{t\_1 \cdot \left(t\_0 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 2e+54)
     (/ (* (+ 1.0 beta) (+ 1.0 alpha)) (* t_1 (* t_0 t_1)))
     (/
      (* (/ (+ 1.0 alpha) t_0) (+ 1.0 (/ (- -1.0 alpha) beta)))
      (+ beta (+ alpha 2.0))))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+54) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_1 * (t_0 * t_1));
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-1.0 - alpha) / beta))) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 2d+54) then
        tmp = ((1.0d0 + beta) * (1.0d0 + alpha)) / (t_1 * (t_0 * t_1))
    else
        tmp = (((1.0d0 + alpha) / t_0) * (1.0d0 + (((-1.0d0) - alpha) / beta))) / (beta + (alpha + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+54) {
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_1 * (t_0 * t_1));
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-1.0 - alpha) / beta))) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2e+54:
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_1 * (t_0 * t_1))
	else:
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-1.0 - alpha) / beta))) / (beta + (alpha + 2.0))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2e+54)
		tmp = Float64(Float64(Float64(1.0 + beta) * Float64(1.0 + alpha)) / Float64(t_1 * Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta))) / Float64(beta + Float64(alpha + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2e+54)
		tmp = ((1.0 + beta) * (1.0 + alpha)) / (t_1 * (t_0 * t_1));
	else
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-1.0 - alpha) / beta))) / (beta + (alpha + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+54], N[(N[(N[(1.0 + beta), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0000000000000002e54

   1. Initial program 98.7%

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

   3. Simplified 92.7%

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

   if 2.0000000000000002e54 < beta

   1. Initial program 75.7%

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

   3. Simplified 60.8%

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

      1. times-frac 88.4%

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

      2. +-commutative 88.4%

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

   6. Applied egg-rr 88.4%

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

      1. expm1-log1p-u 86.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\alpha + \left(\beta + 2\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      2. log1p-define 86.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\alpha + \left(\beta + 2\right)\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. +-commutative 86.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \left(\beta + 2\right)\right) + 1\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      4. associate-+r+ 86.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} + 1\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      5. associate-+l+ 86.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      6. metadata-eval 86.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      7. associate-+r+ 86.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   8. Applied egg-rr 86.1%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 86.1%

         \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)}} \]

      2. +-commutative 86.1%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)} \]

      3. expm1-undefine 86.1%

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

      4. add-exp-log 88.4%

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

   10. Applied egg-rr 88.4%

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

       1. associate-*r/ 72.9%

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

       2. *-commutative 72.9%

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

       3. times-frac 99.8%

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

       4. associate--l+ 99.8%

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

       5. +-commutative 99.8%

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

       6. +-commutative 99.8%

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

       7. associate-+r+ 99.8%

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

       8. +-commutative 99.8%

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

       9. associate-+r+ 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in beta around inf 88.5%

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

       1. associate-*r/ 88.5%

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

       2. mul-1-neg 88.5%

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

   15. Simplified 88.5%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \alpha}{t\_1} \cdot \frac{1 + \beta}{t\_0 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t\_0} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\beta + \left(\alpha + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 2e+16)
     (* (/ (+ 1.0 alpha) t_1) (/ (+ 1.0 beta) (* t_0 t_1)))
     (/
      (* (/ (+ 1.0 alpha) t_0) (+ 1.0 (/ (- -1.0 alpha) beta)))
      (+ beta (+ alpha 2.0))))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+16) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-1.0 - alpha) / beta))) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 2d+16) then
        tmp = ((1.0d0 + alpha) / t_1) * ((1.0d0 + beta) / (t_0 * t_1))
    else
        tmp = (((1.0d0 + alpha) / t_0) * (1.0d0 + (((-1.0d0) - alpha) / beta))) / (beta + (alpha + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 2e+16) {
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	} else {
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-1.0 - alpha) / beta))) / (beta + (alpha + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 2e+16:
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1))
	else:
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-1.0 - alpha) / beta))) / (beta + (alpha + 2.0))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 2e+16)
		tmp = Float64(Float64(Float64(1.0 + alpha) / t_1) * Float64(Float64(1.0 + beta) / Float64(t_0 * t_1)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta))) / Float64(beta + Float64(alpha + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 2e+16)
		tmp = ((1.0 + alpha) / t_1) * ((1.0 + beta) / (t_0 * t_1));
	else
		tmp = (((1.0 + alpha) / t_0) * (1.0 + ((-1.0 - alpha) / beta))) / (beta + (alpha + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+16], N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2e16

   1. Initial program 99.8%

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

   3. Simplified 93.3%

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

      1. times-frac 99.3%

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

      2. +-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

   if 2e16 < beta

   1. Initial program 78.0%

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

   3. Simplified 66.2%

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

      1. times-frac 90.6%

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

      2. +-commutative 90.6%

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

   6. Applied egg-rr 90.6%

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

      1. expm1-log1p-u 88.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\alpha + \left(\beta + 2\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      2. log1p-define 88.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\alpha + \left(\beta + 2\right)\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. +-commutative 88.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \left(\beta + 2\right)\right) + 1\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      4. associate-+r+ 88.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} + 1\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      5. associate-+l+ 88.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      6. metadata-eval 88.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      7. associate-+r+ 88.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   8. Applied egg-rr 88.1%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 88.1%

         \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)}} \]

      2. +-commutative 88.1%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)} \]

      3. expm1-undefine 88.1%

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

      4. add-exp-log 90.6%

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

   10. Applied egg-rr 90.6%

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

       1. associate-*r/ 75.8%

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

       2. *-commutative 75.8%

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

       3. times-frac 99.7%

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

       4. associate--l+ 99.7%

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

       5. +-commutative 99.7%

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

       6. +-commutative 99.7%

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

       7. associate-+r+ 99.7%

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

       8. +-commutative 99.7%

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

       9. associate-+r+ 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in beta around inf 84.5%

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

       1. associate-*r/ 84.5%

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

       2. mul-1-neg 84.5%

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

   15. Simplified 84.5%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 350000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= beta 350000000000.0)
     (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))) t_0)
     (/
      (*
       (/ (+ 1.0 alpha) (+ alpha (+ beta 3.0)))
       (+ 1.0 (/ (- -1.0 alpha) beta)))
      t_0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (beta <= 350000000000.0) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = (((1.0 + alpha) / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    if (beta <= 350000000000.0d0) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0))) / t_0
    else
        tmp = (((1.0d0 + alpha) / (alpha + (beta + 3.0d0))) * (1.0d0 + (((-1.0d0) - alpha) / beta))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (beta <= 350000000000.0) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	} else {
		tmp = (((1.0 + alpha) / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta))) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if beta <= 350000000000.0:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0
	else:
		tmp = (((1.0 + alpha) / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta))) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (beta <= 350000000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 3.0))) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (beta <= 350000000000.0)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / t_0;
	else
		tmp = (((1.0 + alpha) / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 350000000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5e11

   1. Initial program 99.8%

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

   3. Simplified 93.2%

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

      1. times-frac 99.3%

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

      2. +-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. expm1-log1p-u 96.7%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\alpha + \left(\beta + 2\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      2. log1p-define 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\alpha + \left(\beta + 2\right)\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. +-commutative 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \left(\beta + 2\right)\right) + 1\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      4. associate-+r+ 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} + 1\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      5. associate-+l+ 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      6. metadata-eval 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      7. associate-+r+ 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   8. Applied egg-rr 98.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 98.0%

         \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)}} \]

      2. +-commutative 98.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)} \]

      3. expm1-undefine 97.1%

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

      4. add-exp-log 99.2%

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

   10. Applied egg-rr 99.2%

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

       1. associate-*r/ 99.3%

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

       2. *-commutative 99.3%

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

       3. times-frac 99.8%

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

       4. associate--l+ 99.8%

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

       5. +-commutative 99.8%

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

       6. +-commutative 99.8%

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

       7. associate-+r+ 99.8%

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

       8. +-commutative 99.8%

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

       9. associate-+r+ 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in alpha around 0 62.7%

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

   if 3.5e11 < beta

   1. Initial program 78.6%

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

   3. Simplified 67.1%

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

      1. times-frac 90.9%

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

      2. +-commutative 90.9%

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

   6. Applied egg-rr 90.9%

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

      1. expm1-log1p-u 88.3%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\alpha + \left(\beta + 2\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      2. log1p-define 88.3%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\alpha + \left(\beta + 2\right)\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. +-commutative 88.3%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \left(\beta + 2\right)\right) + 1\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      4. associate-+r+ 88.3%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} + 1\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      5. associate-+l+ 88.3%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      6. metadata-eval 88.3%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      7. associate-+r+ 88.3%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   8. Applied egg-rr 88.3%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 88.3%

         \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)}} \]

      2. +-commutative 88.3%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)} \]

      3. expm1-undefine 88.3%

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

      4. add-exp-log 90.9%

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

   10. Applied egg-rr 90.9%

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

       1. associate-*r/ 76.4%

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

       2. *-commutative 76.4%

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

       3. times-frac 99.7%

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

       4. associate--l+ 99.7%

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

       5. +-commutative 99.7%

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

       6. +-commutative 99.7%

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

       7. associate-+r+ 99.7%

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

       8. +-commutative 99.7%

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

       9. associate-+r+ 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in beta around inf 84.8%

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

       1. associate-*r/ 84.8%

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

       2. mul-1-neg 84.8%

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

   15. Simplified 84.8%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 350000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 900000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{2 \cdot \left(\alpha + 2\right)}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 900000000000.0)
   (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))) (+ beta (+ alpha 2.0)))
   (*
    (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0)))
    (/ (- 1.0 (/ (* 2.0 (+ alpha 2.0)) beta)) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 900000000000.0) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - ((2.0 * (alpha + 2.0)) / 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 <= 900000000000.0d0) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0))) / (beta + (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) * ((1.0d0 - ((2.0d0 * (alpha + 2.0d0)) / beta)) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 900000000000.0) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - ((2.0 * (alpha + 2.0)) / beta)) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 900000000000.0:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + (alpha + 2.0))
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - ((2.0 * (alpha + 2.0)) / beta)) / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 900000000000.0)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))) / Float64(beta + Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) * Float64(Float64(1.0 - Float64(Float64(2.0 * Float64(alpha + 2.0)) / beta)) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 900000000000.0)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) * ((1.0 - ((2.0 * (alpha + 2.0)) / beta)) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 900000000000.0], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(2.0 * N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9e11

   1. Initial program 99.8%

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

   3. Simplified 93.2%

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

      1. times-frac 99.3%

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

      2. +-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. expm1-log1p-u 96.7%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\alpha + \left(\beta + 2\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      2. log1p-define 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\alpha + \left(\beta + 2\right)\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. +-commutative 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \left(\beta + 2\right)\right) + 1\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      4. associate-+r+ 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} + 1\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      5. associate-+l+ 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      6. metadata-eval 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      7. associate-+r+ 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   8. Applied egg-rr 98.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 98.0%

         \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)}} \]

      2. +-commutative 98.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)} \]

      3. expm1-undefine 97.1%

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

      4. add-exp-log 99.2%

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

   10. Applied egg-rr 99.2%

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

       1. associate-*r/ 99.3%

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

       2. *-commutative 99.3%

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

       3. times-frac 99.8%

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

       4. associate--l+ 99.8%

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

       5. +-commutative 99.8%

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

       6. +-commutative 99.8%

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

       7. associate-+r+ 99.8%

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

       8. +-commutative 99.8%

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

       9. associate-+r+ 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in alpha around 0 62.9%

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

   if 9e11 < beta

   1. Initial program 78.3%

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

   3. Simplified 66.7%

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

      1. times-frac 90.7%

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

      2. +-commutative 90.7%

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

   6. Applied egg-rr 90.7%

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

      1. expm1-log1p-u 88.2%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\alpha + \left(\beta + 2\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      2. log1p-define 88.2%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\alpha + \left(\beta + 2\right)\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. +-commutative 88.2%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \left(\beta + 2\right)\right) + 1\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      4. associate-+r+ 88.2%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} + 1\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      5. associate-+l+ 88.2%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      6. metadata-eval 88.2%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      7. associate-+r+ 88.2%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   8. Applied egg-rr 88.2%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   9. Taylor expanded in beta around inf 84.3%

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

       1. mul-1-neg 84.3%

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

       2. metadata-eval 84.3%

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

       3. distribute-lft-in 84.3%

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

   11. Simplified 84.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 900000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 - \frac{2 \cdot \left(\alpha + 2\right)}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Simplified 85.1%

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

   1. times-frac 96.7%

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

   2. +-commutative 96.7%

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

6. Applied egg-rr 96.7%

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

   1. expm1-log1p-u 94.1%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\alpha + \left(\beta + 2\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   2. log1p-define 95.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\alpha + \left(\beta + 2\right)\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   3. +-commutative 95.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \left(\beta + 2\right)\right) + 1\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   4. associate-+r+ 95.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} + 1\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   5. associate-+l+ 95.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   6. metadata-eval 95.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   7. associate-+r+ 95.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

8. Applied egg-rr 95.0%

   \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
9. Step-by-step derivation

   1. associate-*l/ 95.0%

      \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)}} \]

   2. +-commutative 95.0%

      \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)} \]

   3. expm1-undefine 94.4%

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

   4. add-exp-log 96.7%

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

10. Applied egg-rr 96.7%

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

    1. associate-*r/ 92.2%

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

    2. *-commutative 92.2%

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

    3. times-frac 99.8%

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

    4. associate--l+ 99.8%

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

    5. +-commutative 99.8%

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

    6. +-commutative 99.8%

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

    7. associate-+r+ 99.8%

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

    8. +-commutative 99.8%

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

    9. associate-+r+ 99.8%

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

12. Simplified 99.8%

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

13. Taylor expanded in alpha around 0 99.8%

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

    1. +-commutative 99.8%

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

    2. associate-+r+ 99.8%

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

    3. +-commutative 99.8%

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

15. Simplified 99.8%

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

16. Final simplification 99.8%

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

Alternative 7: 72.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.45e+15)
   (/ (/ (+ 1.0 beta) (* (+ beta 2.0) (+ beta 3.0))) (+ beta (+ alpha 2.0)))
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ beta alpha))) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.45e+15) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.45d+15) then
        tmp = ((1.0d0 + beta) / ((beta + 2.0d0) * (beta + 3.0d0))) / (beta + (alpha + 2.0d0))
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.45e+15) {
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + (alpha + 2.0));
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.45e+15:
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + (alpha + 2.0))
	else:
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.45e+15)
		tmp = Float64(Float64(Float64(1.0 + beta) / Float64(Float64(beta + 2.0) * Float64(beta + 3.0))) / Float64(beta + Float64(alpha + 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.45e+15)
		tmp = ((1.0 + beta) / ((beta + 2.0) * (beta + 3.0))) / (beta + (alpha + 2.0));
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.45e+15], N[(N[(N[(1.0 + beta), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.45e15

   1. Initial program 99.8%

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

   3. Simplified 93.3%

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

      1. times-frac 99.3%

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

      2. +-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. expm1-log1p-u 96.7%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\alpha + \left(\beta + 2\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      2. log1p-define 97.9%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\alpha + \left(\beta + 2\right)\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. +-commutative 97.9%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \left(\beta + 2\right)\right) + 1\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      4. associate-+r+ 97.9%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} + 1\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      5. associate-+l+ 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      6. metadata-eval 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      7. associate-+r+ 98.0%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   8. Applied egg-rr 98.0%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 97.9%

         \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)}} \]

      2. +-commutative 97.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)} \]

      3. expm1-undefine 97.1%

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

      4. add-exp-log 99.2%

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

   10. Applied egg-rr 99.2%

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

       1. associate-*r/ 99.3%

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

       2. *-commutative 99.3%

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

       3. times-frac 99.8%

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

       4. associate--l+ 99.8%

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

       5. +-commutative 99.8%

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

       6. +-commutative 99.8%

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

       7. associate-+r+ 99.8%

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

       8. +-commutative 99.8%

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

       9. associate-+r+ 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in alpha around 0 63.1%

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

   if 1.45e15 < beta

   1. Initial program 78.0%

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

      1. associate-/l/ 75.8%

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

      2. +-commutative 75.8%

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

      3. associate-+l+ 75.8%

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

      4. *-commutative 75.8%

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

      5. metadata-eval 75.8%

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

      6. associate-+l+ 75.8%

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

      7. metadata-eval 75.8%

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

      8. associate-+l+ 75.8%

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

      9. metadata-eval 75.8%

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

      10. metadata-eval 75.8%

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

      11. associate-+l+ 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in beta around inf 89.6%

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

      1. *-un-lft-identity 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-+r+ 89.6%

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

   7. Applied egg-rr 89.6%

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

      1. *-lft-identity 89.6%

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

      2. associate-/r* 84.5%

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

      3. associate-+r+ 84.5%

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

      4. +-commutative 84.5%

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

      5. +-commutative 84.5%

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

   9. Simplified 84.5%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 3

   1. Initial program 99.8%

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

   3. Simplified 93.5%

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

      1. times-frac 99.3%

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

      2. +-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. expm1-log1p-u 96.7%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\alpha + \left(\beta + 2\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      2. log1p-define 98.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(\alpha + \left(\beta + 2\right)\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      3. +-commutative 98.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \left(\beta + 2\right)\right) + 1\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      4. associate-+r+ 98.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} + 1\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      5. associate-+l+ 98.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\left(\alpha + \beta\right) + \left(2 + 1\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      6. metadata-eval 98.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\left(\alpha + \beta\right) + \color{blue}{3}\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

      7. associate-+r+ 98.1%

         \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \color{blue}{\left(\alpha + \left(\beta + 3\right)\right)}\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   8. Applied egg-rr 98.1%

      \[\leadsto \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \beta}{\color{blue}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 98.1%

         \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)}} \]

      2. +-commutative 98.1%

         \[\leadsto \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 + \beta}{\mathsf{expm1}\left(\log \left(\alpha + \left(\beta + 3\right)\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\alpha + \left(\beta + 2\right)} \]

      3. expm1-undefine 97.2%

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

      4. add-exp-log 99.2%

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

   10. Applied egg-rr 99.2%

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

       1. associate-*r/ 99.3%

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

       2. *-commutative 99.3%

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

       3. times-frac 99.8%

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

       4. associate--l+ 99.8%

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

       5. +-commutative 99.8%

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

       6. +-commutative 99.8%

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

       7. associate-+r+ 99.8%

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

       8. +-commutative 99.8%

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

       9. associate-+r+ 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in beta around 0 98.3%

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

       1. +-commutative 98.3%

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

       2. +-commutative 98.3%

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

   15. Simplified 98.3%

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

   if 3 < beta

   1. Initial program 79.8%

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

      1. associate-/l/ 77.8%

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

      2. +-commutative 77.8%

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

      3. associate-+l+ 77.8%

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

      4. *-commutative 77.8%

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

      5. metadata-eval 77.8%

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

      6. associate-+l+ 77.8%

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

      7. metadata-eval 77.8%

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

      8. associate-+l+ 77.8%

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

      9. metadata-eval 77.8%

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

      10. metadata-eval 77.8%

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

      11. associate-+l+ 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in beta around inf 85.4%

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

      1. *-un-lft-identity 85.4%

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

      2. *-commutative 85.4%

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

      3. associate-+r+ 85.4%

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

   7. Applied egg-rr 85.4%

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

      1. *-lft-identity 85.4%

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

      2. associate-/r* 80.8%

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

      3. associate-+r+ 80.8%

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

      4. +-commutative 80.8%

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

      5. +-commutative 80.8%

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

   9. Simplified 80.8%

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

4. Final simplification 92.5%

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

Alternative 9: 71.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 0.95:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 0.95)
   0.08333333333333333
   (/ (/ (+ 1.0 alpha) (+ 2.0 (+ beta alpha))) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 0.95) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 0.95d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = ((1.0d0 + alpha) / (2.0d0 + (beta + alpha))) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 0.95) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 0.95:
		tmp = 0.08333333333333333
	else:
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 0.95)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(2.0 + Float64(beta + alpha))) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 0.95)
		tmp = 0.08333333333333333;
	else
		tmp = ((1.0 + alpha) / (2.0 + (beta + alpha))) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 0.95], 0.08333333333333333, N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 0.94999999999999996

   1. Initial program 99.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in beta around 0 92.4%

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

   6. Taylor expanded in alpha around 0 60.3%

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

      1. *-commutative 60.3%

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

   8. Simplified 60.3%

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

   9. Taylor expanded in beta around 0 60.4%

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

   10. Taylor expanded in alpha around 0 60.9%

       \[\leadsto \color{blue}{0.08333333333333333} \]

   if 0.94999999999999996 < beta

   1. Initial program 79.8%

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

      1. associate-/l/ 77.8%

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

      2. +-commutative 77.8%

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

      3. associate-+l+ 77.8%

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

      4. *-commutative 77.8%

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

      5. metadata-eval 77.8%

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

      6. associate-+l+ 77.8%

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

      7. metadata-eval 77.8%

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

      8. associate-+l+ 77.8%

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

      9. metadata-eval 77.8%

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

      10. metadata-eval 77.8%

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

      11. associate-+l+ 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in beta around inf 85.4%

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

      1. *-un-lft-identity 85.4%

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

      2. *-commutative 85.4%

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

      3. associate-+r+ 85.4%

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

   7. Applied egg-rr 85.4%

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

      1. *-lft-identity 85.4%

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

      2. associate-/r* 80.8%

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

      3. associate-+r+ 80.8%

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

      4. +-commutative 80.8%

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

      5. +-commutative 80.8%

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

   9. Simplified 80.8%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 0.95:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.3)
   0.08333333333333333
   (/ (/ (+ 1.0 alpha) beta) (+ 1.0 (+ 2.0 (+ beta alpha))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (beta + alpha)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.3d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = ((1.0d0 + alpha) / beta) / (1.0d0 + (2.0d0 + (beta + alpha)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.3) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (beta + alpha)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.3:
		tmp = 0.08333333333333333
	else:
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (beta + alpha)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.3)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) / Float64(1.0 + Float64(2.0 + Float64(beta + alpha))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.3)
		tmp = 0.08333333333333333;
	else
		tmp = ((1.0 + alpha) / beta) / (1.0 + (2.0 + (beta + alpha)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.3], 0.08333333333333333, N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(1.0 + N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.2999999999999998

   1. Initial program 99.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in beta around 0 92.4%

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

   6. Taylor expanded in alpha around 0 60.3%

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

      1. *-commutative 60.3%

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

   8. Simplified 60.3%

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

   9. Taylor expanded in beta around 0 60.4%

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

   10. Taylor expanded in alpha around 0 60.9%

       \[\leadsto \color{blue}{0.08333333333333333} \]

   if 2.2999999999999998 < beta

   1. Initial program 79.8%

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

   3. Taylor expanded in beta around inf 79.9%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6)
   0.08333333333333333
   (/ (/ (+ 1.0 alpha) (+ alpha (+ beta 2.0))) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.6d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = ((1.0d0 + alpha) / (alpha + (beta + 2.0d0))) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.6:
		tmp = 0.08333333333333333
	else:
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / Float64(alpha + Float64(beta + 2.0))) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6)
		tmp = 0.08333333333333333;
	else
		tmp = ((1.0 + alpha) / (alpha + (beta + 2.0))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.6], 0.08333333333333333, N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in beta around 0 92.4%

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

   6. Taylor expanded in alpha around 0 60.3%

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

      1. *-commutative 60.3%

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

   8. Simplified 60.3%

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

   9. Taylor expanded in beta around 0 60.4%

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

   10. Taylor expanded in alpha around 0 60.9%

       \[\leadsto \color{blue}{0.08333333333333333} \]

   if 2.60000000000000009 < beta

   1. Initial program 79.8%

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

   3. Simplified 67.9%

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

      1. times-frac 91.4%

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

      2. +-commutative 91.4%

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

   6. Applied egg-rr 91.4%

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

   7. Taylor expanded in beta around inf 79.7%

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

      1. un-div-inv 79.8%

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

      2. +-commutative 79.8%

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

   9. Applied egg-rr 79.8%

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

Alternative 12: 70.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.6)
   0.08333333333333333
   (* (/ (+ 1.0 alpha) beta) (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3.6d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = ((1.0d0 + alpha) / beta) * (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3.6:
		tmp = 0.08333333333333333
	else:
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.6)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(Float64(Float64(1.0 + alpha) / beta) * Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3.6)
		tmp = 0.08333333333333333;
	else
		tmp = ((1.0 + alpha) / beta) * (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.6], 0.08333333333333333, N[(N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.60000000000000009

   1. Initial program 99.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in beta around 0 92.4%

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

   6. Taylor expanded in alpha around 0 60.3%

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

      1. *-commutative 60.3%

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

   8. Simplified 60.3%

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

   9. Taylor expanded in beta around 0 60.4%

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

   10. Taylor expanded in alpha around 0 60.9%

       \[\leadsto \color{blue}{0.08333333333333333} \]

   if 3.60000000000000009 < beta

   1. Initial program 79.8%

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

   3. Simplified 67.9%

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

      1. times-frac 91.4%

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

      2. +-commutative 91.4%

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

   6. Applied egg-rr 91.4%

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

   7. Taylor expanded in beta around inf 79.7%

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

   8. Taylor expanded in beta around inf 79.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.6) 0.08333333333333333 (/ 1.0 (* beta (+ beta 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * (beta + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.6d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / (beta * (beta + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.6) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / (beta * (beta + 2.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.6:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / (beta * (beta + 2.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.6)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / Float64(beta * Float64(beta + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.6)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / (beta * (beta + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.6], 0.08333333333333333, N[(1.0 / N[(beta * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.60000000000000009

   1. Initial program 99.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in beta around 0 92.4%

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

   6. Taylor expanded in alpha around 0 60.3%

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

      1. *-commutative 60.3%

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

   8. Simplified 60.3%

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

   9. Taylor expanded in beta around 0 60.4%

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

   10. Taylor expanded in alpha around 0 60.9%

       \[\leadsto \color{blue}{0.08333333333333333} \]

   if 2.60000000000000009 < beta

   1. Initial program 79.8%

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

   3. Simplified 67.9%

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

      1. times-frac 91.4%

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

      2. +-commutative 91.4%

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

   6. Applied egg-rr 91.4%

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

   7. Taylor expanded in beta around inf 79.7%

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

   8. Taylor expanded in alpha around 0 72.4%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 12:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 12.0) 0.08333333333333333 (/ 1.0 beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 12.0d0) then
        tmp = 0.08333333333333333d0
    else
        tmp = 1.0d0 / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 12.0) {
		tmp = 0.08333333333333333;
	} else {
		tmp = 1.0 / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 12.0:
		tmp = 0.08333333333333333
	else:
		tmp = 1.0 / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = Float64(1.0 / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 12.0)
		tmp = 0.08333333333333333;
	else
		tmp = 1.0 / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 12.0], 0.08333333333333333, N[(1.0 / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 12

   1. Initial program 99.8%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in beta around 0 92.4%

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

   6. Taylor expanded in alpha around 0 60.3%

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

      1. *-commutative 60.3%

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

   8. Simplified 60.3%

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

   9. Taylor expanded in beta around 0 60.4%

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

   10. Taylor expanded in alpha around 0 60.9%

       \[\leadsto \color{blue}{0.08333333333333333} \]

   if 12 < beta

   1. Initial program 79.8%

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

   3. Simplified 67.9%

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

      1. times-frac 91.4%

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

      2. +-commutative 91.4%

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

   6. Applied egg-rr 91.4%

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

   7. Taylor expanded in beta around inf 79.7%

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

   8. Taylor expanded in alpha around inf 7.0%

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

Alternative 15: 45.2% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.08333333333333333

Julia

function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 93.3%

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

3. Simplified 85.1%

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

5. Taylor expanded in beta around 0 65.1%

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

6. Taylor expanded in alpha around 0 41.8%

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

   1. *-commutative 41.8%

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

8. Simplified 41.8%

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

9. Taylor expanded in beta around 0 41.9%

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

10. Taylor expanded in alpha around 0 42.3%

    \[\leadsto \color{blue}{0.08333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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)))