Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.7%

Time: 13.4s

Alternatives: 24

Speedup: 1.3×

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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{t\_0}}{t\_0}}{1 + t\_0}\\ \mathbf{if}\;t\_1 \leq 0.1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(t\_0 \cdot \left(\frac{\frac{2}{\beta + 1} + \left(\frac{\beta}{\beta + 1} + \frac{-1 - \beta}{\left(-1 - \beta\right) \cdot \left(-1 - \beta\right)}\right)}{\alpha} + \frac{1}{\beta + 1}\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0))
        (t_1
         (/
          (/ (/ (+ (+ (+ alpha beta) (* alpha beta)) 1.0) t_0) t_0)
          (+ 1.0 t_0))))
   (if (<= t_1 0.1)
     t_1
     (/
      1.0
      (*
       (*
        t_0
        (+
         (/
          (+
           (/ 2.0 (+ beta 1.0))
           (+
            (/ beta (+ beta 1.0))
            (/ (- -1.0 beta) (* (- -1.0 beta) (- -1.0 beta)))))
          alpha)
         (/ 1.0 (+ beta 1.0))))
       (+ alpha (+ beta 3.0)))))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = (((((alpha + beta) + (alpha * beta)) + 1.0) / t_0) / t_0) / (1.0 + t_0);
	double tmp;
	if (t_1 <= 0.1) {
		tmp = t_1;
	} else {
		tmp = 1.0 / ((t_0 * ((((2.0 / (beta + 1.0)) + ((beta / (beta + 1.0)) + ((-1.0 - beta) / ((-1.0 - beta) * (-1.0 - beta))))) / alpha) + (1.0 / (beta + 1.0)))) * (alpha + (beta + 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + 2.0d0
    t_1 = (((((alpha + beta) + (alpha * beta)) + 1.0d0) / t_0) / t_0) / (1.0d0 + t_0)
    if (t_1 <= 0.1d0) then
        tmp = t_1
    else
        tmp = 1.0d0 / ((t_0 * ((((2.0d0 / (beta + 1.0d0)) + ((beta / (beta + 1.0d0)) + (((-1.0d0) - beta) / (((-1.0d0) - beta) * ((-1.0d0) - beta))))) / alpha) + (1.0d0 / (beta + 1.0d0)))) * (alpha + (beta + 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = (((((alpha + beta) + (alpha * beta)) + 1.0) / t_0) / t_0) / (1.0 + t_0);
	double tmp;
	if (t_1 <= 0.1) {
		tmp = t_1;
	} else {
		tmp = 1.0 / ((t_0 * ((((2.0 / (beta + 1.0)) + ((beta / (beta + 1.0)) + ((-1.0 - beta) / ((-1.0 - beta) * (-1.0 - beta))))) / alpha) + (1.0 / (beta + 1.0)))) * (alpha + (beta + 3.0)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	t_1 = (((((alpha + beta) + (alpha * beta)) + 1.0) / t_0) / t_0) / (1.0 + t_0)
	tmp = 0
	if t_1 <= 0.1:
		tmp = t_1
	else:
		tmp = 1.0 / ((t_0 * ((((2.0 / (beta + 1.0)) + ((beta / (beta + 1.0)) + ((-1.0 - beta) / ((-1.0 - beta) * (-1.0 - beta))))) / alpha) + (1.0 / (beta + 1.0)))) * (alpha + (beta + 3.0)))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(alpha * beta)) + 1.0) / t_0) / t_0) / Float64(1.0 + t_0))
	tmp = 0.0
	if (t_1 <= 0.1)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(Float64(t_0 * Float64(Float64(Float64(Float64(2.0 / Float64(beta + 1.0)) + Float64(Float64(beta / Float64(beta + 1.0)) + Float64(Float64(-1.0 - beta) / Float64(Float64(-1.0 - beta) * Float64(-1.0 - beta))))) / alpha) + Float64(1.0 / Float64(beta + 1.0)))) * Float64(alpha + Float64(beta + 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	t_1 = (((((alpha + beta) + (alpha * beta)) + 1.0) / t_0) / t_0) / (1.0 + t_0);
	tmp = 0.0;
	if (t_1 <= 0.1)
		tmp = t_1;
	else
		tmp = 1.0 / ((t_0 * ((((2.0 / (beta + 1.0)) + ((beta / (beta + 1.0)) + ((-1.0 - beta) / ((-1.0 - beta) * (-1.0 - beta))))) / alpha) + (1.0 / (beta + 1.0)))) * (alpha + (beta + 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.1], t$95$1, N[(1.0 / N[(N[(t$95$0 * N[(N[(N[(N[(2.0 / N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(beta / N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - beta), $MachinePrecision] / N[(N[(-1.0 - beta), $MachinePrecision] * N[(-1.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(1.0 / N[(beta + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 99.9%

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

   if 0.10000000000000001 < (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) #s(literal 1 binary64)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) #s(literal 1 binary64))) #s(literal 1 binary64)))

   1. Initial program 1.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 1.6%

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

   5. Taylor expanded in alpha around -inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{1 + \left(\left(\alpha + \beta\right) + 2\right)} \leq 0.1:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\frac{2}{\beta + 1} + \left(\frac{\beta}{\beta + 1} + \frac{-1 - \beta}{\left(-1 - \beta\right) \cdot \left(-1 - \beta\right)}\right)}{\alpha} + \frac{1}{\beta + 1}\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{t\_0}}{t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\beta} + \left(\alpha + \frac{\alpha}{\beta}\right)\right) + \left(1 + \left(-1 - \alpha\right) \cdot \frac{\alpha + 2}{\beta}\right)}{t\_0}}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 2.3e+82)
     (/
      (/ (+ 1.0 (fma alpha beta (+ alpha beta))) t_0)
      (* t_0 (+ alpha (+ beta 3.0))))
     (/
      (/
       (+
        (+ (/ 1.0 beta) (+ alpha (/ alpha beta)))
        (+ 1.0 (* (- -1.0 alpha) (/ (+ alpha 2.0) beta))))
       t_0)
      (+ 1.0 t_0)))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 2.3e+82) {
		tmp = ((1.0 + fma(alpha, beta, (alpha + beta))) / t_0) / (t_0 * (alpha + (beta + 3.0)));
	} else {
		tmp = ((((1.0 / beta) + (alpha + (alpha / beta))) + (1.0 + ((-1.0 - alpha) * ((alpha + 2.0) / beta)))) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 2.3e+82)
		tmp = Float64(Float64(Float64(1.0 + fma(alpha, beta, Float64(alpha + beta))) / t_0) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / beta) + Float64(alpha + Float64(alpha / beta))) + Float64(1.0 + Float64(Float64(-1.0 - alpha) * Float64(Float64(alpha + 2.0) / beta)))) / t_0) / Float64(1.0 + t_0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 2.3e+82], N[(N[(N[(1.0 + N[(alpha * beta + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.29999999999999988e82

   1. Initial program 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

   4. Applied rewrites 98.3%

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

   if 2.29999999999999988e82 < beta

   1. Initial program 75.0%

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

   3. Taylor expanded in beta around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. associate-/l* N/A

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

      13. distribute-lft-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. lower--.f64 N/A

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

      21. lower-/.f64 N/A

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

   5. Applied rewrites 84.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\beta} + \left(\alpha + \frac{\alpha}{\beta}\right)\right) + \left(1 + \left(-1 - \alpha\right) \cdot \frac{\alpha + 2}{\beta}\right)}{\left(\alpha + \beta\right) + 2}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{t\_1}}{t\_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + 1\right) + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right) + \left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ (+ alpha beta) 2.0)))
   (if (<= beta 2.3e+82)
     (/ (/ (+ 1.0 (fma alpha beta (+ alpha beta))) t_1) (* t_1 t_0))
     (/
      (/
       (+
        (+ (+ alpha 1.0) (+ (/ 1.0 beta) (/ alpha beta)))
        (* (- -1.0 alpha) (/ (fma 2.0 alpha 4.0) beta)))
       beta)
      t_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 <= 2.3e+82) {
		tmp = ((1.0 + fma(alpha, beta, (alpha + beta))) / t_1) / (t_1 * t_0);
	} else {
		tmp = ((((alpha + 1.0) + ((1.0 / beta) + (alpha / beta))) + ((-1.0 - alpha) * (fma(2.0, alpha, 4.0) / beta))) / beta) / t_0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	t_1 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 2.3e+82)
		tmp = Float64(Float64(Float64(1.0 + fma(alpha, beta, Float64(alpha + beta))) / t_1) / Float64(t_1 * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + 1.0) + Float64(Float64(1.0 / beta) + Float64(alpha / beta))) + Float64(Float64(-1.0 - alpha) * Float64(fma(2.0, alpha, 4.0) / beta))) / beta) / t_0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 2.3e+82], N[(N[(N[(1.0 + N[(alpha * beta + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] + N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - alpha), $MachinePrecision] * N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.29999999999999988e82

   1. Initial program 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

   4. Applied rewrites 98.3%

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

   if 2.29999999999999988e82 < beta

   1. Initial program 75.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 85.4

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

   5. Applied rewrites 85.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 85.4

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

   7. Applied rewrites 85.4%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 84.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + 1\right) + \left(\frac{1}{\beta} + \frac{\alpha}{\beta}\right)\right) + \left(-1 - \alpha\right) \cdot \frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{t\_0}}{t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 2.3e+82)
     (/
      (/ (+ 1.0 (fma alpha beta (+ alpha beta))) t_0)
      (* t_0 (+ alpha (+ beta 3.0))))
     (/ (/ (+ alpha 1.0) t_0) (+ 1.0 t_0)))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 2.3e+82) {
		tmp = ((1.0 + fma(alpha, beta, (alpha + beta))) / t_0) / (t_0 * (alpha + (beta + 3.0)));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 2.3e+82)
		tmp = Float64(Float64(Float64(1.0 + fma(alpha, beta, Float64(alpha + beta))) / t_0) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(1.0 + t_0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 2.3e+82], N[(N[(N[(1.0 + N[(alpha * beta + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.29999999999999988e82

   1. Initial program 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

   4. Applied rewrites 98.3%

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

   if 2.29999999999999988e82 < beta

   1. Initial program 75.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 85.8

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

   5. Applied rewrites 85.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \beta\right) + 2}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{t\_0 \cdot t\_0}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 1.35e+76)
     (/
      (/ (+ 1.0 (fma alpha beta (+ alpha beta))) (* t_0 t_0))
      (+ alpha (+ beta 3.0)))
     (/ (/ (+ alpha 1.0) t_0) (+ 1.0 t_0)))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 1.35e+76) {
		tmp = ((1.0 + fma(alpha, beta, (alpha + beta))) / (t_0 * t_0)) / (alpha + (beta + 3.0));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 1.35e+76)
		tmp = Float64(Float64(Float64(1.0 + fma(alpha, beta, Float64(alpha + beta))) / Float64(t_0 * t_0)) / Float64(alpha + Float64(beta + 3.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(1.0 + t_0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.35e+76], N[(N[(N[(1.0 + N[(alpha * beta + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.34999999999999995e76

   1. Initial program 98.8%

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

   4. Applied rewrites 98.2%

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

   if 1.34999999999999995e76 < beta

   1. Initial program 76.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \beta\right) + 2}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 1.02 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{t\_0 \cdot \left(t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \cdot \left(1 + \left(\beta + \mathsf{fma}\left(\alpha, \beta, \alpha\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 1.02e+76)
     (*
      (/ 1.0 (* t_0 (* t_0 (+ alpha (+ beta 3.0)))))
      (+ 1.0 (+ beta (fma alpha beta alpha))))
     (/ (/ (+ alpha 1.0) t_0) (+ 1.0 t_0)))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 1.02e+76) {
		tmp = (1.0 / (t_0 * (t_0 * (alpha + (beta + 3.0))))) * (1.0 + (beta + fma(alpha, beta, alpha)));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 1.02e+76)
		tmp = Float64(Float64(1.0 / Float64(t_0 * Float64(t_0 * Float64(alpha + Float64(beta + 3.0))))) * Float64(1.0 + Float64(beta + fma(alpha, beta, alpha))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(1.0 + t_0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.02e+76], N[(N[(1.0 / N[(t$95$0 * N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(beta + N[(alpha * beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.02000000000000007e76

   1. Initial program 98.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

   6. Applied rewrites 94.7%

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

   if 1.02000000000000007e76 < beta

   1. Initial program 76.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.02 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)} \cdot \left(1 + \left(\beta + \mathsf{fma}\left(\alpha, \beta, \alpha\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \beta\right) + 2}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 1.02 \cdot 10^{+76}:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{t\_0 \cdot \left(t\_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)))
   (if (<= beta 1.02e+76)
     (/
      (+ 1.0 (fma alpha beta (+ alpha beta)))
      (* t_0 (* t_0 (+ alpha (+ beta 3.0)))))
     (/ (/ (+ alpha 1.0) t_0) (+ 1.0 t_0)))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double tmp;
	if (beta <= 1.02e+76) {
		tmp = (1.0 + fma(alpha, beta, (alpha + beta))) / (t_0 * (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = ((alpha + 1.0) / t_0) / (1.0 + t_0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (beta <= 1.02e+76)
		tmp = Float64(Float64(1.0 + fma(alpha, beta, Float64(alpha + beta))) / Float64(t_0 * Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / Float64(1.0 + t_0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 1.02e+76], N[(N[(1.0 + N[(alpha * beta + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.02000000000000007e76

   1. Initial program 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 94.7%

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

   if 1.02000000000000007e76 < beta

   1. Initial program 76.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.02 \cdot 10^{+76}:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \beta\right) + 2}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))))
   (if (<= beta 1e+16)
     (/ 1.0 (* t_0 (/ (* (+ beta 2.0) (+ beta 2.0)) (+ beta 1.0))))
     (/ (/ (+ alpha 1.0) beta) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 1e+16) {
		tmp = 1.0 / (t_0 * (((beta + 2.0) * (beta + 2.0)) / (beta + 1.0)));
	} else {
		tmp = ((alpha + 1.0) / 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 = alpha + (beta + 3.0d0)
    if (beta <= 1d+16) then
        tmp = 1.0d0 / (t_0 * (((beta + 2.0d0) * (beta + 2.0d0)) / (beta + 1.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 1e+16) {
		tmp = 1.0 / (t_0 * (((beta + 2.0) * (beta + 2.0)) / (beta + 1.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 3.0)
	tmp = 0
	if beta <= 1e+16:
		tmp = 1.0 / (t_0 * (((beta + 2.0) * (beta + 2.0)) / (beta + 1.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 1e+16)
		tmp = Float64(1.0 / Float64(t_0 * Float64(Float64(Float64(beta + 2.0) * Float64(beta + 2.0)) / Float64(beta + 1.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 3.0);
	tmp = 0.0;
	if (beta <= 1e+16)
		tmp = 1.0 / (t_0 * (((beta + 2.0) * (beta + 2.0)) / (beta + 1.0)));
	else
		tmp = ((alpha + 1.0) / beta) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 1e16

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 65.7

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

   7. Applied rewrites 65.7%

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

   if 1e16 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 82.1

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

   5. Applied rewrites 82.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 82.1

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

   7. Applied rewrites 82.1%

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

4. Final simplification 71.5%

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

Alternative 9: 73.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{t\_0}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0)) (t_1 (+ 1.0 t_0)))
   (if (<= beta 7.2e+15)
     (/ (/ (+ beta 1.0) (* (+ beta 2.0) (+ beta 2.0))) t_1)
     (/ (/ (+ alpha 1.0) t_0) t_1))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = 1.0 + t_0;
	double tmp;
	if (beta <= 7.2e+15) {
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / t_1;
	} else {
		tmp = ((alpha + 1.0) / t_0) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + 2.0d0
    t_1 = 1.0d0 + t_0
    if (beta <= 7.2d+15) then
        tmp = ((beta + 1.0d0) / ((beta + 2.0d0) * (beta + 2.0d0))) / t_1
    else
        tmp = ((alpha + 1.0d0) / t_0) / t_1
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = 1.0 + t_0;
	double tmp;
	if (beta <= 7.2e+15) {
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / t_1;
	} else {
		tmp = ((alpha + 1.0) / t_0) / t_1;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	t_1 = 1.0 + t_0
	tmp = 0
	if beta <= 7.2e+15:
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / t_1
	else:
		tmp = ((alpha + 1.0) / t_0) / t_1
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	t_1 = Float64(1.0 + t_0)
	tmp = 0.0
	if (beta <= 7.2e+15)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(Float64(beta + 2.0) * Float64(beta + 2.0))) / t_1);
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / t_0) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (alpha + beta) + 2.0;
	t_1 = 1.0 + t_0;
	tmp = 0.0;
	if (beta <= 7.2e+15)
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / t_1;
	else
		tmp = ((alpha + 1.0) / t_0) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[beta, 7.2e+15], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.2e15

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 65.7

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

   5. Applied rewrites 65.7%

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

   if 7.2e15 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. lower-+.f64 82.6

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

   5. Applied rewrites 82.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta + 2\right)}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + \beta\right) + 2}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.7% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+16)
   (/
    (/ (+ beta 1.0) (* (+ beta 2.0) (+ beta 2.0)))
    (+ 1.0 (+ (+ alpha beta) 2.0)))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+16) {
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + ((alpha + beta) + 2.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1d+16) then
        tmp = ((beta + 1.0d0) / ((beta + 2.0d0) * (beta + 2.0d0))) / (1.0d0 + ((alpha + beta) + 2.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+16) {
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + ((alpha + beta) + 2.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1e+16:
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + ((alpha + beta) + 2.0))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+16)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(Float64(beta + 2.0) * Float64(beta + 2.0))) / Float64(1.0 + Float64(Float64(alpha + beta) + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1e+16)
		tmp = ((beta + 1.0) / ((beta + 2.0) * (beta + 2.0))) / (1.0 + ((alpha + beta) + 2.0));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1e+16], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e16

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 65.7

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

   5. Applied rewrites 65.7%

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

   if 1e16 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 82.1

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

   5. Applied rewrites 82.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 82.1

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

   7. Applied rewrites 82.1%

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

4. Final simplification 71.5%

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

Alternative 11: 72.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+16}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\mathsf{fma}\left(\beta, \beta + 4, 4\right)}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+16)
   (/ (/ (+ beta 1.0) (fma beta (+ beta 4.0) 4.0)) (+ beta 3.0))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+16) {
		tmp = ((beta + 1.0) / fma(beta, (beta + 4.0), 4.0)) / (beta + 3.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+16)
		tmp = Float64(Float64(Float64(beta + 1.0) / fma(beta, Float64(beta + 4.0), 4.0)) / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1e+16], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(beta * N[(beta + 4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e16

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 4.0

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

   5. Applied rewrites 4.0%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 3.9

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

   8. Applied rewrites 3.9%

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

   9. Taylor expanded in alpha around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 63.8

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

   11. Applied rewrites 63.8%

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

   12. Taylor expanded in beta around 0

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

   14. Applied rewrites 63.8%

       \[\leadsto \frac{\frac{\beta + 1}{\mathsf{fma}\left(\beta, \color{blue}{4 + \beta}, 4\right)}}{\beta + 3} \]

   if 1e16 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 82.1

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

   5. Applied rewrites 82.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 82.1

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

   7. Applied rewrites 82.1%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+16}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\mathsf{fma}\left(\beta, \beta + 4, 4\right)}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+16)
   (/ (+ beta 1.0) (* (+ (+ alpha beta) 2.0) (* (+ beta 3.0) (+ beta 2.0))))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+16) {
		tmp = (beta + 1.0) / (((alpha + beta) + 2.0) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1d+16) then
        tmp = (beta + 1.0d0) / (((alpha + beta) + 2.0d0) * ((beta + 3.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+16) {
		tmp = (beta + 1.0) / (((alpha + beta) + 2.0) * ((beta + 3.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1e+16:
		tmp = (beta + 1.0) / (((alpha + beta) + 2.0) * ((beta + 3.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+16)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(Float64(alpha + beta) + 2.0) * Float64(Float64(beta + 3.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1e+16)
		tmp = (beta + 1.0) / (((alpha + beta) + 2.0) * ((beta + 3.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1e+16], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(beta + 3.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e16

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-+.f64 83.3

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

   7. Applied rewrites 83.3%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 64.8

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

   10. Applied rewrites 64.8%

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

   if 1e16 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 82.1

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

   5. Applied rewrites 82.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 82.1

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

   7. Applied rewrites 82.1%

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

4. Final simplification 70.9%

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

Alternative 13: 72.6% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1e+16)
   (/ (+ beta 1.0) (* (+ beta 3.0) (* (+ beta 2.0) (+ beta 2.0))))
   (/ (/ (+ alpha 1.0) beta) (+ alpha (+ beta 3.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+16) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1d+16) then
        tmp = (beta + 1.0d0) / ((beta + 3.0d0) * ((beta + 2.0d0) * (beta + 2.0d0)))
    else
        tmp = ((alpha + 1.0d0) / beta) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1e+16) {
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	} else {
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1e+16:
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)))
	else:
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1e+16)
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(beta + 3.0) * Float64(Float64(beta + 2.0) * Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1e+16)
		tmp = (beta + 1.0) / ((beta + 3.0) * ((beta + 2.0) * (beta + 2.0)));
	else
		tmp = ((alpha + 1.0) / beta) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1e+16], N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(beta + 3.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e16

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 63.8

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

   5. Applied rewrites 63.8%

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

   if 1e16 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 82.1

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

   5. Applied rewrites 82.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 82.1

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

   7. Applied rewrites 82.1%

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

4. Final simplification 70.2%

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

Alternative 14: 72.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 3.1:\\ \;\;\;\;\frac{1}{t\_0 \cdot \mathsf{fma}\left(\beta, \beta, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))))
   (if (<= beta 3.1)
     (/ 1.0 (* t_0 (fma beta beta 4.0)))
     (/ (/ (+ alpha 1.0) beta) t_0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 3.0);
	double tmp;
	if (beta <= 3.1) {
		tmp = 1.0 / (t_0 * fma(beta, beta, 4.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / t_0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 3.0))
	tmp = 0.0
	if (beta <= 3.1)
		tmp = Float64(1.0 / Float64(t_0 * fma(beta, beta, 4.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / t_0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.1], N[(1.0 / N[(t$95$0 * N[(beta * beta + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.10000000000000009

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 65.1

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

   7. Applied rewrites 65.1%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 64.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\beta, \color{blue}{\beta}, 4\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   if 3.10000000000000009 < beta

   1. Initial program 78.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 81.4

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

   5. Applied rewrites 81.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 81.4

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

   7. Applied rewrites 81.4%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.1:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \mathsf{fma}\left(\beta, \beta, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot 4}\\ \mathbf{elif}\;\beta \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0)
   (/ 1.0 (* (+ alpha (+ beta 3.0)) 4.0))
   (if (<= beta 8e+153)
     (/ (+ alpha 1.0) (* beta beta))
     (/ (/ alpha beta) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 1.0 / ((alpha + (beta + 3.0)) * 4.0);
	} else if (beta <= 8e+153) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 1.0d0 / ((alpha + (beta + 3.0d0)) * 4.0d0)
    else if (beta <= 8d+153) then
        tmp = (alpha + 1.0d0) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 1.0 / ((alpha + (beta + 3.0)) * 4.0);
	} else if (beta <= 8e+153) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 1.0 / ((alpha + (beta + 3.0)) * 4.0)
	elif beta <= 8e+153:
		tmp = (alpha + 1.0) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 3.0)) * 4.0));
	elseif (beta <= 8e+153)
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 1.0 / ((alpha + (beta + 3.0)) * 4.0);
	elseif (beta <= 8e+153)
		tmp = (alpha + 1.0) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(1.0 / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 8e+153], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 65.1

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

   7. Applied rewrites 65.1%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 64.5%

       \[\leadsto \frac{1}{4 \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   if 6 < beta < 8e153

   1. Initial program 93.5%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if 8e153 < beta

   1. Initial program 63.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 85.1%

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

   10. Applied rewrites 83.5%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot 4}\\ \mathbf{elif}\;\beta \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{elif}\;\beta \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0)
   (/ 0.25 (+ beta 3.0))
   (if (<= beta 8e+153)
     (/ (+ alpha 1.0) (* beta beta))
     (/ (/ alpha beta) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else if (beta <= 8e+153) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else if (beta <= 8d+153) then
        tmp = (alpha + 1.0d0) / (beta * beta)
    else
        tmp = (alpha / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else if (beta <= 8e+153) {
		tmp = (alpha + 1.0) / (beta * beta);
	} else {
		tmp = (alpha / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.25 / (beta + 3.0)
	elif beta <= 8e+153:
		tmp = (alpha + 1.0) / (beta * beta)
	else:
		tmp = (alpha / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	elseif (beta <= 8e+153)
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	else
		tmp = Float64(Float64(alpha / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.25 / (beta + 3.0);
	elseif (beta <= 8e+153)
		tmp = (alpha + 1.0) / (beta * beta);
	else
		tmp = (alpha / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 8e+153], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(N[(alpha / beta), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 6

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.9

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in alpha around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 63.1

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

   11. Applied rewrites 63.1%

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

   12. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{4}}{\beta + 3} \]
   13. Step-by-step derivation

   14. Applied rewrites 62.5%

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

   if 6 < beta < 8e153

   1. Initial program 93.5%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if 8e153 < beta

   1. Initial program 63.6%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 85.1%

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

   10. Applied rewrites 83.5%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{elif}\;\beta \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 72.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.1:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \mathsf{fma}\left(\beta, \beta, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.1)
   (/ 1.0 (* (+ alpha (+ beta 3.0)) (fma beta beta 4.0)))
   (/ (/ (+ alpha 1.0) beta) (+ beta 3.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.1) {
		tmp = 1.0 / ((alpha + (beta + 3.0)) * fma(beta, beta, 4.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / (beta + 3.0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.1)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 3.0)) * fma(beta, beta, 4.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / Float64(beta + 3.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.1], N[(1.0 / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(beta * beta + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.10000000000000009

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 65.1

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

   7. Applied rewrites 65.1%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 64.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\beta, \color{blue}{\beta}, 4\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   if 3.10000000000000009 < beta

   1. Initial program 78.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 81.2

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

   8. Applied rewrites 81.2%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.1:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \mathsf{fma}\left(\beta, \beta, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta + 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 72.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \mathsf{fma}\left(\beta, \beta, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 4.8)
   (/ 1.0 (* (+ alpha (+ beta 3.0)) (fma beta beta 4.0)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 4.8) {
		tmp = 1.0 / ((alpha + (beta + 3.0)) * fma(beta, beta, 4.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 4.8)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 3.0)) * fma(beta, beta, 4.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 4.8], N[(1.0 / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * N[(beta * beta + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999982

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 65.1

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

   7. Applied rewrites 65.1%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 64.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\beta, \color{blue}{\beta}, 4\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   if 4.79999999999999982 < beta

   1. Initial program 78.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 81.2%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \mathsf{fma}\left(\beta, \beta, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 72.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0)
   (/ 0.25 (+ 1.0 (+ (+ alpha beta) 2.0)))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (1.0 + ((alpha + beta) + 2.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.25d0 / (1.0d0 + ((alpha + beta) + 2.0d0))
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (1.0 + ((alpha + beta) + 2.0));
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.25 / (1.0 + ((alpha + beta) + 2.0))
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(0.25 / Float64(1.0 + Float64(Float64(alpha + beta) + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.25 / (1.0 + ((alpha + beta) + 2.0));
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(0.25 / N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 65.1

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 64.5%

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

   if 6 < beta

   1. Initial program 78.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 81.2%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 72.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0)
   (/ 1.0 (* (+ alpha (+ beta 3.0)) 4.0))
   (/ (/ (+ alpha 1.0) beta) beta)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 1.0 / ((alpha + (beta + 3.0)) * 4.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 1.0d0 / ((alpha + (beta + 3.0d0)) * 4.0d0)
    else
        tmp = ((alpha + 1.0d0) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 1.0 / ((alpha + (beta + 3.0)) * 4.0);
	} else {
		tmp = ((alpha + 1.0) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 1.0 / ((alpha + (beta + 3.0)) * 4.0)
	else:
		tmp = ((alpha + 1.0) / beta) / beta
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(1.0 / Float64(Float64(alpha + Float64(beta + 3.0)) * 4.0));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 1.0 / ((alpha + (beta + 3.0)) * 4.0);
	else
		tmp = ((alpha + 1.0) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(1.0 / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 65.1

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

   7. Applied rewrites 65.1%

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

   8. Taylor expanded in beta around 0

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

   10. Applied rewrites 64.5%

       \[\leadsto \frac{1}{4 \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   if 6 < beta

   1. Initial program 78.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 81.2%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 3\right)\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 70.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0) (/ 0.25 (+ beta 3.0)) (/ (+ alpha 1.0) (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = (alpha + 1.0d0) / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = (alpha + 1.0) / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = (alpha + 1.0) / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(Float64(alpha + 1.0) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = (alpha + 1.0) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(alpha + 1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.9

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in alpha around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 63.1

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

   11. Applied rewrites 63.1%

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

   12. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{4}}{\beta + 3} \]
   13. Step-by-step derivation

   14. Applied rewrites 62.5%

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

   if 6 < beta

   1. Initial program 78.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 69.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.0) (/ 0.25 (+ beta 3.0)) (/ 1.0 (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 6.0d0) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = 1.0d0 / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6.0) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6.0:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = 1.0 / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6.0)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6.0)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = 1.0 / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6.0], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 2.9

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in alpha around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 63.1

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

   11. Applied rewrites 63.1%

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

   12. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{4}}{\beta + 3} \]
   13. Step-by-step derivation

   14. Applied rewrites 62.5%

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

   if 6 < beta

   1. Initial program 78.7%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 74.4%

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

Alternative 23: 61.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{0.25}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.2e+29) (/ 0.25 (+ beta 3.0)) (/ alpha (* beta beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2e+29) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.2d+29) then
        tmp = 0.25d0 / (beta + 3.0d0)
    else
        tmp = alpha / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.2e+29) {
		tmp = 0.25 / (beta + 3.0);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.2e+29:
		tmp = 0.25 / (beta + 3.0)
	else:
		tmp = alpha / (beta * beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.2e+29)
		tmp = Float64(0.25 / Float64(beta + 3.0));
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.2e+29)
		tmp = 0.25 / (beta + 3.0);
	else
		tmp = alpha / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.2e+29], N[(0.25 / N[(beta + 3.0), $MachinePrecision]), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.2e29

   1. Initial program 99.9%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 4.5

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

   8. Applied rewrites 4.5%

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

   9. Taylor expanded in alpha around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 64.0

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

   11. Applied rewrites 64.0%

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

   12. Taylor expanded in beta around 0

       \[\leadsto \frac{\frac{1}{4}}{\beta + 3} \]
   13. Step-by-step derivation

   14. Applied rewrites 61.3%

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

   if 1.2e29 < beta

   1. Initial program 77.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 51.7%

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

Alternative 24: 46.7% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

3. Taylor expanded in beta around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 31.5

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

5. Applied rewrites 31.5%

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

6. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 31.3

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

8. Applied rewrites 31.3%

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

9. Taylor expanded in alpha around 0

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-+.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 N/A

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

    6. +-commutative N/A

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

    7. lower-+.f64 N/A

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

    8. +-commutative N/A

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

    9. lower-+.f64 67.7

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

11. Applied rewrites 67.7%

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

12. Taylor expanded in beta around 0

    \[\leadsto \frac{\frac{1}{4}}{\beta + 3} \]
13. Step-by-step derivation

14. Applied rewrites 42.2%

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

Reproduce

herbie shell --seed 2024223 
(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)))