Octave 3.8, jcobi/1

Percentage Accurate: 74.9% → 99.8%

Time: 23.1s

Alternatives: 12

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\left(-2 - \beta\right) - \beta}{\alpha} \cdot \frac{\beta + 2}{\alpha} + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 + {t_0}^{3}}\right)}{{t_0}^{2} + \left(1 - t_0\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999996)
     (/
      (+
       (* (/ (- (- -2.0 beta) beta) alpha) (/ (+ beta 2.0) alpha))
       (/ (+ beta (- beta -2.0)) alpha))
      2.0)
     (/
      (/ (log (exp (+ 1.0 (pow t_0 3.0)))) (+ (pow t_0 2.0) (- 1.0 t_0)))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (beta + (alpha + 2.0));
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = (log(exp((1.0 + pow(t_0, 3.0)))) / (pow(t_0, 2.0) + (1.0 - t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / (beta + (alpha + 2.0d0))
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999996d0)) then
        tmp = ((((((-2.0d0) - beta) - beta) / alpha) * ((beta + 2.0d0) / alpha)) + ((beta + (beta - (-2.0d0))) / alpha)) / 2.0d0
    else
        tmp = (log(exp((1.0d0 + (t_0 ** 3.0d0)))) / ((t_0 ** 2.0d0) + (1.0d0 - t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (beta + (alpha + 2.0));
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = (Math.log(Math.exp((1.0 + Math.pow(t_0, 3.0)))) / (Math.pow(t_0, 2.0) + (1.0 - t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / (beta + (alpha + 2.0))
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996:
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0
	else:
		tmp = (math.log(math.exp((1.0 + math.pow(t_0, 3.0)))) / (math.pow(t_0, 2.0) + (1.0 - t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999996)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-2.0 - beta) - beta) / alpha) * Float64(Float64(beta + 2.0) / alpha)) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(log(exp(Float64(1.0 + (t_0 ^ 3.0)))) / Float64((t_0 ^ 2.0) + Float64(1.0 - t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / (beta + (alpha + 2.0));
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996)
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	else
		tmp = (log(exp((1.0 + (t_0 ^ 3.0)))) / ((t_0 ^ 2.0) + (1.0 - t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999996], N[(N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Log[N[Exp[N[(1.0 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999995999999999996

   1. Initial program 6.4%

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

      1. +-commutative 6.4%

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

   3. Simplified 6.4%

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

   4. Taylor expanded in alpha around -inf 95.6%

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

   6. Simplified 100.0%

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

   if -0.999995999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. flip3-+ 99.6%

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

      2. pow3 99.6%

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

      3. metadata-eval 99.6%

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

      4. +-commutative 99.6%

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

      5. pow3 99.6%

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

      6. div-inv 99.5%

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

      7. div-inv 99.6%

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

      8. associate-+l+ 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. add-log-exp 99.7%

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\left(-2 - \beta\right) - \beta}{\alpha} \cdot \frac{\beta + 2}{\alpha} + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 + {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}}\right)}{{\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2} + \left(1 - \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\\ t_1 := \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\left(-2 - \beta\right) - \beta}{\alpha} \cdot \frac{\beta + 2}{\alpha} + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(1 + \left({t_1}^{3} + -1\right)\right)}{\left(1 - t_1\right) + t_0 \cdot t_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ alpha (+ beta 2.0))))
        (t_1 (/ (- beta alpha) (+ beta (+ alpha 2.0)))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999996)
     (/
      (+
       (* (/ (- (- -2.0 beta) beta) alpha) (/ (+ beta 2.0) alpha))
       (/ (+ beta (- beta -2.0)) alpha))
      2.0)
     (/
      (/ (+ 1.0 (+ 1.0 (+ (pow t_1 3.0) -1.0))) (+ (- 1.0 t_1) (* t_0 t_0)))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (alpha + (beta + 2.0));
	double t_1 = (beta - alpha) / (beta + (alpha + 2.0));
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 + (pow(t_1, 3.0) + -1.0))) / ((1.0 - t_1) + (t_0 * t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (beta - alpha) / (alpha + (beta + 2.0d0))
    t_1 = (beta - alpha) / (beta + (alpha + 2.0d0))
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999996d0)) then
        tmp = ((((((-2.0d0) - beta) - beta) / alpha) * ((beta + 2.0d0) / alpha)) + ((beta + (beta - (-2.0d0))) / alpha)) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 + ((t_1 ** 3.0d0) + (-1.0d0)))) / ((1.0d0 - t_1) + (t_0 * t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (alpha + (beta + 2.0));
	double t_1 = (beta - alpha) / (beta + (alpha + 2.0));
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 + (Math.pow(t_1, 3.0) + -1.0))) / ((1.0 - t_1) + (t_0 * t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / (alpha + (beta + 2.0))
	t_1 = (beta - alpha) / (beta + (alpha + 2.0))
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996:
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0
	else:
		tmp = ((1.0 + (1.0 + (math.pow(t_1, 3.0) + -1.0))) / ((1.0 - t_1) + (t_0 * t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + 2.0)))
	t_1 = Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999996)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-2.0 - beta) - beta) / alpha) * Float64(Float64(beta + 2.0) / alpha)) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 + Float64((t_1 ^ 3.0) + -1.0))) / Float64(Float64(1.0 - t_1) + Float64(t_0 * t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / (alpha + (beta + 2.0));
	t_1 = (beta - alpha) / (beta + (alpha + 2.0));
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996)
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	else
		tmp = ((1.0 + (1.0 + ((t_1 ^ 3.0) + -1.0))) / ((1.0 - t_1) + (t_0 * t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999996], N[(N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 + N[(N[Power[t$95$1, 3.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999995999999999996

   1. Initial program 6.4%

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

      1. +-commutative 6.4%

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

   3. Simplified 6.4%

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

   4. Taylor expanded in alpha around -inf 95.6%

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

   6. Simplified 100.0%

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

   if -0.999995999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. flip3-+ 99.6%

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

      2. pow3 99.6%

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

      3. metadata-eval 99.6%

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

      4. +-commutative 99.6%

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

      5. pow3 99.6%

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

      6. div-inv 99.5%

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

      7. div-inv 99.6%

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

      8. associate-+l+ 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. expm1-log1p-u 99.6%

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

      2. expm1-udef 99.6%

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

      3. log1p-udef 99.6%

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

      4. add-exp-log 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

   9. Simplified 99.7%

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

       1. unpow2 99.7%

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

       2. +-commutative 99.7%

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

       3. associate-+l+ 99.7%

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

       4. +-commutative 99.7%

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

       5. associate-+l+ 99.7%

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

   11. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\left(-2 - \beta\right) - \beta}{\alpha} \cdot \frac{\beta + 2}{\alpha} + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(1 + \left({\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3} + -1\right)\right)}{\left(1 - \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}}{2}\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\left(-2 - \beta\right) - \beta}{\alpha} \cdot \frac{\beta + 2}{\alpha} + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999996)
   (/
    (+
     (* (/ (- (- -2.0 beta) beta) alpha) (/ (+ beta 2.0) alpha))
     (/ (+ beta (- beta -2.0)) alpha))
    2.0)
   (/ (+ 1.0 (* (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999996d0)) then
        tmp = ((((((-2.0d0) - beta) - beta) / alpha) * ((beta + 2.0d0) / alpha)) + ((beta + (beta - (-2.0d0))) / alpha)) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) * (1.0d0 / (beta + (alpha + 2.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996:
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999996)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-2.0 - beta) - beta) / alpha) * Float64(Float64(beta + 2.0) / alpha)) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) * Float64(1.0 / Float64(beta + Float64(alpha + 2.0))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996)
		tmp = (((((-2.0 - beta) - beta) / alpha) * ((beta + 2.0) / alpha)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999996], N[(N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999995999999999996

   1. Initial program 6.4%

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

      1. +-commutative 6.4%

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

   3. Simplified 6.4%

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

   4. Taylor expanded in alpha around -inf 95.6%

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

   6. Simplified 100.0%

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

   if -0.999995999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. associate-+l+ 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\left(-2 - \beta\right) - \beta}{\alpha} \cdot \frac{\beta + 2}{\alpha} + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999996)
   (/ (+ (* 2.0 (/ beta alpha)) (* 2.0 (/ 1.0 alpha))) 2.0)
   (/ (+ 1.0 (* (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999996d0)) then
        tmp = ((2.0d0 * (beta / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) * (1.0d0 / (beta + (alpha + 2.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996:
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999996)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) * Float64(1.0 / Float64(beta + Float64(alpha + 2.0))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996)
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999996], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999995999999999996

   1. Initial program 6.4%

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

      1. +-commutative 6.4%

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

   3. Simplified 6.4%

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

   4. Taylor expanded in alpha around -inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. sub-neg 99.4%

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

      3. mul-1-neg 99.4%

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

      4. distribute-lft-in 99.4%

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

      5. neg-mul-1 99.4%

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

      6. mul-1-neg 99.4%

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

      7. remove-double-neg 99.4%

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

      8. neg-mul-1 99.4%

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

      9. mul-1-neg 99.4%

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

      10. remove-double-neg 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in beta around 0 99.4%

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

   if -0.999995999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. associate-+l+ 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]

Alternative 5: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.999996:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= (/ (- beta alpha) t_0) -0.999996)
     (/ (+ (* 2.0 (/ beta alpha)) (* 2.0 (/ 1.0 alpha))) 2.0)
     (/ (- 1.0 (/ (- alpha beta) t_0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.999996) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    if (((beta - alpha) / t_0) <= (-0.999996d0)) then
        tmp = ((2.0d0 * (beta / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    else
        tmp = (1.0d0 - ((alpha - beta) / t_0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.999996) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if ((beta - alpha) / t_0) <= -0.999996:
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	else:
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.999996)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(alpha - beta) / t_0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (((beta - alpha) / t_0) <= -0.999996)
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	else
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.999996], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(N[(alpha - beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999995999999999996

   1. Initial program 6.4%

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

      1. +-commutative 6.4%

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

   3. Simplified 6.4%

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

   4. Taylor expanded in alpha around -inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. sub-neg 99.4%

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

      3. mul-1-neg 99.4%

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

      4. distribute-lft-in 99.4%

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

      5. neg-mul-1 99.4%

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

      6. mul-1-neg 99.4%

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

      7. remove-double-neg 99.4%

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

      8. neg-mul-1 99.4%

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

      9. mul-1-neg 99.4%

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

      10. remove-double-neg 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in beta around 0 99.4%

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

   if -0.999995999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array} \]

Alternative 6: 93.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3100:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3100.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ (* 2.0 (/ beta alpha)) (* 2.0 (/ 1.0 alpha))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3100.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 3100.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 * (beta / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3100.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 3100.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3100.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 3100.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 3100.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3100

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 99.3%

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

   if 3100 < alpha

   1. Initial program 23.5%

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

      1. +-commutative 23.5%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in alpha around -inf 83.0%

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

      1. associate-*r/ 83.0%

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

      2. sub-neg 83.0%

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

      3. mul-1-neg 83.0%

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

      4. distribute-lft-in 83.0%

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

      5. neg-mul-1 83.0%

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

      6. mul-1-neg 83.0%

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

      7. remove-double-neg 83.0%

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

      8. neg-mul-1 83.0%

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

      9. mul-1-neg 83.0%

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

      10. remove-double-neg 83.0%

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

   6. Simplified 83.0%

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

   7. Taylor expanded in beta around 0 83.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3100:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 430:\\ \;\;\;\;\frac{1 - \left(\beta - \alpha\right) \cdot \frac{-1}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 430.0)
   (/ (- 1.0 (* (- beta alpha) (/ -1.0 (+ beta 2.0)))) 2.0)
   (/ (+ (* 2.0 (/ beta alpha)) (* 2.0 (/ 1.0 alpha))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 430.0) {
		tmp = (1.0 - ((beta - alpha) * (-1.0 / (beta + 2.0)))) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 430.0d0) then
        tmp = (1.0d0 - ((beta - alpha) * ((-1.0d0) / (beta + 2.0d0)))) / 2.0d0
    else
        tmp = ((2.0d0 * (beta / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 430.0) {
		tmp = (1.0 - ((beta - alpha) * (-1.0 / (beta + 2.0)))) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 430.0:
		tmp = (1.0 - ((beta - alpha) * (-1.0 / (beta + 2.0)))) / 2.0
	else:
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 430.0)
		tmp = Float64(Float64(1.0 - Float64(Float64(beta - alpha) * Float64(-1.0 / Float64(beta + 2.0)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 430.0)
		tmp = (1.0 - ((beta - alpha) * (-1.0 / (beta + 2.0)))) / 2.0;
	else
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 430.0], N[(N[(1.0 - N[(N[(beta - alpha), $MachinePrecision] * N[(-1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 430

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. associate-+l+ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in alpha around 0 99.7%

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

   if 430 < alpha

   1. Initial program 23.5%

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

      1. +-commutative 23.5%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in alpha around -inf 83.0%

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

      1. associate-*r/ 83.0%

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

      2. sub-neg 83.0%

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

      3. mul-1-neg 83.0%

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

      4. distribute-lft-in 83.0%

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

      5. neg-mul-1 83.0%

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

      6. mul-1-neg 83.0%

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

      7. remove-double-neg 83.0%

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

      8. neg-mul-1 83.0%

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

      9. mul-1-neg 83.0%

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

      10. remove-double-neg 83.0%

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

   6. Simplified 83.0%

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

   7. Taylor expanded in beta around 0 83.0%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 430:\\ \;\;\;\;\frac{1 - \left(\beta - \alpha\right) \cdot \frac{-1}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 51.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 880:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} - \frac{2}{\alpha \cdot \alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 880.0) 1.0 (- (/ 1.0 alpha) (/ 2.0 (* alpha alpha)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 880.0) {
		tmp = 1.0;
	} else {
		tmp = (1.0 / alpha) - (2.0 / (alpha * alpha));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 880.0d0) then
        tmp = 1.0d0
    else
        tmp = (1.0d0 / alpha) - (2.0d0 / (alpha * alpha))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 880.0) {
		tmp = 1.0;
	} else {
		tmp = (1.0 / alpha) - (2.0 / (alpha * alpha));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 880.0:
		tmp = 1.0
	else:
		tmp = (1.0 / alpha) - (2.0 / (alpha * alpha))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 880.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(1.0 / alpha) - Float64(2.0 / Float64(alpha * alpha)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 880.0)
		tmp = 1.0;
	else
		tmp = (1.0 / alpha) - (2.0 / (alpha * alpha));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 880.0], 1.0, N[(N[(1.0 / alpha), $MachinePrecision] - N[(2.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 880

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in beta around inf 46.4%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 880 < alpha

   1. Initial program 23.5%

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

      1. +-commutative 23.5%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in alpha around -inf 79.8%

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

   6. Simplified 83.5%

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

   7. Taylor expanded in beta around 0 69.2%

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

      1. associate-*r/ 69.2%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{\alpha}} - 4 \cdot \frac{1}{{\alpha}^{2}}}{2} \]

      2. metadata-eval 69.2%

         \[\leadsto \frac{\frac{\color{blue}{2}}{\alpha} - 4 \cdot \frac{1}{{\alpha}^{2}}}{2} \]

      3. associate-*r/ 69.2%

         \[\leadsto \frac{\frac{2}{\alpha} - \color{blue}{\frac{4 \cdot 1}{{\alpha}^{2}}}}{2} \]

      4. metadata-eval 69.2%

         \[\leadsto \frac{\frac{2}{\alpha} - \frac{\color{blue}{4}}{{\alpha}^{2}}}{2} \]

      5. unpow2 69.2%

         \[\leadsto \frac{\frac{2}{\alpha} - \frac{4}{\color{blue}{\alpha \cdot \alpha}}}{2} \]

   9. Simplified 69.2%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}}}{2} \]

   10. Taylor expanded in alpha around 0 69.2%

       \[\leadsto \color{blue}{\frac{1}{\alpha} - 2 \cdot \frac{1}{{\alpha}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 69.2%

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

       2. metadata-eval 69.2%

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

       3. unpow2 69.2%

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

   12. Simplified 69.2%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 880:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} - \frac{2}{\alpha \cdot \alpha}\\ \end{array} \]

Alternative 9: 87.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2400:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} - \frac{2}{\alpha \cdot \alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2400.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (- (/ 1.0 alpha) (/ 2.0 (* alpha alpha)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2400.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (1.0 / alpha) - (2.0 / (alpha * alpha));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2400.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (1.0d0 / alpha) - (2.0d0 / (alpha * alpha))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2400.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (1.0 / alpha) - (2.0 / (alpha * alpha));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2400.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (1.0 / alpha) - (2.0 / (alpha * alpha))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2400.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(1.0 / alpha) - Float64(2.0 / Float64(alpha * alpha)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2400.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (1.0 / alpha) - (2.0 / (alpha * alpha));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2400.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 / alpha), $MachinePrecision] - N[(2.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2400

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 99.3%

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

   if 2400 < alpha

   1. Initial program 23.5%

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

      1. +-commutative 23.5%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in alpha around -inf 79.8%

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

   6. Simplified 83.5%

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

   7. Taylor expanded in beta around 0 69.2%

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

      1. associate-*r/ 69.2%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{\alpha}} - 4 \cdot \frac{1}{{\alpha}^{2}}}{2} \]

      2. metadata-eval 69.2%

         \[\leadsto \frac{\frac{\color{blue}{2}}{\alpha} - 4 \cdot \frac{1}{{\alpha}^{2}}}{2} \]

      3. associate-*r/ 69.2%

         \[\leadsto \frac{\frac{2}{\alpha} - \color{blue}{\frac{4 \cdot 1}{{\alpha}^{2}}}}{2} \]

      4. metadata-eval 69.2%

         \[\leadsto \frac{\frac{2}{\alpha} - \frac{\color{blue}{4}}{{\alpha}^{2}}}{2} \]

      5. unpow2 69.2%

         \[\leadsto \frac{\frac{2}{\alpha} - \frac{4}{\color{blue}{\alpha \cdot \alpha}}}{2} \]

   9. Simplified 69.2%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}}}{2} \]

   10. Taylor expanded in alpha around 0 69.2%

       \[\leadsto \color{blue}{\frac{1}{\alpha} - 2 \cdot \frac{1}{{\alpha}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 69.2%

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

       2. metadata-eval 69.2%

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

       3. unpow2 69.2%

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

   12. Simplified 69.2%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2400:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} - \frac{2}{\alpha \cdot \alpha}\\ \end{array} \]

Alternative 10: 93.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3100:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3100.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3100.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 3100.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3100.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 3100.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3100.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 3100.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 3100.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3100

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 99.3%

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

   if 3100 < alpha

   1. Initial program 23.5%

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

      1. +-commutative 23.5%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in alpha around -inf 83.0%

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

      1. associate-*r/ 83.0%

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

      2. sub-neg 83.0%

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

      3. mul-1-neg 83.0%

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

      4. distribute-lft-in 83.0%

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

      5. neg-mul-1 83.0%

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

      6. mul-1-neg 83.0%

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

      7. remove-double-neg 83.0%

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

      8. neg-mul-1 83.0%

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

      9. mul-1-neg 83.0%

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

      10. remove-double-neg 83.0%

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

   6. Simplified 83.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3100:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 11: 51.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2700:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2700.0) 1.0 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2700.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2700.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2700.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2700.0:
		tmp = 1.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2700.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2700.0)
		tmp = 1.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2700.0], 1.0, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2700

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in beta around inf 46.4%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 2700 < alpha

   1. Initial program 23.5%

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

      1. +-commutative 23.5%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in alpha around -inf 79.8%

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

   6. Simplified 83.5%

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

   7. Taylor expanded in beta around 0 69.2%

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

      1. associate-*r/ 69.2%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{\alpha}} - 4 \cdot \frac{1}{{\alpha}^{2}}}{2} \]

      2. metadata-eval 69.2%

         \[\leadsto \frac{\frac{\color{blue}{2}}{\alpha} - 4 \cdot \frac{1}{{\alpha}^{2}}}{2} \]

      3. associate-*r/ 69.2%

         \[\leadsto \frac{\frac{2}{\alpha} - \color{blue}{\frac{4 \cdot 1}{{\alpha}^{2}}}}{2} \]

      4. metadata-eval 69.2%

         \[\leadsto \frac{\frac{2}{\alpha} - \frac{\color{blue}{4}}{{\alpha}^{2}}}{2} \]

      5. unpow2 69.2%

         \[\leadsto \frac{\frac{2}{\alpha} - \frac{4}{\color{blue}{\alpha \cdot \alpha}}}{2} \]

   9. Simplified 69.2%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}}}{2} \]

   10. Taylor expanded in alpha around inf 67.8%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2700:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 12: 23.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\alpha} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 1.0 alpha))

C

double code(double alpha, double beta) {
	return 1.0 / alpha;
}

Fortran

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

Java

public static double code(double alpha, double beta) {
	return 1.0 / alpha;
}

Python

def code(alpha, beta):
	return 1.0 / alpha

Julia

function code(alpha, beta)
	return Float64(1.0 / alpha)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 1.0 / alpha;
end

Wolfram

code[alpha_, beta_] := N[(1.0 / alpha), $MachinePrecision]

Derivation

1. Initial program 74.9%

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

   1. +-commutative 74.9%

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

3. Simplified 74.9%

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

4. Taylor expanded in alpha around -inf 26.7%

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

6. Simplified 27.9%

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

7. Taylor expanded in beta around 0 23.4%

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

   1. associate-*r/ 23.4%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{\alpha}} - 4 \cdot \frac{1}{{\alpha}^{2}}}{2} \]

   2. metadata-eval 23.4%

      \[\leadsto \frac{\frac{\color{blue}{2}}{\alpha} - 4 \cdot \frac{1}{{\alpha}^{2}}}{2} \]

   3. associate-*r/ 23.4%

      \[\leadsto \frac{\frac{2}{\alpha} - \color{blue}{\frac{4 \cdot 1}{{\alpha}^{2}}}}{2} \]

   4. metadata-eval 23.4%

      \[\leadsto \frac{\frac{2}{\alpha} - \frac{\color{blue}{4}}{{\alpha}^{2}}}{2} \]

   5. unpow2 23.4%

      \[\leadsto \frac{\frac{2}{\alpha} - \frac{4}{\color{blue}{\alpha \cdot \alpha}}}{2} \]

9. Simplified 23.4%

   \[\leadsto \frac{\color{blue}{\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}}}{2} \]

10. Taylor expanded in alpha around inf 24.5%

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

11. Final simplification 24.5%

    \[\leadsto \frac{1}{\alpha} \]

Reproduce

herbie shell --seed 2023283 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))