Octave 3.8, jcobi/1

Percentage Accurate: 74.4% → 99.9%

Time: 9.5s

Alternatives: 13

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: 74.4% 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta}{t_0}}\right) + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999995)
     (/
      (fma
       (/ (- (- -2.0 beta) beta) alpha)
       (/ (+ beta 2.0) alpha)
       (/ (+ beta (+ beta 2.0)) alpha))
      2.0)
     (/ (+ (log (exp (/ beta t_0))) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999995) {
		tmp = fma((((-2.0 - beta) - beta) / alpha), ((beta + 2.0) / alpha), ((beta + (beta + 2.0)) / alpha)) / 2.0;
	} else {
		tmp = (log(exp((beta / t_0))) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999995)
		tmp = Float64(fma(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(beta / t_0))) + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999995], N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Log[N[Exp[N[(beta / t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[(alpha / 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)) < -0.99999499999999997

   1. Initial program 6.2%

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

      1. +-commutative 6.2%

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

   3. Simplified 6.2%

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

   4. Taylor expanded in alpha around -inf 98.7%

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

   6. Simplified 100.0%

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

   if -0.99999499999999997 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. add-log-exp 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta}{\beta + \left(\alpha + 2\right)}}\right) + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} - \frac{\frac{\frac{\alpha}{t_1}}{\frac{t_1}{\alpha}} + -1}{\frac{\alpha}{t_0} + 1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999995)
     (/
      (fma
       (/ (- (- -2.0 beta) beta) alpha)
       (/ (+ beta 2.0) alpha)
       (/ (+ beta (+ beta 2.0)) alpha))
      2.0)
     (/
      (-
       (/ beta t_0)
       (/ (+ (/ (/ alpha t_1) (/ t_1 alpha)) -1.0) (+ (/ alpha t_0) 1.0)))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999995) {
		tmp = fma((((-2.0 - beta) - beta) / alpha), ((beta + 2.0) / alpha), ((beta + (beta + 2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((beta / t_0) - ((((alpha / t_1) / (t_1 / alpha)) + -1.0) / ((alpha / t_0) + 1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999995)
		tmp = Float64(fma(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(beta / t_0) - Float64(Float64(Float64(Float64(alpha / t_1) / Float64(t_1 / alpha)) + -1.0) / Float64(Float64(alpha / t_0) + 1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999995], N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] - N[(N[(N[(N[(alpha / t$95$1), $MachinePrecision] / N[(t$95$1 / alpha), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(alpha / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999499999999997

   1. Initial program 6.2%

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

      1. +-commutative 6.2%

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

   3. Simplified 6.2%

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

   4. Taylor expanded in alpha around -inf 98.7%

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

   6. Simplified 100.0%

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

   if -0.99999499999999997 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. flip-- 99.9%

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

      2. pow2 99.9%

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

      3. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. unpow2 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\frac{\frac{\alpha}{\alpha + \left(\beta + 2\right)}}{\frac{\alpha + \left(\beta + 2\right)}{\alpha}} + -1}{\frac{\alpha}{\beta + \left(\alpha + 2\right)} + 1}}{2}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} - \frac{\frac{\frac{\alpha}{t_1}}{\frac{t_1}{\alpha}} + -1}{\frac{\alpha}{t_0} + 1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999995)
     (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
     (/
      (-
       (/ beta t_0)
       (/ (+ (/ (/ alpha t_1) (/ t_1 alpha)) -1.0) (+ (/ alpha t_0) 1.0)))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999995) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) - ((((alpha / t_1) / (t_1 / alpha)) + -1.0) / ((alpha / t_0) + 1.0))) / 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 + 2.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999995d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((beta / t_0) - ((((alpha / t_1) / (t_1 / alpha)) + (-1.0d0)) / ((alpha / t_0) + 1.0d0))) / 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 t_1 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999995) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) - ((((alpha / t_1) / (t_1 / alpha)) + -1.0) / ((alpha / t_0) + 1.0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999995:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = ((beta / t_0) - ((((alpha / t_1) / (t_1 / alpha)) + -1.0) / ((alpha / t_0) + 1.0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999995)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) - Float64(Float64(Float64(Float64(alpha / t_1) / Float64(t_1 / alpha)) + -1.0) / Float64(Float64(alpha / t_0) + 1.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999995)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = ((beta / t_0) - ((((alpha / t_1) / (t_1 / alpha)) + -1.0) / ((alpha / t_0) + 1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999995], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] - N[(N[(N[(N[(alpha / t$95$1), $MachinePrecision] / N[(t$95$1 / alpha), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(alpha / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999499999999997

   1. Initial program 6.2%

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

      1. +-commutative 6.2%

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

   3. Simplified 6.2%

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

   4. Taylor expanded in alpha around -inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. sub-neg 99.6%

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

      3. mul-1-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. neg-mul-1 99.6%

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

      6. mul-1-neg 99.6%

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

      7. remove-double-neg 99.6%

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

      8. neg-mul-1 99.6%

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

      9. mul-1-neg 99.6%

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

      10. remove-double-neg 99.6%

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

      11. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   if -0.99999499999999997 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. flip-- 99.9%

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

      2. pow2 99.9%

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

      3. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. unpow2 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\frac{\frac{\alpha}{\alpha + \left(\beta + 2\right)}}{\frac{\alpha + \left(\beta + 2\right)}{\alpha}} + -1}{\frac{\alpha}{\beta + \left(\alpha + 2\right)} + 1}}{2}\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999995)
     (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
     (/ (+ (/ beta t_0) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999995) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / 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) / ((beta + alpha) + 2.0d0)) <= (-0.999995d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((beta / t_0) + (1.0d0 - (alpha / 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) / ((beta + alpha) + 2.0)) <= -0.999995) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999995:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999995)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(1.0 - Float64(alpha / 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) / ((beta + alpha) + 2.0)) <= -0.999995)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999995], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / 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)) < -0.99999499999999997

   1. Initial program 6.2%

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

      1. +-commutative 6.2%

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

   3. Simplified 6.2%

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

   4. Taylor expanded in alpha around -inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. sub-neg 99.6%

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

      3. mul-1-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. neg-mul-1 99.6%

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

      6. mul-1-neg 99.6%

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

      7. remove-double-neg 99.6%

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

      8. neg-mul-1 99.6%

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

      9. mul-1-neg 99.6%

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

      10. remove-double-neg 99.6%

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

      11. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   if -0.99999499999999997 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]

Alternative 5: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999995)
     (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999995) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 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 (t_0 <= (-0.999995d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 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 (t_0 <= -0.999995) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.999995:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.999995)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 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 (t_0 <= -0.999995)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.999995], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999499999999997

   1. Initial program 6.2%

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

      1. +-commutative 6.2%

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

   3. Simplified 6.2%

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

   4. Taylor expanded in alpha around -inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. sub-neg 99.6%

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

      3. mul-1-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. neg-mul-1 99.6%

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

      6. mul-1-neg 99.6%

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

      7. remove-double-neg 99.6%

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

      8. neg-mul-1 99.6%

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

      9. mul-1-neg 99.6%

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

      10. remove-double-neg 99.6%

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

      11. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   if -0.99999499999999997 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 6: 69.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq -7.8 \cdot 10^{-124}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq -1.5 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.12 \cdot 10^{-30}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 15000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-2}{\alpha}}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha -7.8e-124)
   (+ 0.5 (* alpha -0.25))
   (if (<= alpha -1.5e-132)
     1.0
     (if (<= alpha 1.12e-30)
       0.5
       (if (<= alpha 15000000.0) 1.0 (/ (+ 1.0 (/ -2.0 alpha)) alpha))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -7.8e-124) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= -1.5e-132) {
		tmp = 1.0;
	} else if (alpha <= 1.12e-30) {
		tmp = 0.5;
	} else if (alpha <= 15000000.0) {
		tmp = 1.0;
	} else {
		tmp = (1.0 + (-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 <= (-7.8d-124)) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else if (alpha <= (-1.5d-132)) then
        tmp = 1.0d0
    else if (alpha <= 1.12d-30) then
        tmp = 0.5d0
    else if (alpha <= 15000000.0d0) then
        tmp = 1.0d0
    else
        tmp = (1.0d0 + ((-2.0d0) / alpha)) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -7.8e-124) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= -1.5e-132) {
		tmp = 1.0;
	} else if (alpha <= 1.12e-30) {
		tmp = 0.5;
	} else if (alpha <= 15000000.0) {
		tmp = 1.0;
	} else {
		tmp = (1.0 + (-2.0 / alpha)) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= -7.8e-124:
		tmp = 0.5 + (alpha * -0.25)
	elif alpha <= -1.5e-132:
		tmp = 1.0
	elif alpha <= 1.12e-30:
		tmp = 0.5
	elif alpha <= 15000000.0:
		tmp = 1.0
	else:
		tmp = (1.0 + (-2.0 / alpha)) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= -7.8e-124)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	elseif (alpha <= -1.5e-132)
		tmp = 1.0;
	elseif (alpha <= 1.12e-30)
		tmp = 0.5;
	elseif (alpha <= 15000000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(1.0 + Float64(-2.0 / alpha)) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= -7.8e-124)
		tmp = 0.5 + (alpha * -0.25);
	elseif (alpha <= -1.5e-132)
		tmp = 1.0;
	elseif (alpha <= 1.12e-30)
		tmp = 0.5;
	elseif (alpha <= 15000000.0)
		tmp = 1.0;
	else
		tmp = (1.0 + (-2.0 / alpha)) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, -7.8e-124], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[alpha, -1.5e-132], 1.0, If[LessEqual[alpha, 1.12e-30], 0.5, If[LessEqual[alpha, 15000000.0], 1.0, N[(N[(1.0 + N[(-2.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < -7.79999999999999986e-124

   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 0 69.0%

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

      1. +-commutative 69.0%

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

   6. Simplified 69.0%

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

   7. Taylor expanded in alpha around 0 66.3%

      \[\leadsto \color{blue}{0.5 + -0.25 \cdot \alpha} \]
   8. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto 0.5 + \color{blue}{\alpha \cdot -0.25} \]

   9. Simplified 66.3%

      \[\leadsto \color{blue}{0.5 + \alpha \cdot -0.25} \]

   if -7.79999999999999986e-124 < alpha < -1.5e-132 or 1.12e-30 < alpha < 1.5e7

   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 73.3%

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

   if -1.5e-132 < alpha < 1.12e-30

   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 0 75.3%

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

      1. +-commutative 75.3%

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

   6. Simplified 75.3%

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

   7. Taylor expanded in alpha around 0 75.3%

      \[\leadsto \color{blue}{0.5} \]

   if 1.5e7 < alpha

   1. Initial program 14.8%

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

      1. +-commutative 14.8%

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

   3. Simplified 14.8%

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

   4. Taylor expanded in alpha around -inf 90.2%

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

   6. Simplified 91.4%

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

   7. Taylor expanded in beta around 0 76.2%

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

      1. associate-*r/ 76.2%

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

      2. metadata-eval 76.2%

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

      3. associate-*r/ 76.2%

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

      4. metadata-eval 76.2%

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

      5. unpow2 76.2%

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

   9. Simplified 76.2%

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

       1. div-inv 76.2%

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

       2. associate-/r* 76.2%

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

       3. sub-div 76.2%

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

       4. metadata-eval 76.2%

          \[\leadsto \frac{2 - \frac{4}{\alpha}}{\alpha} \cdot \color{blue}{0.5} \]

   11. Applied egg-rr 76.2%

       \[\leadsto \color{blue}{\frac{2 - \frac{4}{\alpha}}{\alpha} \cdot 0.5} \]
   12. Step-by-step derivation

       1. *-commutative 76.2%

          \[\leadsto \color{blue}{0.5 \cdot \frac{2 - \frac{4}{\alpha}}{\alpha}} \]

       2. associate-*r/ 76.2%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \left(2 - \frac{4}{\alpha}\right)}{\alpha}} \]

       3. sub-neg 76.2%

          \[\leadsto \frac{0.5 \cdot \color{blue}{\left(2 + \left(-\frac{4}{\alpha}\right)\right)}}{\alpha} \]

       4. distribute-rgt-in 76.2%

          \[\leadsto \frac{\color{blue}{2 \cdot 0.5 + \left(-\frac{4}{\alpha}\right) \cdot 0.5}}{\alpha} \]

       5. metadata-eval 76.2%

          \[\leadsto \frac{\color{blue}{1} + \left(-\frac{4}{\alpha}\right) \cdot 0.5}{\alpha} \]

       6. metadata-eval 76.2%

          \[\leadsto \frac{1 + \left(-\frac{\color{blue}{2 \cdot 2}}{\alpha}\right) \cdot 0.5}{\alpha} \]

       7. associate-*r/ 76.2%

          \[\leadsto \frac{1 + \left(-\color{blue}{2 \cdot \frac{2}{\alpha}}\right) \cdot 0.5}{\alpha} \]

       8. distribute-lft-neg-in 76.2%

          \[\leadsto \frac{1 + \color{blue}{\left(\left(-2\right) \cdot \frac{2}{\alpha}\right)} \cdot 0.5}{\alpha} \]

       9. metadata-eval 76.2%

          \[\leadsto \frac{1 + \left(\color{blue}{-2} \cdot \frac{2}{\alpha}\right) \cdot 0.5}{\alpha} \]

       10. associate-*r* 76.2%

           \[\leadsto \frac{1 + \color{blue}{-2 \cdot \left(\frac{2}{\alpha} \cdot 0.5\right)}}{\alpha} \]

       11. associate-*l/ 76.2%

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

       12. metadata-eval 76.2%

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

       13. associate-*r/ 76.2%

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

       14. metadata-eval 76.2%

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

       15. metadata-eval 76.2%

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

       16. distribute-neg-frac 76.2%

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

       17. distribute-neg-frac 76.2%

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

       18. metadata-eval 76.2%

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

   13. Simplified 76.2%

       \[\leadsto \color{blue}{\frac{1 + \frac{-2}{\alpha}}{\alpha}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -7.8 \cdot 10^{-124}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq -1.5 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.12 \cdot 10^{-30}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 15000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-2}{\alpha}}{\alpha}\\ \end{array} \]

Alternative 7: 69.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq -7.8 \cdot 10^{-124}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq -2.35 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2.75 \cdot 10^{-30}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 52000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha -7.8e-124)
   (+ 0.5 (* alpha -0.25))
   (if (<= alpha -2.35e-132)
     1.0
     (if (<= alpha 2.75e-30)
       0.5
       (if (<= alpha 52000000.0) 1.0 (/ 1.0 alpha))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -7.8e-124) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= -2.35e-132) {
		tmp = 1.0;
	} else if (alpha <= 2.75e-30) {
		tmp = 0.5;
	} else if (alpha <= 52000000.0) {
		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 <= (-7.8d-124)) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else if (alpha <= (-2.35d-132)) then
        tmp = 1.0d0
    else if (alpha <= 2.75d-30) then
        tmp = 0.5d0
    else if (alpha <= 52000000.0d0) 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 <= -7.8e-124) {
		tmp = 0.5 + (alpha * -0.25);
	} else if (alpha <= -2.35e-132) {
		tmp = 1.0;
	} else if (alpha <= 2.75e-30) {
		tmp = 0.5;
	} else if (alpha <= 52000000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= -7.8e-124:
		tmp = 0.5 + (alpha * -0.25)
	elif alpha <= -2.35e-132:
		tmp = 1.0
	elif alpha <= 2.75e-30:
		tmp = 0.5
	elif alpha <= 52000000.0:
		tmp = 1.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= -7.8e-124)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	elseif (alpha <= -2.35e-132)
		tmp = 1.0;
	elseif (alpha <= 2.75e-30)
		tmp = 0.5;
	elseif (alpha <= 52000000.0)
		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 <= -7.8e-124)
		tmp = 0.5 + (alpha * -0.25);
	elseif (alpha <= -2.35e-132)
		tmp = 1.0;
	elseif (alpha <= 2.75e-30)
		tmp = 0.5;
	elseif (alpha <= 52000000.0)
		tmp = 1.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, -7.8e-124], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[alpha, -2.35e-132], 1.0, If[LessEqual[alpha, 2.75e-30], 0.5, If[LessEqual[alpha, 52000000.0], 1.0, N[(1.0 / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < -7.79999999999999986e-124

   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 0 69.0%

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

      1. +-commutative 69.0%

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

   6. Simplified 69.0%

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

   7. Taylor expanded in alpha around 0 66.3%

      \[\leadsto \color{blue}{0.5 + -0.25 \cdot \alpha} \]
   8. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto 0.5 + \color{blue}{\alpha \cdot -0.25} \]

   9. Simplified 66.3%

      \[\leadsto \color{blue}{0.5 + \alpha \cdot -0.25} \]

   if -7.79999999999999986e-124 < alpha < -2.3500000000000001e-132 or 2.74999999999999988e-30 < alpha < 5.2e7

   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 73.3%

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

   if -2.3500000000000001e-132 < alpha < 2.74999999999999988e-30

   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 0 75.3%

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

      1. +-commutative 75.3%

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

   6. Simplified 75.3%

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

   7. Taylor expanded in alpha around 0 75.3%

      \[\leadsto \color{blue}{0.5} \]

   if 5.2e7 < alpha

   1. Initial program 14.8%

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

      1. +-commutative 14.8%

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

   3. Simplified 14.8%

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

   4. Taylor expanded in alpha around -inf 91.2%

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

      1. associate-*r/ 91.2%

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

      2. sub-neg 91.2%

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

      3. mul-1-neg 91.2%

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

      4. distribute-lft-in 91.2%

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

      5. neg-mul-1 91.2%

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

      6. mul-1-neg 91.2%

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

      7. remove-double-neg 91.2%

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

      8. neg-mul-1 91.2%

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

      9. mul-1-neg 91.2%

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

      10. remove-double-neg 91.2%

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

      11. +-commutative 91.2%

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

   6. Simplified 91.2%

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

   7. Taylor expanded in beta around 0 75.9%

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

   8. Taylor expanded in alpha around 0 75.9%

      \[\leadsto \color{blue}{\frac{1}{\alpha}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -7.8 \cdot 10^{-124}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq -2.35 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2.75 \cdot 10^{-30}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 52000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 8: 67.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.18 \cdot 10^{-124}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 0.00146:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.18e-124)
   (/ (+ 1.0 (* beta 0.5)) 2.0)
   (if (<= beta 0.00146) (/ 1.0 alpha) (/ (- 2.0 (/ 2.0 beta)) 2.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.18e-124) {
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	} else if (beta <= 0.00146) {
		tmp = 1.0 / alpha;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.18d-124) then
        tmp = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    else if (beta <= 0.00146d0) then
        tmp = 1.0d0 / alpha
    else
        tmp = (2.0d0 - (2.0d0 / beta)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.18e-124) {
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	} else if (beta <= 0.00146) {
		tmp = 1.0 / alpha;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.18e-124:
		tmp = (1.0 + (beta * 0.5)) / 2.0
	elif beta <= 0.00146:
		tmp = 1.0 / alpha
	else:
		tmp = (2.0 - (2.0 / beta)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.18e-124)
		tmp = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0);
	elseif (beta <= 0.00146)
		tmp = Float64(1.0 / alpha);
	else
		tmp = Float64(Float64(2.0 - Float64(2.0 / beta)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.18e-124)
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	elseif (beta <= 0.00146)
		tmp = 1.0 / alpha;
	else
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.18e-124], N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 0.00146], N[(1.0 / alpha), $MachinePrecision], N[(N[(2.0 - N[(2.0 / beta), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.17999999999999992e-124

   1. Initial program 71.3%

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

      1. +-commutative 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in alpha around 0 67.2%

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

   5. Taylor expanded in beta around 0 66.9%

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

      1. *-commutative 66.9%

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

   7. Simplified 66.9%

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

   if 1.17999999999999992e-124 < beta < 0.0014599999999999999

   1. Initial program 41.3%

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

      1. +-commutative 41.3%

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

   3. Simplified 41.3%

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

   4. Taylor expanded in alpha around -inf 63.3%

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

      1. associate-*r/ 63.3%

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

      2. sub-neg 63.3%

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

      3. mul-1-neg 63.3%

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

      4. distribute-lft-in 63.3%

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

      5. neg-mul-1 63.3%

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

      6. mul-1-neg 63.3%

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

      7. remove-double-neg 63.3%

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

      8. neg-mul-1 63.3%

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

      9. mul-1-neg 63.3%

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

      10. remove-double-neg 63.3%

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

      11. +-commutative 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in beta around 0 62.2%

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

   8. Taylor expanded in alpha around 0 62.2%

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

   if 0.0014599999999999999 < beta

   1. Initial program 85.0%

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

      1. +-commutative 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in alpha around 0 83.5%

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

   5. Taylor expanded in beta around inf 82.0%

      \[\leadsto \frac{\color{blue}{2 - 2 \cdot \frac{1}{\beta}}}{2} \]
   6. Step-by-step derivation

      1. associate-*r/ 82.0%

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

      2. metadata-eval 82.0%

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

   7. Simplified 82.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.18 \cdot 10^{-124}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 0.00146:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]

Alternative 9: 88.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 122000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} - \frac{2}{\alpha \cdot \alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 122000000.0)
   (/ (+ 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 <= 122000000.0) {
		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 <= 122000000.0d0) 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 <= 122000000.0) {
		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 <= 122000000.0:
		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 <= 122000000.0)
		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 <= 122000000.0)
		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, 122000000.0], 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 < 1.22e8

   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 97.0%

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

   if 1.22e8 < alpha

   1. Initial program 14.8%

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

      1. +-commutative 14.8%

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

   3. Simplified 14.8%

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

   4. Taylor expanded in beta around 0 5.9%

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

      1. +-commutative 5.9%

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

   6. Simplified 5.9%

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

   7. Taylor expanded in alpha around inf 76.2%

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

      1. associate-*r/ 76.2%

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

      2. metadata-eval 76.2%

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

      3. unpow2 76.2%

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

   9. Simplified 76.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 122000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} - \frac{2}{\alpha \cdot \alpha}\\ \end{array} \]

Alternative 10: 93.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 50000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 50000000.0)
   (/ (+ 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 <= 50000000.0) {
		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 <= 50000000.0d0) 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 <= 50000000.0) {
		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 <= 50000000.0:
		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 <= 50000000.0)
		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 <= 50000000.0)
		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, 50000000.0], 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 < 5e7

   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 97.0%

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

   if 5e7 < alpha

   1. Initial program 14.8%

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

      1. +-commutative 14.8%

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

   3. Simplified 14.8%

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

   4. Taylor expanded in alpha around -inf 91.2%

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

      1. associate-*r/ 91.2%

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

      2. sub-neg 91.2%

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

      3. mul-1-neg 91.2%

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

      4. distribute-lft-in 91.2%

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

      5. neg-mul-1 91.2%

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

      6. mul-1-neg 91.2%

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

      7. remove-double-neg 91.2%

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

      8. neg-mul-1 91.2%

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

      9. mul-1-neg 91.2%

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

      10. remove-double-neg 91.2%

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

      11. +-commutative 91.2%

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

   6. Simplified 91.2%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 50000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 11: 67.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{-124}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.1e-124) 0.5 (if (<= beta 1.7e-5) (/ 1.0 alpha) 1.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.1e-124) {
		tmp = 0.5;
	} else if (beta <= 1.7e-5) {
		tmp = 1.0 / alpha;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.1d-124) then
        tmp = 0.5d0
    else if (beta <= 1.7d-5) then
        tmp = 1.0d0 / alpha
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.1e-124) {
		tmp = 0.5;
	} else if (beta <= 1.7e-5) {
		tmp = 1.0 / alpha;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.1e-124:
		tmp = 0.5
	elif beta <= 1.7e-5:
		tmp = 1.0 / alpha
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.1e-124)
		tmp = 0.5;
	elseif (beta <= 1.7e-5)
		tmp = Float64(1.0 / alpha);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.1e-124)
		tmp = 0.5;
	elseif (beta <= 1.7e-5)
		tmp = 1.0 / alpha;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.1e-124], 0.5, If[LessEqual[beta, 1.7e-5], N[(1.0 / alpha), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if beta < 1.0999999999999999e-124

   1. Initial program 71.3%

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

      1. +-commutative 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in beta around 0 70.2%

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

      1. +-commutative 70.2%

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

   6. Simplified 70.2%

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

   7. Taylor expanded in alpha around 0 66.2%

      \[\leadsto \color{blue}{0.5} \]

   if 1.0999999999999999e-124 < beta < 1.7e-5

   1. Initial program 41.3%

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

      1. +-commutative 41.3%

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

   3. Simplified 41.3%

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

   4. Taylor expanded in alpha around -inf 63.3%

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

      1. associate-*r/ 63.3%

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

      2. sub-neg 63.3%

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

      3. mul-1-neg 63.3%

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

      4. distribute-lft-in 63.3%

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

      5. neg-mul-1 63.3%

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

      6. mul-1-neg 63.3%

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

      7. remove-double-neg 63.3%

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

      8. neg-mul-1 63.3%

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

      9. mul-1-neg 63.3%

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

      10. remove-double-neg 63.3%

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

      11. +-commutative 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in beta around 0 62.2%

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

   8. Taylor expanded in alpha around 0 62.2%

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

   if 1.7e-5 < beta

   1. Initial program 85.0%

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

      1. +-commutative 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in beta around inf 81.4%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{-124}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 68.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 15000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 15000000.0) 0.5 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 15000000.0) {
		tmp = 0.5;
	} 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 <= 15000000.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 15000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 15000000.0:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 15000000.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 15000000.0)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 15000000.0], 0.5, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.5e7

   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 0 69.4%

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

      1. +-commutative 69.4%

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

   6. Simplified 69.4%

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

   7. Taylor expanded in alpha around 0 66.6%

      \[\leadsto \color{blue}{0.5} \]

   if 1.5e7 < alpha

   1. Initial program 14.8%

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

      1. +-commutative 14.8%

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

   3. Simplified 14.8%

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

   4. Taylor expanded in alpha around -inf 91.2%

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

      1. associate-*r/ 91.2%

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

      2. sub-neg 91.2%

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

      3. mul-1-neg 91.2%

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

      4. distribute-lft-in 91.2%

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

      5. neg-mul-1 91.2%

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

      6. mul-1-neg 91.2%

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

      7. remove-double-neg 91.2%

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

      8. neg-mul-1 91.2%

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

      9. mul-1-neg 91.2%

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

      10. remove-double-neg 91.2%

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

      11. +-commutative 91.2%

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

   6. Simplified 91.2%

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

   7. Taylor expanded in beta around 0 75.9%

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

   8. Taylor expanded in alpha around 0 75.9%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 15000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 13: 48.9% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

double code(double alpha, double beta) {
	return 0.5;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.5d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.5;
}

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.5;
end

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 72.0%

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

   1. +-commutative 72.0%

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

3. Simplified 72.0%

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

4. Taylor expanded in beta around 0 48.6%

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

   1. +-commutative 48.6%

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

6. Simplified 48.6%

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

7. Taylor expanded in alpha around 0 46.7%

   \[\leadsto \color{blue}{0.5} \]

8. Final simplification 46.7%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023230 
(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))