Octave 3.8, jcobi/1

Percentage Accurate: 74.9% → 99.9%

Time: 11.2s

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: 74.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2 - \beta\right) - \beta\\ t_1 := \beta + \left(\alpha + 2\right)\\ t_2 := \frac{-1}{t_1}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot t_0 - \frac{t_0}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, 1 + \frac{\beta}{t_1}, \alpha \cdot t_2\right) + \mathsf{fma}\left(t_2, \alpha, \alpha \cdot \frac{1}{t_1}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (- -2.0 beta) beta))
        (t_1 (+ beta (+ alpha 2.0)))
        (t_2 (/ -1.0 t_1)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999)
     (/ (- (* (/ (+ beta 2.0) (pow alpha 2.0)) t_0) (/ t_0 alpha)) 2.0)
     (/
      (+
       (fma 1.0 (+ 1.0 (/ beta t_1)) (* alpha t_2))
       (fma t_2 alpha (* alpha (/ 1.0 t_1))))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (-2.0 - beta) - beta;
	double t_1 = beta + (alpha + 2.0);
	double t_2 = -1.0 / t_1;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999) {
		tmp = ((((beta + 2.0) / pow(alpha, 2.0)) * t_0) - (t_0 / alpha)) / 2.0;
	} else {
		tmp = (fma(1.0, (1.0 + (beta / t_1)), (alpha * t_2)) + fma(t_2, alpha, (alpha * (1.0 / t_1)))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(-2.0 - beta) - beta)
	t_1 = Float64(beta + Float64(alpha + 2.0))
	t_2 = Float64(-1.0 / t_1)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999)
		tmp = Float64(Float64(Float64(Float64(Float64(beta + 2.0) / (alpha ^ 2.0)) * t_0) - Float64(t_0 / alpha)) / 2.0);
	else
		tmp = Float64(Float64(fma(1.0, Float64(1.0 + Float64(beta / t_1)), Float64(alpha * t_2)) + fma(t_2, alpha, Float64(alpha * Float64(1.0 / t_1)))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 / t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(N[(N[(N[(beta + 2.0), $MachinePrecision] / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 * N[(1.0 + N[(beta / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(alpha * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * alpha + N[(alpha * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999990000000000046

   1. Initial program 7.0%

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

      1. +-commutative 7.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in alpha around -inf 93.6%

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

   7. Simplified 99.9%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. div-sub 99.6%

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

      3. associate-+r- 99.7%

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

      4. associate-+l+ 99.7%

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

      5. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. div-inv 99.7%

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

      3. prod-diff 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot \left(\left(-2 - \beta\right) - \beta\right) - \frac{\left(-2 - \beta\right) - \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, 1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}, \alpha \cdot \frac{-1}{\beta + \left(\alpha + 2\right)}\right) + \mathsf{fma}\left(\frac{-1}{\beta + \left(\alpha + 2\right)}, \alpha, \alpha \cdot \frac{1}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2 - \beta\right) - \beta\\ t_1 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot t_0 - \frac{t_0}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_1} - \mathsf{fma}\left(\alpha, \frac{1}{t_1}, -1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (- -2.0 beta) beta)) (t_1 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999)
     (/ (- (* (/ (+ beta 2.0) (pow alpha 2.0)) t_0) (/ t_0 alpha)) 2.0)
     (/ (- (/ beta t_1) (fma alpha (/ 1.0 t_1) -1.0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (-2.0 - beta) - beta;
	double t_1 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999) {
		tmp = ((((beta + 2.0) / pow(alpha, 2.0)) * t_0) - (t_0 / alpha)) / 2.0;
	} else {
		tmp = ((beta / t_1) - fma(alpha, (1.0 / t_1), -1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(-2.0 - beta) - beta)
	t_1 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999)
		tmp = Float64(Float64(Float64(Float64(Float64(beta + 2.0) / (alpha ^ 2.0)) * t_0) - Float64(t_0 / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_1) - fma(alpha, Float64(1.0 / t_1), -1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(N[(N[(N[(beta + 2.0), $MachinePrecision] / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$1), $MachinePrecision] - N[(alpha * N[(1.0 / t$95$1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999990000000000046

   1. Initial program 7.0%

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

      1. +-commutative 7.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in alpha around -inf 93.6%

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

   7. Simplified 99.9%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. div-sub 99.6%

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

      2. associate-+l- 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. div-inv 99.7%

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

      2. fma-neg 99.7%

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

      3. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot \left(\left(-2 - \beta\right) - \beta\right) - \frac{\left(-2 - \beta\right) - \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \mathsf{fma}\left(\alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, -1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} - \mathsf{fma}\left(\alpha, \frac{1}{t_0}, -1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999999)
     (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
     (/ (- (/ beta t_0) (fma alpha (/ 1.0 t_0) -1.0)) 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)) <= -0.99999999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) - fma(alpha, (1.0 / t_0), -1.0)) / 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)) <= -0.99999999)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) - fma(alpha, Float64(1.0 / 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]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999999], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] - N[(alpha * N[(1.0 / t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999998999999995

   1. Initial program 6.3%

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

      1. +-commutative 6.3%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in alpha around -inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. sub-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. neg-mul-1 99.5%

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

      6. mul-1-neg 99.5%

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

      7. remove-double-neg 99.5%

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

      8. neg-mul-1 99.5%

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

      9. mul-1-neg 99.5%

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

      10. remove-double-neg 99.5%

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

   7. Simplified 99.5%

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

   if -0.99999998999999995 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

   3. Simplified 99.4%

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

      1. div-sub 99.4%

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

      2. associate-+l- 99.5%

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

      3. associate-+l+ 99.5%

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

      4. associate-+l+ 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. div-inv 99.5%

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

      2. fma-neg 99.5%

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

      3. metadata-eval 99.5%

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

   8. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \mathsf{fma}\left(\alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, -1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{t_0}\right) - \frac{\alpha}{t_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999999)
     (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
     (/ (- (+ 1.0 (/ beta t_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)) <= -0.99999999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_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)) <= (-0.99999999d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((1.0d0 + (beta / t_0)) - (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)) <= -0.99999999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_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)) <= -0.99999999:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = ((1.0 + (beta / t_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)) <= -0.99999999)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(beta / t_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)) <= -0.99999999)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = ((1.0 + (beta / t_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], -0.99999999], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999998999999995

   1. Initial program 6.3%

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

      1. +-commutative 6.3%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in alpha around -inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. sub-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. neg-mul-1 99.5%

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

      6. mul-1-neg 99.5%

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

      7. remove-double-neg 99.5%

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

      8. neg-mul-1 99.5%

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

      9. mul-1-neg 99.5%

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

      10. remove-double-neg 99.5%

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

   7. Simplified 99.5%

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

   if -0.99999998999999995 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. div-sub 99.4%

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

      3. associate-+r- 99.5%

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

      4. associate-+l+ 99.5%

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

      5. associate-+l+ 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) - \frac{\alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -1.0)
     (/ (/ (+ 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)) <= -1.0) {
		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)) <= (-1.0d0)) 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)) <= -1.0) {
		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)) <= -1.0:
		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)) <= -1.0)
		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)) <= -1.0)
		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], -1.0], 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)) < -1

   1. Initial program 5.6%

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

      1. +-commutative 5.6%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in alpha around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. mul-1-neg 100.0%

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

      7. remove-double-neg 100.0%

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

      8. neg-mul-1 100.0%

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

      9. mul-1-neg 100.0%

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

      10. remove-double-neg 100.0%

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

   7. Simplified 100.0%

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

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

   3. Simplified 99.2%

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

      1. div-sub 99.2%

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

      2. associate-+l- 99.3%

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

      3. associate-+l+ 99.3%

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

      4. associate-+l+ 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 99.5%

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

Alternative 6: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999999)
     (/ (/ (+ 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 <= -0.99999999) {
		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 <= (-0.99999999d0)) 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 <= -0.99999999) {
		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 <= -0.99999999:
		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 <= -0.99999999)
		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 <= -0.99999999)
		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, -0.99999999], 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)) < -0.99999998999999995

   1. Initial program 6.3%

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

      1. +-commutative 6.3%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in alpha around -inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. sub-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. neg-mul-1 99.5%

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

      6. mul-1-neg 99.5%

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

      7. remove-double-neg 99.5%

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

      8. neg-mul-1 99.5%

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

      9. mul-1-neg 99.5%

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

      10. remove-double-neg 99.5%

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

   7. Simplified 99.5%

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

   if -0.99999998999999995 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 4.2e+19)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta 2.0) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 4.2e+19) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((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 <= 4.2d+19) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((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 <= 4.2e+19) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 4.2e+19:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + 2.0) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 4.2e+19)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + 2.0) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 4.2e+19)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + 2.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 4.2e+19], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.2e19

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in alpha around 0 96.5%

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

   if 4.2e19 < alpha

   1. Initial program 16.4%

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

      1. +-commutative 16.4%

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

   3. Simplified 16.4%

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

      1. div-inv 16.7%

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

      2. fma-def 15.6%

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

      3. associate-+l+ 15.6%

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

   6. Applied egg-rr 15.6%

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

   7. Taylor expanded in beta around 0 5.4%

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

      1. +-commutative 5.4%

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

   9. Simplified 5.4%

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

   10. Taylor expanded in alpha around inf 70.1%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 4.2e+19)
   (/ (+ 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 <= 4.2e+19) {
		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 <= 4.2d+19) 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 <= 4.2e+19) {
		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 <= 4.2e+19:
		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 <= 4.2e+19)
		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 <= 4.2e+19)
		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, 4.2e+19], 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 < 4.2e19

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in alpha around 0 96.5%

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

   if 4.2e19 < alpha

   1. Initial program 16.4%

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

      1. +-commutative 16.4%

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

   3. Simplified 16.4%

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

   5. Taylor expanded in alpha around -inf 89.9%

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

      1. associate-*r/ 89.9%

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

      2. sub-neg 89.9%

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

      3. mul-1-neg 89.9%

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

      4. distribute-lft-in 89.9%

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

      5. neg-mul-1 89.9%

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

      6. mul-1-neg 89.9%

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

      7. remove-double-neg 89.9%

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

      8. neg-mul-1 89.9%

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

      9. mul-1-neg 89.9%

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

      10. remove-double-neg 89.9%

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

   7. Simplified 89.9%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.00345:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.00345) (/ (+ 1.0 (* beta 0.5)) 2.0) (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.00345) {
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	} else {
		tmp = (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 <= 0.00345d0) then
        tmp = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.00345) {
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.00345:
		tmp = (1.0 + (beta * 0.5)) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.00345)
		tmp = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0);
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.00345)
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.00345], N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.0034499999999999999

   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. Add Preprocessing
   5. Step-by-step derivation

      1. div-inv 100.0%

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

      2. fma-def 100.0%

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

      3. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in beta around 0 76.7%

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

      1. +-commutative 76.7%

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

   9. Simplified 76.7%

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

   10. Taylor expanded in alpha around 0 74.2%

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

       1. *-commutative 74.2%

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

   12. Simplified 74.2%

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

   if 0.0034499999999999999 < alpha

   1. Initial program 21.2%

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

      1. +-commutative 21.2%

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

   3. Simplified 21.2%

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

   5. Taylor expanded in beta around 0 7.0%

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

      1. +-commutative 7.0%

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

   7. Simplified 7.0%

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

   8. Taylor expanded in alpha around inf 64.0%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 0.00345:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.98:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.98) (/ (- 1.0 (* alpha 0.5)) 2.0) (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.98) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else {
		tmp = (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 <= 0.98d0) then
        tmp = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.98) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.98:
		tmp = (1.0 - (alpha * 0.5)) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.98)
		tmp = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0);
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.98)
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.98], N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.97999999999999998

   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. Add Preprocessing

   5. Taylor expanded in beta around 0 78.8%

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

      1. +-commutative 78.8%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in alpha around 0 77.9%

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

      1. *-commutative 77.9%

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

   10. Simplified 77.9%

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

   if 0.97999999999999998 < alpha

   1. Initial program 20.3%

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

      1. +-commutative 20.3%

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

   3. Simplified 20.3%

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

   5. Taylor expanded in beta around 0 6.9%

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

      1. +-commutative 6.9%

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

   7. Simplified 6.9%

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

   8. Taylor expanded in alpha around inf 64.5%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 0.98:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if alpha < 2

   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. Add Preprocessing

   5. Taylor expanded in beta around 0 78.8%

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

      1. +-commutative 78.8%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in alpha around 0 77.9%

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

      1. *-commutative 77.9%

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

   10. Simplified 77.9%

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

   if 2 < alpha

   1. Initial program 20.3%

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

      1. +-commutative 20.3%

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

   3. Simplified 20.3%

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

      1. div-inv 20.6%

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

      2. fma-def 19.6%

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

      3. associate-+l+ 19.6%

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

   6. Applied egg-rr 19.6%

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

   7. Taylor expanded in beta around 0 6.9%

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

      1. +-commutative 6.9%

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

   9. Simplified 6.9%

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

   10. Taylor expanded in alpha around inf 67.6%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 4.2e+19) 1.0 (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 4.2e+19) {
		tmp = 1.0;
	} else {
		tmp = (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 <= 4.2d+19) then
        tmp = 1.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 4.2e+19) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 4.2e+19:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 4.2e+19)
		tmp = 1.0;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 4.2e+19)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 4.2e+19], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.2e19

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in beta around inf 39.1%

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

   if 4.2e19 < alpha

   1. Initial program 16.4%

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

      1. +-commutative 16.4%

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

   3. Simplified 16.4%

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

   5. Taylor expanded in beta around 0 5.3%

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

      1. +-commutative 5.3%

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

   7. Simplified 5.3%

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

   8. Taylor expanded in alpha around inf 66.7%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.3% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 71.1%

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

   1. +-commutative 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in beta around inf 30.1%

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

6. Final simplification 30.1%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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))