Octave 3.8, jcobi/1

Percentage Accurate: 74.4% → 99.8%

Time: 10.0s

Alternatives: 11

Speedup: 0.9×

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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\\ \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.9998:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha} + \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + {t\_0}^{3}}{1 + \left({t\_0}^{2} + \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))))
   (if (<= (/ (- beta alpha) (+ 2.0 (+ beta alpha))) -0.9998)
     (/
      (*
       0.5
       (+
        (* (+ beta 2.0) (/ (- (- -2.0 beta) beta) alpha))
        (fma beta 2.0 2.0)))
      alpha)
     (/
      (/
       (+ 1.0 (pow t_0 3.0))
       (+ 1.0 (+ (pow t_0 2.0) (/ (- beta alpha) (- -2.0 (+ beta alpha))))))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (beta + (alpha + 2.0));
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.9998) {
		tmp = (0.5 * (((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)) + fma(beta, 2.0, 2.0))) / alpha;
	} else {
		tmp = ((1.0 + pow(t_0, 3.0)) / (1.0 + (pow(t_0, 2.0) + ((beta - alpha) / (-2.0 - (beta + alpha)))))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))) <= -0.9998)
		tmp = Float64(Float64(0.5 * Float64(Float64(Float64(beta + 2.0) * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)) + fma(beta, 2.0, 2.0))) / alpha);
	else
		tmp = Float64(Float64(Float64(1.0 + (t_0 ^ 3.0)) / Float64(1.0 + Float64((t_0 ^ 2.0) + Float64(Float64(beta - alpha) / Float64(-2.0 - Float64(beta + alpha)))))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.9998], N[(N[(0.5 * N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(beta * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(1.0 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[(beta - alpha), $MachinePrecision] / N[(-2.0 - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99980000000000002

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in alpha around inf 92.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(2 + 2 \cdot \beta\right) + 0.5 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}}{\alpha}} \]
   6. Step-by-step derivation

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha} + \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha}} \]

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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. Add Preprocessing
   5. Step-by-step derivation

      1. flip3-+ 99.9%

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

      2. div-inv 99.9%

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

      3. pow3 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. pow3 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l+ 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-neg-in 99.9%

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

      12. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.9998:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha} + \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}}{1 + \left({\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2} + \frac{\beta - \alpha}{-2 - \left(\beta + \alpha\right)}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t\_0 \leq -0.9998:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha} + \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.9998)
     (/
      (*
       0.5
       (+
        (* (+ beta 2.0) (/ (- (- -2.0 beta) beta) alpha))
        (fma beta 2.0 2.0)))
      alpha)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.9998) {
		tmp = (0.5 * (((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)) + fma(beta, 2.0, 2.0))) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.9998)
		tmp = Float64(Float64(0.5 * Float64(Float64(Float64(beta + 2.0) * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)) + fma(beta, 2.0, 2.0))) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9998], N[(N[(0.5 * N[(N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(beta * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $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) #s(literal 2 binary64))) < -0.99980000000000002

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in alpha around inf 92.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(2 + 2 \cdot \beta\right) + 0.5 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}}{\alpha}} \]
   6. Step-by-step derivation

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha} + \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha}} \]

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.9998:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha} + \mathsf{fma}\left(\beta, 2, 2\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.99999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\frac{\beta}{t\_0} - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ 2.0 (+ beta alpha))) -0.99999995)
     (/ (+ beta 1.0) alpha)
     (/ (+ 1.0 (- (/ beta t_0) (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.99999995) {
		tmp = (beta + 1.0) / alpha;
	} 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 = alpha + (beta + 2.0d0)
    if (((beta - alpha) / (2.0d0 + (beta + alpha))) <= (-0.99999995d0)) then
        tmp = (beta + 1.0d0) / alpha
    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 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.99999995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (1.0 + ((beta / t_0) - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if ((beta - alpha) / (2.0 + (beta + alpha))) <= -0.99999995:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = (1.0 + ((beta / t_0) - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))) <= -0.99999995)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta / t_0) - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.99999995)
		tmp = (beta + 1.0) / alpha;
	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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.99999995], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 + N[(N[(beta / t$95$0), $MachinePrecision] - 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) #s(literal 2 binary64))) < -0.999999949999999971

   1. Initial program 6.4%

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

      1. +-commutative 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in alpha around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in beta around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. *-rgt-identity 99.6%

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

      3. associate-*r/ 99.6%

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

      4. *-lft-identity 99.6%

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

      5. distribute-rgt-in 99.6%

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

      6. associate-*l/ 99.6%

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

      7. *-lft-identity 99.6%

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

      8. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   if -0.999999949999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

   3. Simplified 99.5%

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

      1. div-sub 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.99999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\frac{\beta}{\alpha + \left(\beta + 2\right)} - \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t\_0 \leq -0.99999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.99999995) (/ (+ beta 1.0) alpha) (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.99999995) {
		tmp = (beta + 1.0) / alpha;
	} 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) / (2.0d0 + (beta + alpha))
    if (t_0 <= (-0.99999995d0)) then
        tmp = (beta + 1.0d0) / alpha
    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) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.99999995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / (2.0 + (beta + alpha))
	tmp = 0
	if t_0 <= -0.99999995:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.99999995)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	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) / (2.0 + (beta + alpha));
	tmp = 0.0;
	if (t_0 <= -0.99999995)
		tmp = (beta + 1.0) / alpha;
	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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99999995], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $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) #s(literal 2 binary64))) < -0.999999949999999971

   1. Initial program 6.4%

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

      1. +-commutative 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in alpha around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in beta around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. *-rgt-identity 99.6%

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

      3. associate-*r/ 99.6%

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

      4. *-lft-identity 99.6%

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

      5. distribute-rgt-in 99.6%

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

      6. associate-*l/ 99.6%

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

      7. *-lft-identity 99.6%

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

      8. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   if -0.999999949999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.99999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \alpha \cdot -0.25\\ \mathbf{if}\;\alpha \leq -2.2 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq -2.3 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* alpha -0.25))))
   (if (<= alpha -2.2e-134)
     t_0
     (if (<= alpha -2.3e-195) 1.0 (if (<= alpha 1.1) t_0 (/ 1.0 alpha))))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -2.2e-134) {
		tmp = t_0;
	} else if (alpha <= -2.3e-195) {
		tmp = 1.0;
	} else if (alpha <= 1.1) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (alpha * (-0.25d0))
    if (alpha <= (-2.2d-134)) then
        tmp = t_0
    else if (alpha <= (-2.3d-195)) then
        tmp = 1.0d0
    else if (alpha <= 1.1d0) then
        tmp = t_0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -2.2e-134) {
		tmp = t_0;
	} else if (alpha <= -2.3e-195) {
		tmp = 1.0;
	} else if (alpha <= 1.1) {
		tmp = t_0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (alpha * -0.25)
	tmp = 0
	if alpha <= -2.2e-134:
		tmp = t_0
	elif alpha <= -2.3e-195:
		tmp = 1.0
	elif alpha <= 1.1:
		tmp = t_0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(alpha * -0.25))
	tmp = 0.0
	if (alpha <= -2.2e-134)
		tmp = t_0;
	elseif (alpha <= -2.3e-195)
		tmp = 1.0;
	elseif (alpha <= 1.1)
		tmp = t_0;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (alpha * -0.25);
	tmp = 0.0;
	if (alpha <= -2.2e-134)
		tmp = t_0;
	elseif (alpha <= -2.3e-195)
		tmp = 1.0;
	elseif (alpha <= 1.1)
		tmp = t_0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, -2.2e-134], t$95$0, If[LessEqual[alpha, -2.3e-195], 1.0, If[LessEqual[alpha, 1.1], t$95$0, N[(1.0 / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -2.2e-134 or -2.3000000000000002e-195 < alpha < 1.1000000000000001

   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 76.2%

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

   6. Taylor expanded in alpha around 0 75.7%

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

      1. *-commutative 75.7%

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

   8. Simplified 75.7%

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

   if -2.2e-134 < alpha < -2.3000000000000002e-195

   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%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 74.6%

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

   if 1.1000000000000001 < alpha

   1. Initial program 21.8%

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

      1. +-commutative 21.8%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in beta around 0 6.5%

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

   6. Taylor expanded in alpha around inf 67.8%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -2.2 \cdot 10^{-134}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq -2.3 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.1:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \alpha \cdot -0.25\\ \mathbf{if}\;\alpha \leq -4.8 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq -1.8 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* alpha -0.25))))
   (if (<= alpha -4.8e-132)
     t_0
     (if (<= alpha -1.8e-197)
       1.0
       (if (<= alpha 2.0) t_0 (/ (+ beta 1.0) alpha))))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -4.8e-132) {
		tmp = t_0;
	} else if (alpha <= -1.8e-197) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	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 = 0.5d0 + (alpha * (-0.25d0))
    if (alpha <= (-4.8d-132)) then
        tmp = t_0
    else if (alpha <= (-1.8d-197)) then
        tmp = 1.0d0
    else if (alpha <= 2.0d0) then
        tmp = t_0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -4.8e-132) {
		tmp = t_0;
	} else if (alpha <= -1.8e-197) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (alpha * -0.25)
	tmp = 0
	if alpha <= -4.8e-132:
		tmp = t_0
	elif alpha <= -1.8e-197:
		tmp = 1.0
	elif alpha <= 2.0:
		tmp = t_0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(alpha * -0.25))
	tmp = 0.0
	if (alpha <= -4.8e-132)
		tmp = t_0;
	elseif (alpha <= -1.8e-197)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (alpha * -0.25);
	tmp = 0.0;
	if (alpha <= -4.8e-132)
		tmp = t_0;
	elseif (alpha <= -1.8e-197)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = t_0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, -4.8e-132], t$95$0, If[LessEqual[alpha, -1.8e-197], 1.0, If[LessEqual[alpha, 2.0], t$95$0, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -4.80000000000000031e-132 or -1.7999999999999999e-197 < 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 76.2%

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

   6. Taylor expanded in alpha around 0 75.7%

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

      1. *-commutative 75.7%

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

   8. Simplified 75.7%

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

   if -4.80000000000000031e-132 < alpha < -1.7999999999999999e-197

   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%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 74.6%

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

   if 2 < alpha

   1. Initial program 21.8%

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

      1. +-commutative 21.8%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in alpha around inf 84.8%

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

      1. *-commutative 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in beta around 0 84.8%

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

      1. +-commutative 84.8%

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

      2. *-rgt-identity 84.8%

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

      3. associate-*r/ 84.8%

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

      4. *-lft-identity 84.8%

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

      5. distribute-rgt-in 84.8%

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

      6. associate-*l/ 84.8%

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

      7. *-lft-identity 84.8%

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

      8. +-commutative 84.8%

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

   10. Simplified 84.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -4.8 \cdot 10^{-132}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq -1.8 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq -4.4 \cdot 10^{-134}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -2 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha -4.4e-134)
   0.5
   (if (<= alpha -2e-195) 1.0 (if (<= alpha 2.1) 0.5 (/ 1.0 alpha)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -4.4e-134) {
		tmp = 0.5;
	} else if (alpha <= -2e-195) {
		tmp = 1.0;
	} else if (alpha <= 2.1) {
		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 <= (-4.4d-134)) then
        tmp = 0.5d0
    else if (alpha <= (-2d-195)) then
        tmp = 1.0d0
    else if (alpha <= 2.1d0) 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 <= -4.4e-134) {
		tmp = 0.5;
	} else if (alpha <= -2e-195) {
		tmp = 1.0;
	} else if (alpha <= 2.1) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= -4.4e-134:
		tmp = 0.5
	elif alpha <= -2e-195:
		tmp = 1.0
	elif alpha <= 2.1:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= -4.4e-134)
		tmp = 0.5;
	elseif (alpha <= -2e-195)
		tmp = 1.0;
	elseif (alpha <= 2.1)
		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 <= -4.4e-134)
		tmp = 0.5;
	elseif (alpha <= -2e-195)
		tmp = 1.0;
	elseif (alpha <= 2.1)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, -4.4e-134], 0.5, If[LessEqual[alpha, -2e-195], 1.0, If[LessEqual[alpha, 2.1], 0.5, N[(1.0 / alpha), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -4.3999999999999999e-134 or -2.0000000000000002e-195 < alpha < 2.10000000000000009

   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 alpha around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. +-commutative 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in beta around 0 74.9%

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

   if -4.3999999999999999e-134 < alpha < -2.0000000000000002e-195

   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%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 74.6%

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

   if 2.10000000000000009 < alpha

   1. Initial program 21.8%

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

      1. +-commutative 21.8%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in beta around 0 6.5%

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

   6. Taylor expanded in alpha around inf 67.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -4.4 \cdot 10^{-134}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -2 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7800000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7800000000.0)
   (+ 0.5 (* (- alpha beta) (/ -0.5 (+ beta (+ alpha 2.0)))))
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7800000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	} else {
		tmp = (beta + 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 <= 7800000000.0d0) then
        tmp = 0.5d0 + ((alpha - beta) * ((-0.5d0) / (beta + (alpha + 2.0d0))))
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7800000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 7800000000.0:
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))))
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 7800000000.0)
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) * Float64(-0.5 / Float64(beta + Float64(alpha + 2.0)))));
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 7800000000.0)
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 7800000000.0], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7.8e9

   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%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 99.9%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 99.9%

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

      7. distribute-frac-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. sub-neg 99.9%

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

      10. div-sub 99.9%

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

      11. sub-neg 99.9%

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

      12. metadata-eval 99.9%

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

      13. neg-mul-1 99.9%

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

      14. *-commutative 99.9%

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

      15. +-commutative 99.9%

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

      16. associate-/l/ 99.9%

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

      17. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   if 7.8e9 < 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 alpha around inf 85.3%

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

      1. *-commutative 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in beta around 0 85.3%

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

      1. +-commutative 85.3%

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

      2. *-rgt-identity 85.3%

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

      3. associate-*r/ 85.3%

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

      4. *-lft-identity 85.3%

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

      5. distribute-rgt-in 85.3%

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

      6. associate-*l/ 85.3%

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

      7. *-lft-identity 85.3%

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

      8. +-commutative 85.3%

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

   10. Simplified 85.3%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7800000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1020:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1020.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1020.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 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 <= 1020.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1020.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1020.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1020.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1020.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1020.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1020

   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 alpha around 0 98.8%

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

      1. +-commutative 98.8%

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

      2. +-commutative 98.8%

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

   7. Simplified 98.8%

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

   if 1020 < alpha

   1. Initial program 21.8%

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

      1. +-commutative 21.8%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in alpha around inf 84.8%

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

      1. *-commutative 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in beta around 0 84.8%

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

      1. +-commutative 84.8%

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

      2. *-rgt-identity 84.8%

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

      3. associate-*r/ 84.8%

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

      4. *-lft-identity 84.8%

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

      5. distribute-rgt-in 84.8%

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

      6. associate-*l/ 84.8%

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

      7. *-lft-identity 84.8%

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

      8. +-commutative 84.8%

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

   10. Simplified 84.8%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1020:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1040000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 1040000000.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1040000000.0) {
		tmp = 0.5;
	} 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 <= 1040000000.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1040000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1040000000.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1040000000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1040000000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1040000000.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.04e9

   1. Initial program 64.0%

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

      1. +-commutative 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in alpha around 0 62.0%

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

      1. +-commutative 62.0%

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

      2. +-commutative 62.0%

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

   7. Simplified 62.0%

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

   8. Taylor expanded in beta around 0 61.3%

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

   if 1.04e9 < beta

   1. Initial program 83.0%

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

      1. +-commutative 83.0%

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

      2. sub-neg 83.0%

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

      3. +-commutative 83.0%

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

      4. neg-sub0 83.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 83.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 83.0%

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

      7. distribute-frac-neg 83.0%

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

      8. +-commutative 83.0%

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

      9. sub-neg 83.0%

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

      10. div-sub 83.0%

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

      11. sub-neg 83.0%

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

      12. metadata-eval 83.0%

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

      13. neg-mul-1 83.0%

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

      14. *-commutative 83.0%

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

      15. +-commutative 83.0%

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

      16. associate-/l/ 82.0%

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

      17. associate-*l/ 82.0%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in beta around inf 80.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1040000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.7% 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 70.1%

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

   1. +-commutative 70.1%

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

   2. sub-neg 70.1%

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

   3. +-commutative 70.1%

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

   4. neg-sub0 70.1%

      \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

   5. associate-+l- 70.1%

      \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

   6. sub0-neg 70.1%

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

   7. distribute-frac-neg 70.1%

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

   8. +-commutative 70.1%

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

   9. sub-neg 70.1%

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

   10. div-sub 70.1%

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

   11. sub-neg 70.1%

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

   12. metadata-eval 70.1%

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

   13. neg-mul-1 70.1%

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

   14. *-commutative 70.1%

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

   15. +-commutative 70.1%

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

   16. associate-/l/ 69.8%

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

   17. associate-*l/ 69.8%

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

3. Simplified 70.1%

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

5. Taylor expanded in beta around inf 34.9%

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

6. Final simplification 34.9%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(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))