Octave 3.8, jcobi/1

Percentage Accurate: 74.7% → 99.8%

Time: 10.9s

Alternatives: 10

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: 74.7% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.5

   1. Initial program 7.8%

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

      1. +-commutative 7.8%

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

   3. Simplified 7.8%

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

   5. Taylor expanded in alpha around inf 95.9%

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

      1. +-commutative 95.9%

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

      2. associate--l+ 95.9%

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

      3. +-commutative 95.9%

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

      4. mul-1-neg 95.9%

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

      5. unsub-neg 95.9%

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

      6. sub-neg 95.9%

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

      7. associate-/l* 95.9%

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

      8. distribute-rgt-neg-in 95.9%

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

      9. distribute-frac-neg 95.9%

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

      10. distribute-neg-in 95.9%

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

      11. metadata-eval 95.9%

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

      12. unsub-neg 95.9%

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

   7. Simplified 95.9%

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

      1. clear-num 95.9%

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

      2. inv-pow 95.9%

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

      3. *-commutative 95.9%

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

      4. +-commutative 95.9%

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

      5. +-commutative 95.9%

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

      6. fma-define 95.9%

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

   9. Applied egg-rr 95.9%

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

       1. unpow-1 95.9%

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

       2. associate-/r/ 95.9%

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

       3. *-commutative 95.9%

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

       4. +-commutative 95.9%

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

   11. Simplified 95.9%

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

   12. Taylor expanded in alpha around -inf 96.9%

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

       1. fma-define 96.9%

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

       2. associate-/r* 96.9%

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

       3. cancel-sign-sub-inv 96.9%

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

       4. metadata-eval 96.9%

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

       5. *-lft-identity 96.9%

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

       6. unpow2 96.9%

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

       7. distribute-rgt-in 96.9%

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

       8. associate-/l* 99.9%

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

       9. associate-*r/ 99.9%

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

       10. metadata-eval 99.9%

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

   14. Simplified 99.9%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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. add-log-exp 100.0%

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

      2. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.998:\\ \;\;\;\;\frac{\frac{\left(\beta - \left(-2 - \beta\right)\right) - \left(\beta + \left(\beta + 2\right)\right) \cdot \frac{\beta + 2}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ 2.0 (+ beta alpha))) -0.998)
   (/
    (/
     (-
      (- beta (- -2.0 beta))
      (* (+ beta (+ beta 2.0)) (/ (+ beta 2.0) alpha)))
     alpha)
    2.0)
   (/ (log (exp (+ (/ (- beta alpha) (+ beta (+ alpha 2.0))) 1.0))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.998) {
		tmp = (((beta - (-2.0 - beta)) - ((beta + (beta + 2.0)) * ((beta + 2.0) / alpha))) / alpha) / 2.0;
	} else {
		tmp = log(exp((((beta - alpha) / (beta + (alpha + 2.0))) + 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) :: tmp
    if (((beta - alpha) / (2.0d0 + (beta + alpha))) <= (-0.998d0)) then
        tmp = (((beta - ((-2.0d0) - beta)) - ((beta + (beta + 2.0d0)) * ((beta + 2.0d0) / alpha))) / alpha) / 2.0d0
    else
        tmp = log(exp((((beta - alpha) / (beta + (alpha + 2.0d0))) + 1.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.998) {
		tmp = (((beta - (-2.0 - beta)) - ((beta + (beta + 2.0)) * ((beta + 2.0) / alpha))) / alpha) / 2.0;
	} else {
		tmp = Math.log(Math.exp((((beta - alpha) / (beta + (alpha + 2.0))) + 1.0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / (2.0 + (beta + alpha))) <= -0.998:
		tmp = (((beta - (-2.0 - beta)) - ((beta + (beta + 2.0)) * ((beta + 2.0) / alpha))) / alpha) / 2.0
	else:
		tmp = math.log(math.exp((((beta - alpha) / (beta + (alpha + 2.0))) + 1.0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))) <= -0.998)
		tmp = Float64(Float64(Float64(Float64(beta - Float64(-2.0 - beta)) - Float64(Float64(beta + Float64(beta + 2.0)) * Float64(Float64(beta + 2.0) / alpha))) / alpha) / 2.0);
	else
		tmp = Float64(log(exp(Float64(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))) + 1.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= -0.998)
		tmp = (((beta - (-2.0 - beta)) - ((beta + (beta + 2.0)) * ((beta + 2.0) / alpha))) / alpha) / 2.0;
	else
		tmp = log(exp((((beta - alpha) / (beta + (alpha + 2.0))) + 1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.998], N[(N[(N[(N[(beta - N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision] - N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Log[N[Exp[N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.998

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

   3. Simplified 6.6%

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

      1. add-log-exp 6.5%

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

      2. associate-+l+ 6.5%

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

   6. Applied egg-rr 6.5%

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

   7. Taylor expanded in alpha around -inf 96.7%

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

   9. Simplified 99.7%

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

   if -0.998 < (/.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. add-log-exp 100.0%

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

      2. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.998:\\ \;\;\;\;\frac{\frac{\left(\beta - \left(-2 - \beta\right)\right) - \left(\beta + \left(\beta + 2\right)\right) \cdot \frac{\beta + 2}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t\_0 \leq -0.998:\\ \;\;\;\;\frac{\frac{\left(\beta - \left(-2 - \beta\right)\right) - \left(\beta + \left(\beta + 2\right)\right) \cdot \frac{\beta + 2}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.998)
     (/
      (/
       (-
        (- beta (- -2.0 beta))
        (* (+ beta (+ beta 2.0)) (/ (+ beta 2.0) alpha)))
       alpha)
      2.0)
     (/ (+ 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 <= -0.998) {
		tmp = (((beta - (-2.0 - beta)) - ((beta + (beta + 2.0)) * ((beta + 2.0) / alpha))) / 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) / (2.0d0 + (beta + alpha))
    if (t_0 <= (-0.998d0)) then
        tmp = (((beta - ((-2.0d0) - beta)) - ((beta + (beta + 2.0d0)) * ((beta + 2.0d0) / alpha))) / 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) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.998) {
		tmp = (((beta - (-2.0 - beta)) - ((beta + (beta + 2.0)) * ((beta + 2.0) / alpha))) / alpha) / 2.0;
	} 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 <= -0.998:
		tmp = (((beta - (-2.0 - beta)) - ((beta + (beta + 2.0)) * ((beta + 2.0) / alpha))) / 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(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.998)
		tmp = Float64(Float64(Float64(Float64(beta - Float64(-2.0 - beta)) - Float64(Float64(beta + Float64(beta + 2.0)) * Float64(Float64(beta + 2.0) / alpha))) / 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) / (2.0 + (beta + alpha));
	tmp = 0.0;
	if (t_0 <= -0.998)
		tmp = (((beta - (-2.0 - beta)) - ((beta + (beta + 2.0)) * ((beta + 2.0) / alpha))) / 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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.998], N[(N[(N[(N[(beta - N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision] - N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $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) #s(literal 2 binary64))) < -0.998

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

   3. Simplified 6.6%

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

      1. add-log-exp 6.5%

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

      2. associate-+l+ 6.5%

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

   6. Applied egg-rr 6.5%

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

   7. Taylor expanded in alpha around -inf 96.7%

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

   9. Simplified 99.7%

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

   if -0.998 < (/.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
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.998:\\ \;\;\;\;\frac{\frac{\left(\beta - \left(-2 - \beta\right)\right) - \left(\beta + \left(\beta + 2\right)\right) \cdot \frac{\beta + 2}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t\_0 \leq -0.998:\\ \;\;\;\;\frac{\frac{\left(\beta - \left(-2 - \beta\right)\right) - \left(\beta \cdot 2\right) \cdot \frac{\beta + 2}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.998)
     (/
      (/
       (- (- beta (- -2.0 beta)) (* (* beta 2.0) (/ (+ beta 2.0) alpha)))
       alpha)
      2.0)
     (/ (+ 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 <= -0.998) {
		tmp = (((beta - (-2.0 - beta)) - ((beta * 2.0) * ((beta + 2.0) / alpha))) / 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) / (2.0d0 + (beta + alpha))
    if (t_0 <= (-0.998d0)) then
        tmp = (((beta - ((-2.0d0) - beta)) - ((beta * 2.0d0) * ((beta + 2.0d0) / alpha))) / 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) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.998) {
		tmp = (((beta - (-2.0 - beta)) - ((beta * 2.0) * ((beta + 2.0) / alpha))) / alpha) / 2.0;
	} 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 <= -0.998:
		tmp = (((beta - (-2.0 - beta)) - ((beta * 2.0) * ((beta + 2.0) / alpha))) / 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(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.998)
		tmp = Float64(Float64(Float64(Float64(beta - Float64(-2.0 - beta)) - Float64(Float64(beta * 2.0) * Float64(Float64(beta + 2.0) / alpha))) / 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) / (2.0 + (beta + alpha));
	tmp = 0.0;
	if (t_0 <= -0.998)
		tmp = (((beta - (-2.0 - beta)) - ((beta * 2.0) * ((beta + 2.0) / alpha))) / 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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.998], N[(N[(N[(N[(beta - N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision] - N[(N[(beta * 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $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) #s(literal 2 binary64))) < -0.998

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

   3. Simplified 6.6%

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

      1. add-log-exp 6.5%

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

      2. associate-+l+ 6.5%

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

   6. Applied egg-rr 6.5%

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

   7. Taylor expanded in alpha around -inf 96.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in beta around inf 99.5%

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

   if -0.998 < (/.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
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.998:\\ \;\;\;\;\frac{\frac{\left(\beta - \left(-2 - \beta\right)\right) - \left(\beta \cdot 2\right) \cdot \frac{\beta + 2}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t\_0 \leq -0.999999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.999999)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (+ 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 <= -0.999999) {
		tmp = ((2.0 + (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) / (2.0d0 + (beta + alpha))
    if (t_0 <= (-0.999999d0)) then
        tmp = ((2.0d0 + (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) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.999999) {
		tmp = ((2.0 + (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) / (2.0 + (beta + alpha))
	tmp = 0
	if t_0 <= -0.999999:
		tmp = ((2.0 + (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(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.999999)
		tmp = Float64(Float64(Float64(2.0 + 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) / (2.0 + (beta + alpha));
	tmp = 0.0;
	if (t_0 <= -0.999999)
		tmp = ((2.0 + (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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.999999], N[(N[(N[(2.0 + 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) #s(literal 2 binary64))) < -0.999998999999999971

   1. Initial program 5.5%

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

      1. +-commutative 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in alpha around inf 99.9%

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

   if -0.999998999999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.999999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.55 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.55e+41)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.55e+41) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (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 <= 2.55d+41) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + (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 <= 2.55e+41) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.55e+41:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.55e+41)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.55e+41)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.55e+41], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.54999999999999989e41

   1. Initial program 98.5%

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

      1. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in alpha around 0 96.9%

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

   if 2.54999999999999989e41 < alpha

   1. Initial program 15.6%

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

      1. +-commutative 15.6%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in alpha around inf 89.9%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.55 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.1e+41)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.1e+41) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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 <= 2.1d+41) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 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 <= 2.1e+41) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.1e+41:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.1e+41)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 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 <= 2.1e+41)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.1e+41], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.1e41

   1. Initial program 98.5%

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

      1. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in alpha around 0 96.9%

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

   if 2.1e41 < alpha

   1. Initial program 15.6%

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

      1. +-commutative 15.6%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in alpha around inf 89.9%

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

   6. Taylor expanded in beta around 0 67.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 70.4%

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

      1. +-commutative 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in alpha around 0 68.4%

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

   6. Taylor expanded in beta around 0 68.1%

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

      1. *-commutative 68.1%

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

   8. Simplified 68.1%

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

   if 2 < beta

   1. Initial program 84.5%

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

      1. +-commutative 84.5%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in beta around inf 84.0%

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

4. Final simplification 74.3%

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

Alternative 9: 70.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+14}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 5.5e+14) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.5e+14) {
		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 <= 5.5d+14) 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 <= 5.5e+14) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.5e+14:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.5e+14)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.5e+14)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.5e+14], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5e14

   1. Initial program 69.5%

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

      1. +-commutative 69.5%

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

   3. Simplified 69.5%

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

      1. add-log-exp 69.5%

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

      2. associate-+l+ 69.5%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in beta around 0 68.6%

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

      1. +-commutative 68.6%

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

   9. Simplified 68.6%

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

   10. Taylor expanded in alpha around 0 66.6%

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

   if 5.5e14 < beta

   1. Initial program 86.1%

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

      1. +-commutative 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in beta around inf 85.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+14}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.9% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 75.9%

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

   1. +-commutative 75.9%

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

3. Simplified 75.9%

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

   1. add-log-exp 75.9%

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

   2. associate-+l+ 75.9%

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

6. Applied egg-rr 75.9%

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

7. Taylor expanded in beta around 0 47.9%

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

   1. +-commutative 47.9%

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

9. Simplified 47.9%

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

10. Taylor expanded in alpha around 0 47.6%

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

11. Final simplification 47.6%

    \[\leadsto 0.5 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(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))