Octave 3.8, jcobi/1

Percentage Accurate: 74.6% → 98.3%

Time: 9.0s

Alternatives: 9

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: 74.6% 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: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.8:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 2 - \frac{{\left(\beta + 2\right)}^{2}}{\alpha}\right) + \left(2 + \beta \cdot \frac{-2 - \beta}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.8)
   (/
    (/
     (+
      (- (* beta 2.0) (/ (pow (+ beta 2.0) 2.0) alpha))
      (+ 2.0 (* beta (/ (- -2.0 beta) alpha))))
     alpha)
    2.0)
   (/ (+ 1.0 (* (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.8) {
		tmp = ((((beta * 2.0) - (pow((beta + 2.0), 2.0) / alpha)) + (2.0 + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.8d0)) then
        tmp = ((((beta * 2.0d0) - (((beta + 2.0d0) ** 2.0d0) / alpha)) + (2.0d0 + (beta * (((-2.0d0) - beta) / alpha)))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) * (1.0d0 / (beta + (alpha + 2.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.8) {
		tmp = ((((beta * 2.0) - (Math.pow((beta + 2.0), 2.0) / alpha)) + (2.0 + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.8:
		tmp = ((((beta * 2.0) - (math.pow((beta + 2.0), 2.0) / alpha)) + (2.0 + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.8)
		tmp = Float64(Float64(Float64(Float64(Float64(beta * 2.0) - Float64((Float64(beta + 2.0) ^ 2.0) / alpha)) + Float64(2.0 + Float64(beta * Float64(Float64(-2.0 - beta) / alpha)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) * Float64(1.0 / Float64(beta + Float64(alpha + 2.0))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.8)
		tmp = ((((beta * 2.0) - (((beta + 2.0) ^ 2.0) / alpha)) + (2.0 + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[(N[(N[(beta * 2.0), $MachinePrecision] - N[(N[Power[N[(beta + 2.0), $MachinePrecision], 2.0], $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(beta * N[(N[(-2.0 - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.80000000000000004

   1. Initial program 8.5%

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

      1. +-commutative 8.5%

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

   3. Simplified 8.5%

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

   5. Taylor expanded in alpha around inf 98.3%

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

      1. +-commutative 98.3%

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

      2. associate--l+ 98.3%

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

      3. +-commutative 98.3%

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

      4. mul-1-neg 98.3%

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

      5. unsub-neg 98.3%

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

      6. sub-neg 98.3%

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

      7. associate-/l* 98.3%

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

      8. distribute-rgt-neg-in 98.3%

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

      9. distribute-frac-neg 98.3%

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

      10. distribute-neg-in 98.3%

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

      11. metadata-eval 98.3%

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

      12. unsub-neg 98.3%

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

   7. Simplified 98.3%

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

   if -0.80000000000000004 < (/.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. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.8:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 2 - \frac{{\left(\beta + 2\right)}^{2}}{\alpha}\right) + \left(2 + \beta \cdot \frac{-2 - \beta}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999999)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
   (/ (fma (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = fma((beta - alpha), (1.0 / (beta + (alpha + 2.0))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999999)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999999], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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))) < -0.99999998999999995

   1. Initial program 6.8%

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

      1. +-commutative 6.8%

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

   3. Simplified 6.8%

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

   5. Taylor expanded in alpha around -inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. sub-neg 99.2%

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

      3. distribute-lft-in 99.2%

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

      4. neg-mul-1 99.2%

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

      5. mul-1-neg 99.2%

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

      6. remove-double-neg 99.2%

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

      7. neg-mul-1 99.2%

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

      8. remove-double-neg 99.2%

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

   7. Simplified 99.2%

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

   if -0.99999998999999995 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. div-inv 99.6%

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

      2. fma-define 99.6%

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

      3. associate-+l+ 99.6%

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

   6. Applied egg-rr 99.6%

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

4. Final simplification 99.5%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999999)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
   (/ (+ 1.0 (* (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99999999d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) * (1.0d0 / (beta + (alpha + 2.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999999:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999999)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) * Float64(1.0 / Float64(beta + Float64(alpha + 2.0))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999999)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999999], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99999998999999995

   1. Initial program 6.8%

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

      1. +-commutative 6.8%

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

   3. Simplified 6.8%

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

   5. Taylor expanded in alpha around -inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. sub-neg 99.2%

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

      3. distribute-lft-in 99.2%

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

      4. neg-mul-1 99.2%

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

      5. mul-1-neg 99.2%

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

      6. remove-double-neg 99.2%

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

      7. neg-mul-1 99.2%

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

      8. remove-double-neg 99.2%

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

   7. Simplified 99.2%

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

   if -0.99999998999999995 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. associate-+l+ 99.6%

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

   6. Applied egg-rr 99.6%

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

4. Final simplification 99.5%

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

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999999)
     (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99999999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.99999999d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99999999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.99999999:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.99999999)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.99999999)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99999999], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99999998999999995

   1. Initial program 6.8%

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

      1. +-commutative 6.8%

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

   3. Simplified 6.8%

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

   5. Taylor expanded in alpha around -inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. sub-neg 99.2%

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

      3. distribute-lft-in 99.2%

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

      4. neg-mul-1 99.2%

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

      5. mul-1-neg 99.2%

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

      6. remove-double-neg 99.2%

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

      7. neg-mul-1 99.2%

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

      8. remove-double-neg 99.2%

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

   7. Simplified 99.2%

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

   if -0.99999998999999995 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.6%

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

4. Final simplification 99.5%

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

Alternative 5: 69.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ t_1 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq 3.45 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq 4.5 \cdot 10^{-83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\beta \leq 3.7 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)) (t_1 (/ (/ 2.0 alpha) 2.0)))
   (if (<= beta 3.45e-103)
     t_0
     (if (<= beta 4.5e-83)
       t_1
       (if (<= beta 3.7e-40) t_0 (if (<= beta 2.1e-16) t_1 1.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double t_1 = (2.0 / alpha) / 2.0;
	double tmp;
	if (beta <= 3.45e-103) {
		tmp = t_0;
	} else if (beta <= 4.5e-83) {
		tmp = t_1;
	} else if (beta <= 3.7e-40) {
		tmp = t_0;
	} else if (beta <= 2.1e-16) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    t_1 = (2.0d0 / alpha) / 2.0d0
    if (beta <= 3.45d-103) then
        tmp = t_0
    else if (beta <= 4.5d-83) then
        tmp = t_1
    else if (beta <= 3.7d-40) then
        tmp = t_0
    else if (beta <= 2.1d-16) then
        tmp = t_1
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double t_1 = (2.0 / alpha) / 2.0;
	double tmp;
	if (beta <= 3.45e-103) {
		tmp = t_0;
	} else if (beta <= 4.5e-83) {
		tmp = t_1;
	} else if (beta <= 3.7e-40) {
		tmp = t_0;
	} else if (beta <= 2.1e-16) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (beta * 0.5)) / 2.0
	t_1 = (2.0 / alpha) / 2.0
	tmp = 0
	if beta <= 3.45e-103:
		tmp = t_0
	elif beta <= 4.5e-83:
		tmp = t_1
	elif beta <= 3.7e-40:
		tmp = t_0
	elif beta <= 2.1e-16:
		tmp = t_1
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0)
	t_1 = Float64(Float64(2.0 / alpha) / 2.0)
	tmp = 0.0
	if (beta <= 3.45e-103)
		tmp = t_0;
	elseif (beta <= 4.5e-83)
		tmp = t_1;
	elseif (beta <= 3.7e-40)
		tmp = t_0;
	elseif (beta <= 2.1e-16)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta * 0.5)) / 2.0;
	t_1 = (2.0 / alpha) / 2.0;
	tmp = 0.0;
	if (beta <= 3.45e-103)
		tmp = t_0;
	elseif (beta <= 4.5e-83)
		tmp = t_1;
	elseif (beta <= 3.7e-40)
		tmp = t_0;
	elseif (beta <= 2.1e-16)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, 3.45e-103], t$95$0, If[LessEqual[beta, 4.5e-83], t$95$1, If[LessEqual[beta, 3.7e-40], t$95$0, If[LessEqual[beta, 2.1e-16], t$95$1, 1.0]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.4500000000000001e-103 or 4.49999999999999997e-83 < beta < 3.69999999999999998e-40

   1. Initial program 72.4%

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

      1. +-commutative 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in alpha around 0 70.5%

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

   6. Taylor expanded in beta around 0 70.0%

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

      1. *-commutative 70.0%

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

   8. Simplified 70.0%

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

   if 3.4500000000000001e-103 < beta < 4.49999999999999997e-83 or 3.69999999999999998e-40 < beta < 2.1000000000000001e-16

   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 21.8%

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

      1. +-commutative 21.8%

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

   7. Simplified 21.8%

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

   8. Taylor expanded in alpha around inf 82.9%

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

   if 2.1000000000000001e-16 < beta

   1. Initial program 92.4%

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

      1. +-commutative 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in beta around inf 89.6%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.45 \cdot 10^{-103}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 4.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 3.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 780000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 780000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 780000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 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 <= 780000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 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 <= 780000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 780000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 780000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 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 <= 780000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 780000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7.8e5

   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 99.2%

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

   if 7.8e5 < alpha

   1. Initial program 20.4%

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

      1. +-commutative 20.4%

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

   3. Simplified 20.4%

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

   5. Taylor expanded in beta around 0 6.8%

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

      1. +-commutative 6.8%

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

   7. Simplified 6.8%

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

   8. Taylor expanded in alpha around inf 75.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 780000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7200000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7200000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7200000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 7200000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7200000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 7200000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 7200000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 7200000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 7200000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7.2e6

   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 99.2%

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

   if 7.2e6 < alpha

   1. Initial program 20.4%

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

      1. +-commutative 20.4%

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

   3. Simplified 20.4%

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

   5. Taylor expanded in alpha around -inf 86.1%

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

      1. associate-*r/ 86.1%

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

      2. sub-neg 86.1%

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

      3. distribute-lft-in 86.1%

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

      4. neg-mul-1 86.1%

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

      5. mul-1-neg 86.1%

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

      6. remove-double-neg 86.1%

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

      7. neg-mul-1 86.1%

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

      8. remove-double-neg 86.1%

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

   7. Simplified 86.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7200000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 235000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 235000.0) 1.0 (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 235000.0) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 235000.0d0) then
        tmp = 1.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 235000.0) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 235000.0:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 235000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 235000.0)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 235000.0], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 235000

   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 inf 50.8%

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

   if 235000 < alpha

   1. Initial program 20.4%

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

      1. +-commutative 20.4%

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

   3. Simplified 20.4%

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

   5. Taylor expanded in beta around 0 6.8%

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

      1. +-commutative 6.8%

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

   7. Simplified 6.8%

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

   8. Taylor expanded in alpha around inf 75.7%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 235000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.9% 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 76.0%

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

   1. +-commutative 76.0%

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

3. Simplified 76.0%

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

5. Taylor expanded in beta around inf 40.5%

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

6. Final simplification 40.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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))