Octave 3.8, jcobi/1

Percentage Accurate: 75.3% → 99.8%

Time: 10.2s

Alternatives: 11

Speedup: 0.9×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.3% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \beta \cdot 2\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.998:\\ \;\;\;\;\frac{-0.5 \cdot \left(\frac{t\_0}{\alpha} \cdot \left(\beta + 2\right) - t\_0\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.998)
     (/ (* -0.5 (- (* (/ t_0 alpha) (+ beta 2.0)) t_0)) alpha)
     (/ (exp (log1p (/ (- beta alpha) (+ alpha (+ beta 2.0))))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta * 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.998) {
		tmp = (-0.5 * (((t_0 / alpha) * (beta + 2.0)) - t_0)) / alpha;
	} else {
		tmp = exp(log1p(((beta - alpha) / (alpha + (beta + 2.0))))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta * 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.998) {
		tmp = (-0.5 * (((t_0 / alpha) * (beta + 2.0)) - t_0)) / alpha;
	} else {
		tmp = Math.exp(Math.log1p(((beta - alpha) / (alpha + (beta + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 2.0 + (beta * 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.998:
		tmp = (-0.5 * (((t_0 / alpha) * (beta + 2.0)) - t_0)) / alpha
	else:
		tmp = math.exp(math.log1p(((beta - alpha) / (alpha + (beta + 2.0))))) / 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(Float64(beta + alpha) + 2.0)) <= -0.998)
		tmp = Float64(Float64(-0.5 * Float64(Float64(Float64(t_0 / alpha) * Float64(beta + 2.0)) - t_0)) / alpha);
	else
		tmp = Float64(exp(log1p(Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + 2.0))))) / 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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.998], N[(N[(-0.5 * N[(N[(N[(t$95$0 / alpha), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[Exp[N[Log[1 + N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + 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))) < -0.998

   1. Initial program 9.6%

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

      1. +-commutative 9.6%

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

      2. sub-neg 9.6%

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

      3. +-commutative 9.6%

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

      4. neg-sub0 9.6%

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

      5. associate-+l- 9.6%

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

      6. sub0-neg 9.6%

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

      7. distribute-frac-neg 9.6%

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

      8. +-commutative 9.6%

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

      9. sub-neg 9.6%

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

      10. div-sub 9.6%

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

      11. sub-neg 9.6%

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

      12. metadata-eval 9.6%

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

      13. neg-mul-1 9.6%

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

      14. *-commutative 9.6%

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

      15. +-commutative 9.6%

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

      16. associate-/l/ 9.6%

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

      17. associate-*l/ 9.6%

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

   3. Simplified 9.4%

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

   5. Taylor expanded in alpha around -inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. fma-define 98.4%

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

      3. associate-/l* 99.9%

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

      4. +-commutative 99.9%

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

      5. neg-mul-1 99.9%

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

      6. +-commutative 99.9%

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

      7. neg-mul-1 99.9%

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

      8. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around inf 98.5%

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

      1. distribute-lft-out-- 98.5%

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

      2. associate-/l* 100.0%

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

   10. Simplified 100.0%

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

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

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

      2. +-commutative 100.0%

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

      3. log1p-define 100.0%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}}}{2} \]

      4. +-commutative 100.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right)} + 2}\right)}}{2} \]

      5. associate-+l+ 100.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}\right)}}{2} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.998:\\ \;\;\;\;\frac{-0.5 \cdot \left(\frac{2 + \beta \cdot 2}{\alpha} \cdot \left(\beta + 2\right) - \left(2 + \beta \cdot 2\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \beta \cdot 2\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.998:\\ \;\;\;\;\frac{-0.5 \cdot \left(\frac{t\_0}{\alpha} \cdot \left(\beta + 2\right) - t\_0\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_1} + \left(1 - \frac{\alpha}{t\_1}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* beta 2.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.998)
     (/ (* -0.5 (- (* (/ t_0 alpha) (+ beta 2.0)) t_0)) alpha)
     (/ (+ (/ beta t_1) (- 1.0 (/ alpha t_1))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta * 2.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.998) {
		tmp = (-0.5 * (((t_0 / alpha) * (beta + 2.0)) - t_0)) / alpha;
	} else {
		tmp = ((beta / t_1) + (1.0 - (alpha / t_1))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (beta * 2.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.998d0)) then
        tmp = ((-0.5d0) * (((t_0 / alpha) * (beta + 2.0d0)) - t_0)) / alpha
    else
        tmp = ((beta / t_1) + (1.0d0 - (alpha / t_1))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta * 2.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.998) {
		tmp = (-0.5 * (((t_0 / alpha) * (beta + 2.0)) - t_0)) / alpha;
	} else {
		tmp = ((beta / t_1) + (1.0 - (alpha / t_1))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 2.0 + (beta * 2.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.998:
		tmp = (-0.5 * (((t_0 / alpha) * (beta + 2.0)) - t_0)) / alpha
	else:
		tmp = ((beta / t_1) + (1.0 - (alpha / t_1))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta * 2.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.998)
		tmp = Float64(Float64(-0.5 * Float64(Float64(Float64(t_0 / alpha) * Float64(beta + 2.0)) - t_0)) / alpha);
	else
		tmp = Float64(Float64(Float64(beta / t_1) + Float64(1.0 - Float64(alpha / t_1))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta * 2.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.998)
		tmp = (-0.5 * (((t_0 / alpha) * (beta + 2.0)) - t_0)) / alpha;
	else
		tmp = ((beta / t_1) + (1.0 - (alpha / t_1))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.998], N[(N[(-0.5 * N[(N[(N[(t$95$0 / alpha), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta / t$95$1), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$1), $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 9.6%

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

      1. +-commutative 9.6%

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

      2. sub-neg 9.6%

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

      3. +-commutative 9.6%

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

      4. neg-sub0 9.6%

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

      5. associate-+l- 9.6%

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

      6. sub0-neg 9.6%

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

      7. distribute-frac-neg 9.6%

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

      8. +-commutative 9.6%

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

      9. sub-neg 9.6%

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

      10. div-sub 9.6%

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

      11. sub-neg 9.6%

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

      12. metadata-eval 9.6%

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

      13. neg-mul-1 9.6%

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

      14. *-commutative 9.6%

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

      15. +-commutative 9.6%

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

      16. associate-/l/ 9.6%

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

      17. associate-*l/ 9.6%

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

   3. Simplified 9.4%

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

   5. Taylor expanded in alpha around -inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. fma-define 98.4%

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

      3. associate-/l* 99.9%

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

      4. +-commutative 99.9%

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

      5. neg-mul-1 99.9%

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

      6. +-commutative 99.9%

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

      7. neg-mul-1 99.9%

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

      8. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in alpha around inf 98.5%

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

      1. distribute-lft-out-- 98.5%

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

      2. associate-/l* 100.0%

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

   10. Simplified 100.0%

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

   if -0.998 < (/.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. div-sub 100.0%

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

      2. associate-+l- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+l+ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.998:\\ \;\;\;\;\frac{-0.5 \cdot \left(\frac{2 + \beta \cdot 2}{\alpha} \cdot \left(\beta + 2\right) - \left(2 + \beta \cdot 2\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\alpha + \left(\beta + 2\right)} + \left(1 - \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \left(1 - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999999995)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (+ (/ beta t_0) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999995) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999999995d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((beta / t_0) + (1.0d0 - (alpha / t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999995) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999995:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999999995)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999995)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999999995], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99999999500000003

   1. Initial program 8.8%

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

      1. +-commutative 8.8%

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

   3. Simplified 8.8%

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

   5. Taylor expanded in alpha around inf 98.2%

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

      1. *-commutative 98.2%

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

   7. Simplified 98.2%

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

   if -0.99999999500000003 < (/.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
   5. Step-by-step derivation

      1. div-sub 99.8%

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

      2. associate-+l- 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\alpha + \left(\beta + 2\right)} + \left(1 - \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.999999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999999995)
     (/ (/ (+ 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) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999999995) {
		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) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.999999995d0)) 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) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999999995) {
		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) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.999999995:
		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(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.999999995)
		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) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.999999995)
		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[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.999999995], 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))) < -0.99999999500000003

   1. Initial program 8.8%

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

      1. +-commutative 8.8%

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

   3. Simplified 8.8%

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

   5. Taylor expanded in alpha around inf 98.2%

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

      1. *-commutative 98.2%

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

   7. Simplified 98.2%

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

   if -0.99999999500000003 < (/.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. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 75000000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 75000000000.0)
   (+ 0.5 (* (- alpha beta) (/ -0.5 (+ beta (+ alpha 2.0)))))
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 75000000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 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 <= 75000000000.0d0) then
        tmp = 0.5d0 + ((alpha - beta) * ((-0.5d0) / (beta + (alpha + 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 <= 75000000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 75000000000.0:
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))))
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 75000000000.0)
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) * Float64(-0.5 / Float64(beta + Float64(alpha + 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 <= 75000000000.0)
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 75000000000.0], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7.5e10

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. sub-neg 99.3%

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

      3. +-commutative 99.3%

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

      4. neg-sub0 99.3%

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

      5. associate-+l- 99.3%

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

      6. sub0-neg 99.3%

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

      7. distribute-frac-neg 99.3%

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

      8. +-commutative 99.3%

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

      9. sub-neg 99.3%

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

      10. div-sub 99.3%

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

      11. sub-neg 99.3%

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

      12. metadata-eval 99.3%

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

      13. neg-mul-1 99.3%

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

      14. *-commutative 99.3%

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

      15. +-commutative 99.3%

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

      16. associate-/l/ 99.3%

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

      17. associate-*l/ 99.3%

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

   3. Simplified 99.2%

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

   if 7.5e10 < alpha

   1. Initial program 19.8%

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

      1. +-commutative 19.8%

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

   3. Simplified 19.8%

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

   5. Taylor expanded in alpha around inf 86.6%

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

      1. *-commutative 86.6%

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

   7. Simplified 86.6%

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

Alternative 6: 93.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 800:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 800.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 800.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 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 <= 800.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 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 <= 800.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 800.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 800.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 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 <= 800.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 800.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $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 < 800

   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 99.4%

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

      1. +-commutative 99.4%

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

   7. Simplified 99.4%

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

   if 800 < alpha

   1. Initial program 23.3%

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

      1. +-commutative 23.3%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in alpha around inf 84.1%

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

      1. *-commutative 84.1%

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

   7. Simplified 84.1%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 800:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 620:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{2}{\alpha}}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 620.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (- 1.0 (/ 2.0 alpha)) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 620.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (1.0 - (2.0 / alpha)) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 620.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (1.0d0 - (2.0d0 / alpha)) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 620.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (1.0 - (2.0 / alpha)) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 620.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (1.0 - (2.0 / alpha)) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 620.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(2.0 / alpha)) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 620.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (1.0 - (2.0 / alpha)) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 620.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 620

   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 99.4%

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

      1. +-commutative 99.4%

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

   7. Simplified 99.4%

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

   if 620 < alpha

   1. Initial program 23.3%

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

      1. +-commutative 23.3%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in beta around 0 9.3%

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

      1. +-commutative 9.3%

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

   7. Simplified 9.3%

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

   8. Taylor expanded in alpha around inf 73.9%

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

      1. associate-*r/ 73.9%

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

      2. metadata-eval 73.9%

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

   10. Simplified 73.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 620:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{2}{\alpha}}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (+ 0.5 (* beta 0.25)) 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * 0.25);
	} 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 = 0.5d0 + (beta * 0.25d0)
    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 = 0.5 + (beta * 0.25);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5 + (beta * 0.25)
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(beta * 0.25));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.5 + (beta * 0.25);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 68.7%

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

      1. +-commutative 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in alpha around 0 66.1%

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

      1. +-commutative 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in beta around 0 65.8%

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

      1. *-commutative 65.8%

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

   10. Simplified 65.8%

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

   if 2 < beta

   1. Initial program 91.3%

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

      1. +-commutative 91.3%

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

   3. Simplified 91.3%

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

      1. add-exp-log 91.3%

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

      2. +-commutative 91.3%

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

      3. log1p-define 91.3%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}}}{2} \]

      4. +-commutative 91.3%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right)} + 2}\right)}}{2} \]

      5. associate-+l+ 91.3%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}\right)}}{2} \]

   6. Applied egg-rr 91.3%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}}}{2} \]

   7. Taylor expanded in beta around inf 90.6%

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

Alternative 9: 71.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 950000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 950000.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 950000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 950000.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 950000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 950000.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 950000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 950000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 950000.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5e5

   1. Initial program 68.3%

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

      1. +-commutative 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in alpha around 0 65.7%

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

      1. +-commutative 65.7%

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

   7. Simplified 65.7%

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

   8. Taylor expanded in beta around 0 64.8%

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

   if 9.5e5 < 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. Step-by-step derivation

      1. add-exp-log 92.4%

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

      2. +-commutative 92.4%

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

      3. log1p-define 92.4%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}}}{2} \]

      4. +-commutative 92.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right)} + 2}\right)}}{2} \]

      5. associate-+l+ 92.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}\right)}}{2} \]

   6. Applied egg-rr 92.4%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}\right)}}}{2} \]

   7. Taylor expanded in beta around inf 91.6%

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

Alternative 10: 50.3% 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 76.3%

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

   1. +-commutative 76.3%

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

3. Simplified 76.3%

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

5. Taylor expanded in alpha around 0 74.3%

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

   1. +-commutative 74.3%

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

7. Simplified 74.3%

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

8. Taylor expanded in beta around 0 49.2%

   \[\leadsto \color{blue}{0.5} \]
9. Add Preprocessing

Alternative 11: 3.7% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.0)

C

double code(double alpha, double beta) {
	return 0.0;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.0d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.0;
}

Python

def code(alpha, beta):
	return 0.0

Julia

function code(alpha, beta)
	return 0.0
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.0;
end

Wolfram

code[alpha_, beta_] := 0.0

Derivation

1. Initial program 76.3%

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

   1. +-commutative 76.3%

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

   2. sub-neg 76.3%

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

   3. +-commutative 76.3%

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

   4. neg-sub0 76.3%

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

   5. associate-+l- 76.3%

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

   6. sub0-neg 76.3%

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

   7. distribute-frac-neg 76.3%

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

   8. +-commutative 76.3%

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

   9. sub-neg 76.3%

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

   10. div-sub 76.3%

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

   11. sub-neg 76.3%

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

   12. metadata-eval 76.3%

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

   13. neg-mul-1 76.3%

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

   14. *-commutative 76.3%

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

   15. +-commutative 76.3%

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

   16. associate-/l/ 76.3%

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

   17. associate-*l/ 76.3%

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

3. Simplified 76.2%

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

5. Taylor expanded in alpha around inf 3.6%

   \[\leadsto 0.5 + \color{blue}{-0.5} \]
6. Step-by-step derivation

   1. metadata-eval 3.6%

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 3.6%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024177 
(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))