Octave 3.8, jcobi/1

Percentage Accurate: 75.1% → 99.7%

Time: 10.8s

Alternatives: 9

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.1% 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: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99999995d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((beta / t_0) - log(exp(((alpha / t_0) + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999995) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) - Math.log(Math.exp(((alpha / t_0) + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999995)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = ((beta / t_0) - log(exp(((alpha / t_0) + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999995], 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[Log[N[Exp[N[(N[(alpha / t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999999949999999971

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in alpha around inf 99.2%

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

   if -0.999999949999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. div-sub 99.6%

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

      2. associate-+l- 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+l+ 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. add-log-exp 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \log \left(e^{\frac{\alpha}{\beta + \left(\alpha + 2\right)} + -1}\right)}{2}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999995], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta * N[(1.0 / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999999949999999971

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in alpha around inf 99.2%

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

   if -0.999999949999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. div-sub 99.6%

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

      2. associate-+l- 99.6%

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

      3. div-inv 99.6%

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

      4. fma-neg 99.6%

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

      5. associate-+l+ 99.6%

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

      6. associate-+l+ 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \frac{1}{\beta + \left(\alpha + 2\right)}, 1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999995)
     (/ (/ (+ 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 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999995) {
		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 = beta + (alpha + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99999995d0)) 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 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999995) {
		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 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999995:
		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(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999995)
		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 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999995)
		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[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999995], 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) 2)) < -0.999999949999999971

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in alpha around inf 99.2%

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

   if -0.999999949999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. div-sub 99.6%

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

      2. associate-+l- 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+l+ 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.5%

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

Alternative 4: 99.7% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999995)
     (/ (/ (+ 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 <= -0.99999995) {
		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 <= (-0.99999995d0)) 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 <= -0.99999995) {
		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 <= -0.99999995:
		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 <= -0.99999995)
		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 <= -0.99999995)
		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, -0.99999995], 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) 2)) < -0.999999949999999971

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in alpha around inf 99.2%

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

   if -0.999999949999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.6%

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

4. Final simplification 99.5%

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

Alternative 5: 88.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 40000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 40000000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 40000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 40000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 40000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 40000000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 40000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 40000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 40000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4e7

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in alpha around 0 98.1%

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

   if 4e7 < 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. Taylor expanded in beta around 0 6.0%

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

      1. +-commutative 6.0%

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

   6. Simplified 6.0%

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

   7. Taylor expanded in alpha around inf 66.0%

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

   8. Taylor expanded in alpha around 0 66.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 40000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 6: 93.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1150000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1150000.0)
   (/ (+ 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 <= 1150000.0) {
		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 <= 1150000.0d0) 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 <= 1150000.0) {
		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 <= 1150000.0:
		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 <= 1150000.0)
		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 <= 1150000.0)
		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, 1150000.0], 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 < 1.15e6

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in alpha around 0 98.1%

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

   if 1.15e6 < 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. Taylor expanded in alpha around inf 86.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1150000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 68.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.95:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.95) 0.5 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.95) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.95d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.95) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.95:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.95)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.95)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.95], 0.5, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.94999999999999996

   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. Step-by-step derivation

      1. flip-+ 89.1%

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

      2. associate-/r/ 89.1%

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

      3. fma-def 89.1%

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

      4. sub-neg 89.1%

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

      5. pow2 89.1%

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

      6. metadata-eval 89.1%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2} + \left(-\color{blue}{4}\right)}, \left(\beta + \alpha\right) - 2, 1\right)}{2} \]

      7. metadata-eval 89.1%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2} + \color{blue}{-4}}, \left(\beta + \alpha\right) - 2, 1\right)}{2} \]

      8. associate--l+ 89.1%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2} + -4}, \color{blue}{\beta + \left(\alpha - 2\right)}, 1\right)}{2} \]

   5. Applied egg-rr 89.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2} + -4}, \beta + \left(\alpha - 2\right), 1\right)}}{2} \]

   6. Taylor expanded in beta around 0 73.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-1 \cdot \frac{\alpha}{{\alpha}^{2} - 4}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]
   7. Step-by-step derivation

      1. associate-*r/ 73.3%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \alpha}{{\alpha}^{2} - 4}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

      2. mul-1-neg 73.3%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{-\alpha}}{{\alpha}^{2} - 4}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

      3. sub-neg 73.3%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-\alpha}{\color{blue}{{\alpha}^{2} + \left(-4\right)}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

      4. unpow2 73.3%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-\alpha}{\color{blue}{\alpha \cdot \alpha} + \left(-4\right)}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

      5. metadata-eval 73.3%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-\alpha}{\alpha \cdot \alpha + \color{blue}{-4}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

   8. Simplified 73.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-\alpha}{\alpha \cdot \alpha + -4}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

   9. Taylor expanded in alpha around 0 74.8%

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

   if 1.94999999999999996 < 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. Taylor expanded in beta around 0 9.2%

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

      1. +-commutative 9.2%

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

   6. Simplified 9.2%

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

   7. Taylor expanded in alpha around inf 64.0%

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

   8. Taylor expanded in alpha around 0 64.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.95:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 8: 72.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6500:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 6500.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 6500.0) {
		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 <= 6500.0d0) 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 <= 6500.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 6500.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 6500.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 6500.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 6500.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 6500

   1. Initial program 71.0%

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

      1. +-commutative 71.0%

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

   3. Simplified 71.0%

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

      1. flip-+ 70.5%

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

      2. associate-/r/ 70.6%

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

      3. fma-def 70.5%

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

      4. sub-neg 70.5%

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

      5. pow2 70.5%

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

      6. metadata-eval 70.5%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2} + \left(-\color{blue}{4}\right)}, \left(\beta + \alpha\right) - 2, 1\right)}{2} \]

      7. metadata-eval 70.5%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2} + \color{blue}{-4}}, \left(\beta + \alpha\right) - 2, 1\right)}{2} \]

      8. associate--l+ 70.5%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2} + -4}, \color{blue}{\beta + \left(\alpha - 2\right)}, 1\right)}{2} \]

   5. Applied egg-rr 70.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2} + -4}, \beta + \left(\alpha - 2\right), 1\right)}}{2} \]

   6. Taylor expanded in beta around 0 69.6%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-1 \cdot \frac{\alpha}{{\alpha}^{2} - 4}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]
   7. Step-by-step derivation

      1. associate-*r/ 69.6%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \alpha}{{\alpha}^{2} - 4}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

      2. mul-1-neg 69.6%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{-\alpha}}{{\alpha}^{2} - 4}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

      3. sub-neg 69.6%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-\alpha}{\color{blue}{{\alpha}^{2} + \left(-4\right)}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

      4. unpow2 69.6%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-\alpha}{\color{blue}{\alpha \cdot \alpha} + \left(-4\right)}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

      5. metadata-eval 69.6%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-\alpha}{\alpha \cdot \alpha + \color{blue}{-4}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

   8. Simplified 69.6%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-\alpha}{\alpha \cdot \alpha + -4}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

   9. Taylor expanded in alpha around 0 67.7%

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

   if 6500 < beta

   1. Initial program 83.0%

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

      1. +-commutative 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in beta around inf 78.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6500:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 49.4% 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 74.5%

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

   1. +-commutative 74.5%

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

3. Simplified 74.5%

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

   1. flip-+ 63.8%

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

   2. associate-/r/ 63.8%

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

   3. fma-def 63.8%

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

   4. sub-neg 63.8%

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

   5. pow2 63.8%

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

   6. metadata-eval 63.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2} + \left(-\color{blue}{4}\right)}, \left(\beta + \alpha\right) - 2, 1\right)}{2} \]

   7. metadata-eval 63.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2} + \color{blue}{-4}}, \left(\beta + \alpha\right) - 2, 1\right)}{2} \]

   8. associate--l+ 63.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2} + -4}, \color{blue}{\beta + \left(\alpha - 2\right)}, 1\right)}{2} \]

5. Applied egg-rr 63.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2} + -4}, \beta + \left(\alpha - 2\right), 1\right)}}{2} \]

6. Taylor expanded in beta around 0 51.9%

   \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-1 \cdot \frac{\alpha}{{\alpha}^{2} - 4}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]
7. Step-by-step derivation

   1. associate-*r/ 51.9%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \alpha}{{\alpha}^{2} - 4}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

   2. mul-1-neg 51.9%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{-\alpha}}{{\alpha}^{2} - 4}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

   3. sub-neg 51.9%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-\alpha}{\color{blue}{{\alpha}^{2} + \left(-4\right)}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

   4. unpow2 51.9%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-\alpha}{\color{blue}{\alpha \cdot \alpha} + \left(-4\right)}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

   5. metadata-eval 51.9%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-\alpha}{\alpha \cdot \alpha + \color{blue}{-4}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

8. Simplified 51.9%

   \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-\alpha}{\alpha \cdot \alpha + -4}}, \beta + \left(\alpha - 2\right), 1\right)}{2} \]

9. Taylor expanded in alpha around 0 52.5%

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

10. Final simplification 52.5%

    \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023252 
(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))