Octave 3.8, jcobi/1

Percentage Accurate: 74.4% → 99.3%

Time: 10.0s

Alternatives: 11

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -1.0)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta * Float64(1.0 - Float64(alpha / beta))) / 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) / t_0) <= -1.0)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = (1.0 + ((beta * (1.0 - (alpha / beta))) / t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

   1. Initial program 5.3%

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

      1. +-commutative 5.3%

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

   3. Simplified 5.3%

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

   5. Taylor expanded in alpha around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in alpha around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. times-frac 100.0%

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

      8. *-lft-identity 100.0%

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

      9. fma-undefine 100.0%

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

      10. distribute-lft1-in 100.0%

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

      11. *-commutative 100.0%

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

      12. times-frac 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   10. Simplified 100.0%

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

       1. *-un-lft-identity 100.0%

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

       2. +-commutative 100.0%

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

   12. Applied egg-rr 100.0%

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

   if -1 < (/.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. Taylor expanded in beta around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -1.0) (/ (+ beta 1.0) alpha) (/ (+ 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 <= -1.0) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

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

Java

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -1

   1. Initial program 5.3%

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

      1. +-commutative 5.3%

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

   3. Simplified 5.3%

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

   5. Taylor expanded in alpha around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in alpha around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. times-frac 100.0%

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

      8. *-lft-identity 100.0%

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

      9. fma-undefine 100.0%

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

      10. distribute-lft1-in 100.0%

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

      11. *-commutative 100.0%

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

      12. times-frac 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   10. Simplified 100.0%

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

       1. *-un-lft-identity 100.0%

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

       2. +-commutative 100.0%

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

   12. Applied egg-rr 100.0%

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

   if -1 < (/.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. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternative 3: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 26500000000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 26500000000000.0)
   (+ 0.5 (* (- alpha beta) (/ -0.5 (+ beta (+ alpha 2.0)))))
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 26500000000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 26500000000000.0d0) then
        tmp = 0.5d0 + ((alpha - beta) * ((-0.5d0) / (beta + (alpha + 2.0d0))))
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 26500000000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 26500000000000.0:
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))))
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 26500000000000.0)
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) * Float64(-0.5 / Float64(beta + Float64(alpha + 2.0)))));
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 26500000000000.0)
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 26500000000000.0], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.65e13

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 99.5%

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

      17. associate-*l/ 99.5%

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

   3. Simplified 100.0%

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

   if 2.65e13 < alpha

   1. Initial program 22.4%

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

      1. +-commutative 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in alpha around inf 83.3%

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

      1. *-commutative 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in alpha around 0 83.3%

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

      1. associate-*r/ 83.3%

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

      2. +-commutative 83.3%

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

      3. *-commutative 83.3%

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

      4. fma-undefine 83.3%

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

      5. associate-*r/ 83.3%

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

      6. metadata-eval 83.3%

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

      7. times-frac 83.3%

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

      8. *-lft-identity 83.3%

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

      9. fma-undefine 83.3%

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

      10. distribute-lft1-in 83.3%

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

      11. *-commutative 83.3%

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

      12. times-frac 83.3%

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

      13. metadata-eval 83.3%

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

      14. +-commutative 83.3%

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

   10. Simplified 83.3%

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

       1. *-un-lft-identity 83.3%

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

       2. +-commutative 83.3%

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

   12. Applied egg-rr 83.3%

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

Alternative 4: 93.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1020000000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1020000000000.0)
   (+ 0.5 (* (- alpha beta) (/ -0.5 (+ beta 2.0))))
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1020000000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + 2.0)));
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1020000000000.0d0) then
        tmp = 0.5d0 + ((alpha - beta) * ((-0.5d0) / (beta + 2.0d0)))
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1020000000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + 2.0)));
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1020000000000.0:
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + 2.0)))
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1020000000000.0)
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) * Float64(-0.5 / Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1020000000000.0)
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + 2.0)));
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1020000000000.0], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.02e12

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 99.5%

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

      17. associate-*l/ 99.5%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.8%

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

      1. +-commutative 98.8%

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

   7. Simplified 98.8%

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

   if 1.02e12 < alpha

   1. Initial program 22.4%

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

      1. +-commutative 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in alpha around inf 83.3%

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

      1. *-commutative 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in alpha around 0 83.3%

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

      1. associate-*r/ 83.3%

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

      2. +-commutative 83.3%

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

      3. *-commutative 83.3%

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

      4. fma-undefine 83.3%

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

      5. associate-*r/ 83.3%

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

      6. metadata-eval 83.3%

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

      7. times-frac 83.3%

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

      8. *-lft-identity 83.3%

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

      9. fma-undefine 83.3%

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

      10. distribute-lft1-in 83.3%

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

      11. *-commutative 83.3%

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

      12. times-frac 83.3%

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

      13. metadata-eval 83.3%

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

      14. +-commutative 83.3%

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

   10. Simplified 83.3%

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

       1. *-un-lft-identity 83.3%

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

       2. +-commutative 83.3%

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

   12. Applied egg-rr 83.3%

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

Alternative 5: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 9000000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 9000000000000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 9000000000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 9000000000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 9000000000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 9000000000000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 9000000000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 9000000000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 9000000000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 9e12

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.8%

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

      1. +-commutative 98.8%

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

   7. Simplified 98.8%

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

   if 9e12 < alpha

   1. Initial program 22.4%

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

      1. +-commutative 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in alpha around inf 83.3%

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

      1. *-commutative 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in alpha around 0 83.3%

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

      1. associate-*r/ 83.3%

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

      2. +-commutative 83.3%

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

      3. *-commutative 83.3%

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

      4. fma-undefine 83.3%

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

      5. associate-*r/ 83.3%

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

      6. metadata-eval 83.3%

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

      7. times-frac 83.3%

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

      8. *-lft-identity 83.3%

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

      9. fma-undefine 83.3%

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

      10. distribute-lft1-in 83.3%

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

      11. *-commutative 83.3%

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

      12. times-frac 83.3%

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

      13. metadata-eval 83.3%

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

      14. +-commutative 83.3%

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

   10. Simplified 83.3%

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

       1. *-un-lft-identity 83.3%

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

       2. +-commutative 83.3%

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

   12. Applied egg-rr 83.3%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9000000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 22000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 22000000000.0) 0.5 (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 22000000000.0) {
		tmp = 0.5;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 22000000000.0d0) then
        tmp = 0.5d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 22000000000.0) {
		tmp = 0.5;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 22000000000.0:
		tmp = 0.5
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 22000000000.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 22000000000.0)
		tmp = 0.5;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 22000000000.0], 0.5, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.2e10

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 66.2%

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

      1. +-commutative 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in alpha around 0 65.0%

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

      1. *-commutative 65.0%

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

   10. Simplified 65.0%

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

   11. Taylor expanded in alpha around 0 65.2%

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

   if 2.2e10 < alpha

   1. Initial program 22.4%

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

      1. +-commutative 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in alpha around inf 83.3%

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

      1. *-commutative 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in alpha around 0 83.3%

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

      1. associate-*r/ 83.3%

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

      2. +-commutative 83.3%

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

      3. *-commutative 83.3%

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

      4. fma-undefine 83.3%

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

      5. associate-*r/ 83.3%

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

      6. metadata-eval 83.3%

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

      7. times-frac 83.3%

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

      8. *-lft-identity 83.3%

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

      9. fma-undefine 83.3%

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

      10. distribute-lft1-in 83.3%

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

      11. *-commutative 83.3%

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

      12. times-frac 83.3%

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

      13. metadata-eval 83.3%

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

      14. +-commutative 83.3%

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

   10. Simplified 83.3%

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

       1. *-un-lft-identity 83.3%

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

       2. +-commutative 83.3%

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

   12. Applied egg-rr 83.3%

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

Alternative 7: 71.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(beta * 0.25));
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	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 - (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 65.1%

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

      1. +-commutative 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in alpha around 0 63.8%

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

      1. +-commutative 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in beta around 0 63.5%

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

      1. *-commutative 63.5%

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

   10. Simplified 63.5%

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

   if 2 < beta

   1. Initial program 85.5%

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

      1. +-commutative 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in alpha around 0 85.3%

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

      1. +-commutative 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in beta around inf 84.1%

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

Alternative 8: 71.6% 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 65.1%

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

      1. +-commutative 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in alpha around 0 63.8%

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

      1. +-commutative 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in beta around 0 63.5%

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

      1. *-commutative 63.5%

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

   10. Simplified 63.5%

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

   if 2 < beta

   1. Initial program 85.5%

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

      1. +-commutative 85.5%

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

      2. sub-neg 85.5%

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

      3. +-commutative 85.5%

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

      4. neg-sub0 85.5%

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

      5. associate-+l- 85.5%

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

      6. sub0-neg 85.5%

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

      7. distribute-frac-neg 85.5%

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

      8. +-commutative 85.5%

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

      9. sub-neg 85.5%

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

      10. div-sub 85.5%

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

      11. sub-neg 85.5%

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

      12. metadata-eval 85.5%

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

      13. neg-mul-1 85.5%

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

      14. *-commutative 85.5%

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

      15. +-commutative 85.5%

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

      16. associate-/l/ 84.7%

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

      17. associate-*l/ 84.7%

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

   3. Simplified 85.5%

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

      1. +-commutative 85.5%

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

      2. *-commutative 85.5%

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

      3. fma-define 85.3%

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

      4. associate-+r+ 85.3%

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

      5. +-commutative 85.3%

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

      6. associate-+l+ 85.3%

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

   6. Applied egg-rr 85.3%

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

   7. Taylor expanded in beta around inf 83.2%

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

Alternative 9: 71.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 65.1%

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

      1. +-commutative 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in beta around 0 64.4%

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

      1. +-commutative 64.4%

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

   7. Simplified 64.4%

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

   8. Taylor expanded in alpha around 0 61.8%

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

      1. *-commutative 61.8%

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

   10. Simplified 61.8%

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

   11. Taylor expanded in alpha around 0 63.1%

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

   if 2 < beta

   1. Initial program 85.5%

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

      1. +-commutative 85.5%

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

      2. sub-neg 85.5%

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

      3. +-commutative 85.5%

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

      4. neg-sub0 85.5%

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

      5. associate-+l- 85.5%

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

      6. sub0-neg 85.5%

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

      7. distribute-frac-neg 85.5%

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

      8. +-commutative 85.5%

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

      9. sub-neg 85.5%

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

      10. div-sub 85.5%

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

      11. sub-neg 85.5%

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

      12. metadata-eval 85.5%

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

      13. neg-mul-1 85.5%

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

      14. *-commutative 85.5%

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

      15. +-commutative 85.5%

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

      16. associate-/l/ 84.7%

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

      17. associate-*l/ 84.7%

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

   3. Simplified 85.5%

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

      1. +-commutative 85.5%

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

      2. *-commutative 85.5%

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

      3. fma-define 85.3%

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

      4. associate-+r+ 85.3%

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

      5. +-commutative 85.3%

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

      6. associate-+l+ 85.3%

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

   6. Applied egg-rr 85.3%

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

   7. Taylor expanded in beta around inf 83.2%

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

Alternative 10: 49.0% 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 73.0%

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

   1. +-commutative 73.0%

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

3. Simplified 73.0%

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

5. Taylor expanded in beta around 0 44.9%

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

   1. +-commutative 44.9%

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

7. Simplified 44.9%

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

8. Taylor expanded in alpha around 0 42.9%

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

   1. *-commutative 42.9%

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

10. Simplified 42.9%

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

11. Taylor expanded in alpha around 0 45.2%

    \[\leadsto \color{blue}{0.5} \]
12. 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 73.0%

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

   1. +-commutative 73.0%

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

   2. sub-neg 73.0%

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

   3. +-commutative 73.0%

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

   4. neg-sub0 73.0%

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

   5. associate-+l- 73.0%

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

   6. sub0-neg 73.0%

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

   7. distribute-frac-neg 73.0%

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

   8. +-commutative 73.0%

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

   9. sub-neg 73.0%

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

   10. div-sub 73.0%

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

   11. sub-neg 73.0%

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

   12. metadata-eval 73.0%

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

   13. neg-mul-1 73.0%

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

   14. *-commutative 73.0%

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

   15. +-commutative 73.0%

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

   16. associate-/l/ 72.7%

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

   17. associate-*l/ 72.7%

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

3. Simplified 73.0%

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

5. Taylor expanded in alpha around inf 3.8%

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

   1. metadata-eval 3.8%

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

7. Applied egg-rr 3.8%

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

Reproduce

herbie shell --seed 2024172 
(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))