Octave 3.8, jcobi/2

Percentage Accurate: 63.4% → 97.5%

Time: 24.7s

Alternatives: 9

Speedup: 4.8×

Specification

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

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

FPCore

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

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

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

Initial Program: 63.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

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

Alternative 1: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -1:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -1.0)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        (+ alpha (+ beta (fma 2.0 i 2.0))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -1.0) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / (alpha + (beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -1.0)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 1.9%

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

   3. Simplified 15.9%

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

   5. Taylor expanded in beta around 0 15.9%

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

   6. Taylor expanded in alpha around inf 90.3%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]

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

   1. Initial program 83.9%

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

   3. Simplified 100.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -1:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -1:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -1.0)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (* (- beta alpha) (/ beta (+ beta (* 2.0 i))))
        (+ alpha (+ beta (fma 2.0 i 2.0)))))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -1.0) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / (beta + (2.0 * i)))) / (alpha + (beta + fma(2.0, i, 2.0))))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -1.0)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(beta / Float64(beta + Float64(2.0 * i)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0))))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 1.9%

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

   3. Simplified 15.9%

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

   5. Taylor expanded in beta around 0 15.9%

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

   6. Taylor expanded in alpha around inf 90.3%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]

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

   1. Initial program 83.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 99.5%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -1:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{t\_1 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -1.0)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (if (<= t_1 2e-12)
       (/ (+ t_1 1.0) 2.0)
       (/ (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else if (t_1 <= 2e-12) {
		tmp = (t_1 + 1.0) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)
    if (t_1 <= (-1.0d0)) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else if (t_1 <= 2d-12) then
        tmp = (t_1 + 1.0d0) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else if (t_1 <= 2e-12) {
		tmp = (t_1 + 1.0) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)
	tmp = 0
	if t_1 <= -1.0:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	elif t_1 <= 2e-12:
		tmp = (t_1 + 1.0) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	elseif (t_1 <= 2e-12)
		tmp = Float64(Float64(t_1 + 1.0) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	tmp = 0.0;
	if (t_1 <= -1.0)
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	elseif (t_1 <= 2e-12)
		tmp = (t_1 + 1.0) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e-12], N[(N[(t$95$1 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 1.9%

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

   3. Simplified 15.9%

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

   5. Taylor expanded in beta around 0 15.9%

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

   6. Taylor expanded in alpha around inf 90.3%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]

   if -1 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 1.99999999999999996e-12

   1. Initial program 100.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   if 1.99999999999999996e-12 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 37.5%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 94.6%

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

      1. associate-+r+ 94.6%

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

   7. Simplified 94.6%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -1:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.2e+81)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.2e+81) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.2d+81) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.2e+81) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.2e+81:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.2e+81)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.2e+81)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.2e+81], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.19999999999999995e81

   1. Initial program 83.2%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in i around 0 84.8%

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

      1. associate-+r+ 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in alpha around 0 88.4%

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

   if 1.19999999999999995e81 < alpha

   1. Initial program 5.4%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in beta around 0 24.9%

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

   6. Taylor expanded in alpha around inf 81.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8.6 \cdot 10^{+83}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 8.6e+83)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (+ (/ beta alpha) (/ 1.0 alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 8.6e+83) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 8.6d+83) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta / alpha) + (1.0d0 / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 8.6e+83) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 8.6e+83:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta / alpha) + (1.0 / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 8.6e+83)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 8.6e+83)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta / alpha) + (1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 8.6e+83], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.6e83

   1. Initial program 83.2%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in i around 0 84.8%

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

      1. associate-+r+ 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in alpha around 0 88.4%

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

   if 8.6e83 < alpha

   1. Initial program 5.4%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in i around 0 11.4%

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

      1. associate-+r+ 11.4%

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

   7. Simplified 11.4%

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

   8. Taylor expanded in alpha around inf 54.2%

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

   9. Taylor expanded in beta around 0 54.2%

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

       1. +-commutative 54.2%

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

   11. Simplified 54.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8.6 \cdot 10^{+83}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8.6 \cdot 10^{+19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 8.6e+19) 0.5 (+ (/ beta alpha) (/ 1.0 alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 8.6e+19) {
		tmp = 0.5;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 8.6d+19) then
        tmp = 0.5d0
    else
        tmp = (beta / alpha) + (1.0d0 / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 8.6e+19) {
		tmp = 0.5;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 8.6e+19:
		tmp = 0.5
	else:
		tmp = (beta / alpha) + (1.0 / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 8.6e+19)
		tmp = 0.5;
	else
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 8.6e+19)
		tmp = 0.5;
	else
		tmp = (beta / alpha) + (1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 8.6e+19], 0.5, N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.6e19

   1. Initial program 87.7%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 90.0%

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

      1. associate-+r+ 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in beta around 0 78.2%

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

      1. +-commutative 78.2%

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

   10. Simplified 78.2%

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

   11. Taylor expanded in alpha around 0 81.5%

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

   if 8.6e19 < alpha

   1. Initial program 10.0%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in i around 0 14.3%

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

      1. associate-+r+ 14.3%

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

   7. Simplified 14.3%

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

   8. Taylor expanded in alpha around inf 52.5%

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

   9. Taylor expanded in beta around 0 52.5%

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

       1. +-commutative 52.5%

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

   11. Simplified 52.5%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8.6 \cdot 10^{+19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 9.5e+19) 0.5 (/ 1.0 alpha)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 9.5e+19) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 9.5d+19) 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 i) {
	double tmp;
	if (alpha <= 9.5e+19) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 9.5e+19:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 9.5e+19)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 9.5e+19)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 9.5e+19], 0.5, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 9.5e19

   1. Initial program 87.7%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 90.0%

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

      1. associate-+r+ 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in beta around 0 78.2%

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

      1. +-commutative 78.2%

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

   10. Simplified 78.2%

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

   11. Taylor expanded in alpha around 0 81.5%

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

   if 9.5e19 < alpha

   1. Initial program 10.0%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in i around 0 14.3%

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

      1. associate-+r+ 14.3%

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

   7. Simplified 14.3%

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

   8. Taylor expanded in alpha around inf 52.5%

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

   9. Taylor expanded in beta around 0 39.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+17}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 5.5e+17) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.5e+17) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.5d+17) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.5e+17) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.5e+17:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.5e+17)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.5e+17)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 5.5e+17], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 5.5e17

   1. Initial program 67.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in i around 0 60.4%

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

      1. associate-+r+ 60.4%

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

   7. Simplified 60.4%

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

   8. Taylor expanded in beta around 0 61.7%

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

      1. +-commutative 61.7%

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

   10. Simplified 61.7%

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

   11. Taylor expanded in alpha around 0 70.0%

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

   if 5.5e17 < beta

   1. Initial program 34.0%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in beta around inf 61.2%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+17}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.2% accurate, 29.0× speedup

Math

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

FPCore

(FPCore (alpha beta i) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta, i):
	return 0.5

Julia

function code(alpha, beta, i)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 58.0%

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

3. Simplified 73.4%

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

5. Taylor expanded in i around 0 61.0%

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

   1. associate-+r+ 61.0%

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

7. Simplified 61.0%

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

8. Taylor expanded in beta around 0 50.0%

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

   1. +-commutative 50.0%

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

10. Simplified 50.0%

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

11. Taylor expanded in alpha around 0 58.6%

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

12. Final simplification 58.6%

    \[\leadsto 0.5 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))