Octave 3.8, jcobi/2

Percentage Accurate: 63.7% → 98.0%

Time: 14.8s

Alternatives: 13

Speedup: 4.0×

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: 63.7% 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: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha} + \left(\frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right) - -2}{\alpha} + \left(t_1 - {t_1}^{2}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, 1\right)}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (+ 2.0 (fma 2.0 beta (* i 4.0))) alpha)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.99999)
     (/
      (+
       (* (/ beta alpha) (/ beta alpha))
       (+
        (* (/ (fma 2.0 i beta) alpha) (/ (- (fma 2.0 i beta) -2.0) alpha))
        (- t_1 (pow t_1 2.0))))
      2.0)
     (/
      (log
       (exp
        (fma
         (/ (+ alpha beta) (+ alpha (+ beta (fma 2.0 i 2.0))))
         (/ (- beta alpha) (+ beta (fma 2.0 i alpha)))
         1.0)))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (2.0 + fma(2.0, beta, (i * 4.0))) / alpha;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.99999) {
		tmp = (((beta / alpha) * (beta / alpha)) + (((fma(2.0, i, beta) / alpha) * ((fma(2.0, i, beta) - -2.0) / alpha)) + (t_1 - pow(t_1, 2.0)))) / 2.0;
	} else {
		tmp = log(exp(fma(((alpha + beta) / (alpha + (beta + fma(2.0, i, 2.0)))), ((beta - alpha) / (beta + fma(2.0, i, alpha))), 1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(2.0 + fma(2.0, beta, Float64(i * 4.0))) / alpha)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.99999)
		tmp = Float64(Float64(Float64(Float64(beta / alpha) * Float64(beta / alpha)) + Float64(Float64(Float64(fma(2.0, i, beta) / alpha) * Float64(Float64(fma(2.0, i, beta) - -2.0) / alpha)) + Float64(t_1 - (t_1 ^ 2.0)))) / 2.0);
	else
		tmp = Float64(log(exp(fma(Float64(Float64(alpha + beta) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), Float64(Float64(beta - alpha) / Float64(beta + fma(2.0, i, alpha))), 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]}, Block[{t$95$1 = N[(N[(2.0 + N[(2.0 * beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $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], -0.99999], N[(N[(N[(N[(beta / alpha), $MachinePrecision] * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(2.0 * i + beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(N[(2.0 * i + beta), $MachinePrecision] - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Log[N[Exp[N[(N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999990000000000046

   1. Initial program 3.3%

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

   3. Simplified 15.6%

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

   4. Taylor expanded in alpha around inf 84.8%

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

   6. Simplified 92.4%

      \[\leadsto \frac{\color{blue}{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \left(\frac{-2 - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} + \left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} \cdot \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} - \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. sub-neg 92.4%

         \[\leadsto \frac{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \left(\frac{-2 - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} + \color{blue}{\left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} \cdot \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} + \left(-\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)\right)}\right)}{2} \]

      2. pow2 92.4%

         \[\leadsto \frac{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \left(\frac{-2 - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} + \left(\color{blue}{{\left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)}^{2}} + \left(-\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)\right)\right)}{2} \]

   8. Applied egg-rr 92.4%

      \[\leadsto \frac{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \left(\frac{-2 - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} + \color{blue}{\left({\left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)}^{2} + \left(-\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)\right)}\right)}{2} \]
   9. Step-by-step derivation

      1. sub-neg 92.4%

         \[\leadsto \frac{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \left(\frac{-2 - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} + \color{blue}{\left({\left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)}^{2} - \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)}\right)}{2} \]

   10. Simplified 92.4%

       \[\leadsto \frac{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \left(\frac{-2 - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} + \color{blue}{\left({\left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)}^{2} - \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)}\right)}{2} \]

   if -0.999990000000000046 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 79.0%

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

   3. Simplified 99.9%

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

      1. add-log-exp 99.9%

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

      2. fma-def 99.9%

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

      3. associate-+r+ 99.9%

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

      4. fma-udef 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha} + \left(\frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right) - -2}{\alpha} + \left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} - {\left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)}^{2}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, 1\right)}\right)}{2}\\ \end{array} \]

Alternative 2: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, 1\right)}\right)}{2}\\ \end{array} \end{array} \]

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)) -0.99999)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (log
       (exp
        (fma
         (/ (+ alpha beta) (+ alpha (+ beta (fma 2.0 i 2.0))))
         (/ (- beta alpha) (+ beta (fma 2.0 i alpha)))
         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)) <= -0.99999) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = log(exp(fma(((alpha + beta) / (alpha + (beta + fma(2.0, i, 2.0)))), ((beta - alpha) / (beta + fma(2.0, i, alpha))), 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)) <= -0.99999)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(log(exp(fma(Float64(Float64(alpha + beta) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), Float64(Float64(beta - alpha) / Float64(beta + fma(2.0, i, alpha))), 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], -0.99999], 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[Log[N[Exp[N[(N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999990000000000046

   1. Initial program 3.3%

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

   3. Simplified 15.6%

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

   4. Taylor expanded in alpha around inf 84.8%

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

   6. Simplified 92.4%

      \[\leadsto \frac{\color{blue}{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \left(\frac{-2 - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} + \left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} \cdot \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} - \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)\right)}}{2} \]

   7. Taylor expanded in alpha around inf 91.0%

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

   if -0.999990000000000046 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 79.0%

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

   3. Simplified 99.9%

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

      1. add-log-exp 99.9%

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

      2. fma-def 99.9%

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

      3. associate-+r+ 99.9%

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

      4. fma-udef 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. +-commutative 99.9%

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

      9. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, 1\right)}\right)}{2}\\ \end{array} \]

Alternative 3: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

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)) -0.99999)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (fma
       (/ (+ alpha beta) (+ alpha (+ beta (fma 2.0 i 2.0))))
       (/ (- beta alpha) (+ alpha (fma 2.0 i beta)))
       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)) <= -0.99999) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = fma(((alpha + beta) / (alpha + (beta + fma(2.0, i, 2.0)))), ((beta - alpha) / (alpha + fma(2.0, i, beta))), 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)) <= -0.99999)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(alpha + beta) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), Float64(Float64(beta - alpha) / Float64(alpha + fma(2.0, i, beta))), 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], -0.99999], 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[(alpha + beta), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $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 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999990000000000046

   1. Initial program 3.3%

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

   3. Simplified 15.6%

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

   4. Taylor expanded in alpha around inf 84.8%

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

   6. Simplified 92.4%

      \[\leadsto \frac{\color{blue}{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \left(\frac{-2 - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} + \left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} \cdot \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} - \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)\right)}}{2} \]

   7. Taylor expanded in alpha around inf 91.0%

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

   if -0.999990000000000046 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 79.0%

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

   3. Simplified 99.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 4: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2}\\ \end{array} \end{array} \]

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)) -0.99999)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       1.0
       (*
        (/ (+ alpha beta) (+ (+ alpha beta) (fma 2.0 i 2.0)))
        (/ (- beta alpha) (fma 2.0 i (+ alpha beta)))))
      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)) <= -0.99999) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) / ((alpha + beta) + fma(2.0, i, 2.0))) * ((beta - alpha) / fma(2.0, i, (alpha + beta))))) / 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)) <= -0.99999)
		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(alpha + beta) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))) * Float64(Float64(beta - alpha) / fma(2.0, i, Float64(alpha + beta))))) / 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], -0.99999], 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[(alpha + beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $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 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999990000000000046

   1. Initial program 3.3%

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

   3. Simplified 15.6%

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

   4. Taylor expanded in alpha around inf 84.8%

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

   6. Simplified 92.4%

      \[\leadsto \frac{\color{blue}{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \left(\frac{-2 - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} + \left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} \cdot \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} - \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)\right)}}{2} \]

   7. Taylor expanded in alpha around inf 91.0%

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

   if -0.999990000000000046 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 79.0%

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

   3. Simplified 99.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2}\\ \end{array} \]

Alternative 5: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.5:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{t_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.5)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/ (+ 1.0 (/ beta t_1)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / t_1)) / 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 = 2.0d0 + t_0
    if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= (-0.5d0)) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (beta / t_1)) / 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 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / t_1)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = 2.0 + t_0
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = (1.0 + (beta / t_1)) / 2.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = 2.0 + t_0;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5)
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = (1.0 + (beta / t_1)) / 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[(2.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.5], 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[(beta / t$95$1), $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 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.5

   1. Initial program 4.4%

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

   3. Simplified 16.6%

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

   4. Taylor expanded in alpha around inf 84.5%

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

   6. Simplified 92.1%

      \[\leadsto \frac{\color{blue}{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \left(\frac{-2 - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} + \left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} \cdot \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} - \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)\right)}}{2} \]

   7. 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 -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 79.0%

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

   2. Taylor expanded in beta around inf 97.8%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 6: 83.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.5e+106)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ beta (* 2.0 i))))) 2.0)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.5e+106) {
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / 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 <= 2.5d+106) then
        tmp = (1.0d0 + (beta / (2.0d0 + (beta + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.5e+106) {
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.5e+106:
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0
	else:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.5e+106)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(beta + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.5e+106)
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0;
	else
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.5e+106], N[(N[(1.0 + N[(beta / N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.4999999999999999e106

   1. Initial program 75.3%

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

   2. Taylor expanded in beta around inf 89.7%

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

   3. Taylor expanded in alpha around 0 89.7%

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

   if 2.4999999999999999e106 < alpha

   1. Initial program 1.6%

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

   2. Taylor expanded in beta around inf 15.6%

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

      1. mul-1-neg 15.6%

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

      2. sub-neg 15.6%

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

      3. cancel-sign-sub-inv 15.6%

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

      4. metadata-eval 15.6%

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

   4. Simplified 15.6%

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

   5. Taylor expanded in alpha around -inf 56.0%

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

      1. mul-1-neg 56.0%

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

      2. mul-1-neg 56.0%

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

      3. +-commutative 56.0%

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

   7. Simplified 56.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 87.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.2e+29)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ beta (* 2.0 i))))) 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 <= 4.2e+29) {
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 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 <= 4.2d+29) then
        tmp = (1.0d0 + (beta / (2.0d0 + (beta + (2.0d0 * i))))) / 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 <= 4.2e+29) {
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 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 <= 4.2e+29:
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 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 <= 4.2e+29)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(beta + Float64(2.0 * i))))) / 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 <= 4.2e+29)
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 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, 4.2e+29], N[(N[(1.0 + N[(beta / N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $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 < 4.2000000000000003e29

   1. Initial program 81.3%

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

   2. Taylor expanded in beta around inf 96.0%

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

   3. Taylor expanded in alpha around 0 96.0%

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

   if 4.2000000000000003e29 < alpha

   1. Initial program 13.9%

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

   3. Simplified 35.4%

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

   4. Taylor expanded in alpha around inf 63.8%

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

   6. Simplified 70.1%

      \[\leadsto \frac{\color{blue}{\left(0 + \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) - \left(\frac{-2 - \mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} + \left(\frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} \cdot \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha} - \frac{2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)}{\alpha}\right)\right)}}{2} \]

   7. Taylor expanded in alpha around inf 70.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 78.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2e+107)
   (/ (+ 1.0 (/ beta (+ (+ alpha beta) 2.0))) 2.0)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2e+107) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / 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 <= 2d+107) then
        tmp = (1.0d0 + (beta / ((alpha + beta) + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2e+107) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2e+107:
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0
	else:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2e+107)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2e+107)
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	else
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2e+107], N[(N[(1.0 + N[(beta / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.9999999999999999e107

   1. Initial program 75.3%

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

   2. Taylor expanded in beta around inf 89.7%

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

   3. Taylor expanded in i around 0 84.9%

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

      1. +-commutative 84.9%

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

   5. Simplified 84.9%

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

   if 1.9999999999999999e107 < alpha

   1. Initial program 1.6%

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

   2. Taylor expanded in beta around inf 15.6%

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

      1. mul-1-neg 15.6%

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

      2. sub-neg 15.6%

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

      3. cancel-sign-sub-inv 15.6%

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

      4. metadata-eval 15.6%

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

   4. Simplified 15.6%

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

   5. Taylor expanded in alpha around -inf 56.0%

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

      1. mul-1-neg 56.0%

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

      2. mul-1-neg 56.0%

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

      3. +-commutative 56.0%

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

   7. Simplified 56.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 71.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8500000000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+92}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+137}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8500000000000.0)
   0.5
   (if (<= beta 1.5e+92)
     (/ (- 2.0 (/ 2.0 beta)) 2.0)
     (if (<= beta 2.4e+137) 0.5 1.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8500000000000.0) {
		tmp = 0.5;
	} else if (beta <= 1.5e+92) {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	} else if (beta <= 2.4e+137) {
		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 <= 8500000000000.0d0) then
        tmp = 0.5d0
    else if (beta <= 1.5d+92) then
        tmp = (2.0d0 - (2.0d0 / beta)) / 2.0d0
    else if (beta <= 2.4d+137) 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 <= 8500000000000.0) {
		tmp = 0.5;
	} else if (beta <= 1.5e+92) {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	} else if (beta <= 2.4e+137) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 8500000000000.0:
		tmp = 0.5
	elif beta <= 1.5e+92:
		tmp = (2.0 - (2.0 / beta)) / 2.0
	elif beta <= 2.4e+137:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8500000000000.0)
		tmp = 0.5;
	elseif (beta <= 1.5e+92)
		tmp = Float64(Float64(2.0 - Float64(2.0 / beta)) / 2.0);
	elseif (beta <= 2.4e+137)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8500000000000.0)
		tmp = 0.5;
	elseif (beta <= 1.5e+92)
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	elseif (beta <= 2.4e+137)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 8500000000000.0], 0.5, If[LessEqual[beta, 1.5e+92], N[(N[(2.0 - N[(2.0 / beta), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 2.4e+137], 0.5, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if beta < 8.5e12 or 1.50000000000000007e92 < beta < 2.39999999999999983e137

   1. Initial program 70.8%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in i around inf 71.2%

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

   if 8.5e12 < beta < 1.50000000000000007e92

   1. Initial program 62.6%

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

   2. Taylor expanded in beta around inf 64.0%

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

   3. Taylor expanded in alpha around 0 64.0%

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

   4. Taylor expanded in beta around inf 54.1%

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

      1. mul-1-neg 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in i around 0 56.2%

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

      1. associate-*r/ 56.2%

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

      2. metadata-eval 56.2%

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

   9. Simplified 56.2%

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

   if 2.39999999999999983e137 < beta

   1. Initial program 11.5%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in beta around inf 88.5%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8500000000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+92}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+137}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 77.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{+105}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.7e+105)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.7e+105) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 3.7d+105) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.7e+105) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3.7e+105:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3.7e+105)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3.7e+105)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3.7e+105], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.69999999999999985e105

   1. Initial program 75.3%

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

   2. Taylor expanded in beta around inf 89.7%

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

   3. Taylor expanded in alpha around 0 89.7%

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

   4. Taylor expanded in i around 0 84.7%

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

      1. +-commutative 84.7%

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

   6. Simplified 84.7%

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

   if 3.69999999999999985e105 < alpha

   1. Initial program 1.6%

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

   2. Taylor expanded in beta around inf 15.6%

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

      1. mul-1-neg 15.6%

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

      2. sub-neg 15.6%

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

      3. cancel-sign-sub-inv 15.6%

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

      4. metadata-eval 15.6%

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

   4. Simplified 15.6%

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

   5. Taylor expanded in alpha around -inf 56.0%

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

      1. mul-1-neg 56.0%

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

      2. mul-1-neg 56.0%

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

      3. +-commutative 56.0%

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

   7. Simplified 56.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{+105}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 11: 72.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \frac{\beta}{\beta + 2}}{2} \end{array} \]

FPCore

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

C

double code(double alpha, double beta, double i) {
	return (1.0 + (beta / (beta + 2.0))) / 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
    code = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.5%

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

2. Taylor expanded in beta around inf 75.7%

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

3. Taylor expanded in alpha around 0 75.7%

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

4. Taylor expanded in i around 0 70.8%

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

   1. +-commutative 70.8%

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

6. Simplified 70.8%

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

7. Final simplification 70.8%

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

Alternative 12: 71.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8600000000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 7.6 \cdot 10^{+96}:\\ \;\;\;\;1\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+137}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8600000000000.0)
   0.5
   (if (<= beta 7.6e+96) 1.0 (if (<= beta 2.4e+137) 0.5 1.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8600000000000.0) {
		tmp = 0.5;
	} else if (beta <= 7.6e+96) {
		tmp = 1.0;
	} else if (beta <= 2.4e+137) {
		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 <= 8600000000000.0d0) then
        tmp = 0.5d0
    else if (beta <= 7.6d+96) then
        tmp = 1.0d0
    else if (beta <= 2.4d+137) 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 <= 8600000000000.0) {
		tmp = 0.5;
	} else if (beta <= 7.6e+96) {
		tmp = 1.0;
	} else if (beta <= 2.4e+137) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 8600000000000.0:
		tmp = 0.5
	elif beta <= 7.6e+96:
		tmp = 1.0
	elif beta <= 2.4e+137:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8600000000000.0)
		tmp = 0.5;
	elseif (beta <= 7.6e+96)
		tmp = 1.0;
	elseif (beta <= 2.4e+137)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8600000000000.0)
		tmp = 0.5;
	elseif (beta <= 7.6e+96)
		tmp = 1.0;
	elseif (beta <= 2.4e+137)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 8600000000000.0], 0.5, If[LessEqual[beta, 7.6e+96], 1.0, If[LessEqual[beta, 2.4e+137], 0.5, 1.0]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.6e12 or 7.6000000000000003e96 < beta < 2.39999999999999983e137

   1. Initial program 70.8%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in i around inf 71.2%

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

   if 8.6e12 < beta < 7.6000000000000003e96 or 2.39999999999999983e137 < beta

   1. Initial program 27.5%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in beta around inf 78.2%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8600000000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 7.6 \cdot 10^{+96}:\\ \;\;\;\;1\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+137}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 61.7% 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 59.5%

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

3. Simplified 78.2%

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

4. Taylor expanded in i around inf 58.9%

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

5. Final simplification 58.9%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023271 
(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))