Octave 3.8, jcobi/2

Percentage Accurate: 62.7% → 96.5%

Time: 29.6s

Alternatives: 12

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: 62.7% 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: 96.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \mathsf{fma}\left(2, \beta, i \cdot 4\right)\\ t_2 := \left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}\\ t_3 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_4 := 2 + t\_1\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_3}}{2 + t\_3} \leq -0.9995:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 0 + \frac{{\beta}^{2}}{\alpha}\right) - \mathsf{fma}\left(-1, t\_4, \mathsf{fma}\left(-1, \left(2 + t\_0\right) \cdot \frac{t\_0}{\alpha}, t\_4 \cdot \frac{\beta \cdot 0 + \left(t\_1 - -2\right)}{\alpha}\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{t\_2}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \frac{\sqrt[3]{t\_2}}{\sqrt[3]{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i)))
        (t_1 (fma 2.0 beta (* i 4.0)))
        (t_2 (* (+ alpha beta) (/ (- beta alpha) (+ beta (fma 2.0 i alpha)))))
        (t_3 (+ (+ alpha beta) (* 2.0 i)))
        (t_4 (+ 2.0 t_1)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_3) (+ 2.0 t_3)) -0.9995)
     (/
      (/
       (-
        (+ (* beta 0.0) (/ (pow beta 2.0) alpha))
        (fma
         -1.0
         t_4
         (fma
          -1.0
          (* (+ 2.0 t_0) (/ t_0 alpha))
          (* t_4 (/ (+ (* beta 0.0) (- t_1 -2.0)) alpha)))))
       alpha)
      2.0)
     (/
      (fma
       (cbrt (pow (/ t_2 (+ beta (+ alpha (fma 2.0 i 2.0)))) 2.0))
       (/ (cbrt t_2) (cbrt (+ (+ alpha beta) (fma 2.0 i 2.0))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = fma(2.0, beta, (i * 4.0));
	double t_2 = (alpha + beta) * ((beta - alpha) / (beta + fma(2.0, i, alpha)));
	double t_3 = (alpha + beta) + (2.0 * i);
	double t_4 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_3) / (2.0 + t_3)) <= -0.9995) {
		tmp = ((((beta * 0.0) + (pow(beta, 2.0) / alpha)) - fma(-1.0, t_4, fma(-1.0, ((2.0 + t_0) * (t_0 / alpha)), (t_4 * (((beta * 0.0) + (t_1 - -2.0)) / alpha))))) / alpha) / 2.0;
	} else {
		tmp = fma(cbrt(pow((t_2 / (beta + (alpha + fma(2.0, i, 2.0)))), 2.0)), (cbrt(t_2) / cbrt(((alpha + beta) + fma(2.0, i, 2.0)))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = fma(2.0, beta, Float64(i * 4.0))
	t_2 = Float64(Float64(alpha + beta) * Float64(Float64(beta - alpha) / Float64(beta + fma(2.0, i, alpha))))
	t_3 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_4 = Float64(2.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_3) / Float64(2.0 + t_3)) <= -0.9995)
		tmp = Float64(Float64(Float64(Float64(Float64(beta * 0.0) + Float64((beta ^ 2.0) / alpha)) - fma(-1.0, t_4, fma(-1.0, Float64(Float64(2.0 + t_0) * Float64(t_0 / alpha)), Float64(t_4 * Float64(Float64(Float64(beta * 0.0) + Float64(t_1 - -2.0)) / alpha))))) / alpha) / 2.0);
	else
		tmp = Float64(fma(cbrt((Float64(t_2 / Float64(beta + Float64(alpha + fma(2.0, i, 2.0)))) ^ 2.0)), Float64(cbrt(t_2) / cbrt(Float64(Float64(alpha + 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[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(2.0 + t$95$3), $MachinePrecision]), $MachinePrecision], -0.9995], N[(N[(N[(N[(N[(beta * 0.0), $MachinePrecision] + N[(N[Power[beta, 2.0], $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * t$95$4 + N[(-1.0 * N[(N[(2.0 + t$95$0), $MachinePrecision] * N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * N[(N[(N[(beta * 0.0), $MachinePrecision] + N[(t$95$1 - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Power[N[Power[N[(t$95$2 / N[(beta + N[(alpha + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[t$95$2, 1/3], $MachinePrecision] / N[Power[N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision], 1/3], $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))) < -0.99950000000000006

   1. Initial program 3.8%

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

   3. Simplified 18.1%

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

   6. Applied egg-rr 17.8%

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

   8. Simplified 17.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \sqrt[3]{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right)}}{2} \]

   9. Taylor expanded in alpha around inf 76.9%

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

   11. Simplified 89.6%

       \[\leadsto \frac{\color{blue}{\frac{\left(0 \cdot \beta + \frac{{\beta}^{2}}{\alpha}\right) - \mathsf{fma}\left(-1, 2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right), \mathsf{fma}\left(-1, \left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \frac{\beta + 2 \cdot i}{\alpha}, \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right) \cdot \frac{0 \cdot \beta - \left(-2 + \left(-\mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)\right)}{\alpha}\right)\right)}{\alpha}}}{2} \]

   if -0.99950000000000006 < (/.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 80.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 99.9%

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

   8. Simplified 99.9%

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

      1. cbrt-div 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \color{blue}{\frac{\sqrt[3]{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}}{\sqrt[3]{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}}, 1\right)}{2} \]

      2. fma-undefine 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \frac{\sqrt[3]{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\color{blue}{\left(2 \cdot i + \alpha\right)} + \beta}}}{\sqrt[3]{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right)}{2} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \frac{\sqrt[3]{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2 \cdot i\right)} + \beta}}}{\sqrt[3]{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right)}{2} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \frac{\sqrt[3]{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}}}{\sqrt[3]{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right)}{2} \]

      5. +-commutative 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \frac{\sqrt[3]{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}}}{\sqrt[3]{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right)}{2} \]

      6. fma-undefine 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \frac{\sqrt[3]{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}}}{\sqrt[3]{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right)}{2} \]

      7. associate-+r+ 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \frac{\sqrt[3]{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}}{\sqrt[3]{\color{blue}{\left(\beta + \alpha\right) + \mathsf{fma}\left(2, i, 2\right)}}}, 1\right)}{2} \]

      8. +-commutative 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \frac{\sqrt[3]{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}}{\sqrt[3]{\color{blue}{\left(\alpha + \beta\right)} + \mathsf{fma}\left(2, i, 2\right)}}, 1\right)}{2} \]

   10. Applied egg-rr 99.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9995:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 0 + \frac{{\beta}^{2}}{\alpha}\right) - \mathsf{fma}\left(-1, 2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right), \mathsf{fma}\left(-1, \left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \frac{\beta + 2 \cdot i}{\alpha}, \left(2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)\right) \cdot \frac{\beta \cdot 0 + \left(\mathsf{fma}\left(2, \beta, i \cdot 4\right) - -2\right)}{\alpha}\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \frac{\sqrt[3]{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}}{\sqrt[3]{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \mathsf{fma}\left(2, \beta, i \cdot 4\right)\\ t_2 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_3 := 2 + t\_1\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_2}}{2 + t\_2} \leq -0.9995:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 0 + \frac{{\beta}^{2}}{\alpha}\right) - \mathsf{fma}\left(-1, t\_3, \mathsf{fma}\left(-1, \left(2 + t\_0\right) \cdot \frac{t\_0}{\alpha}, t\_3 \cdot \frac{\beta \cdot 0 + \left(t\_1 - -2\right)}{\alpha}\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\sqrt[3]{{\left(1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}\right)}^{3}}\right)}^{3}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i)))
        (t_1 (fma 2.0 beta (* i 4.0)))
        (t_2 (+ (+ alpha beta) (* 2.0 i)))
        (t_3 (+ 2.0 t_1)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_2) (+ 2.0 t_2)) -0.9995)
     (/
      (/
       (-
        (+ (* beta 0.0) (/ (pow beta 2.0) alpha))
        (fma
         -1.0
         t_3
         (fma
          -1.0
          (* (+ 2.0 t_0) (/ t_0 alpha))
          (* t_3 (/ (+ (* beta 0.0) (- t_1 -2.0)) alpha)))))
       alpha)
      2.0)
     (/
      (cbrt
       (pow
        (cbrt
         (pow
          (+
           1.0
           (*
            (+ alpha beta)
            (/
             (/ (- beta alpha) (+ beta (fma 2.0 i alpha)))
             (+ (+ alpha beta) (fma 2.0 i 2.0)))))
          3.0))
        3.0))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = fma(2.0, beta, (i * 4.0));
	double t_2 = (alpha + beta) + (2.0 * i);
	double t_3 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (2.0 + t_2)) <= -0.9995) {
		tmp = ((((beta * 0.0) + (pow(beta, 2.0) / alpha)) - fma(-1.0, t_3, fma(-1.0, ((2.0 + t_0) * (t_0 / alpha)), (t_3 * (((beta * 0.0) + (t_1 - -2.0)) / alpha))))) / alpha) / 2.0;
	} else {
		tmp = cbrt(pow(cbrt(pow((1.0 + ((alpha + beta) * (((beta - alpha) / (beta + fma(2.0, i, alpha))) / ((alpha + beta) + fma(2.0, i, 2.0))))), 3.0)), 3.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = fma(2.0, beta, Float64(i * 4.0))
	t_2 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_3 = Float64(2.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_2) / Float64(2.0 + t_2)) <= -0.9995)
		tmp = Float64(Float64(Float64(Float64(Float64(beta * 0.0) + Float64((beta ^ 2.0) / alpha)) - fma(-1.0, t_3, fma(-1.0, Float64(Float64(2.0 + t_0) * Float64(t_0 / alpha)), Float64(t_3 * Float64(Float64(Float64(beta * 0.0) + Float64(t_1 - -2.0)) / alpha))))) / alpha) / 2.0);
	else
		tmp = Float64(cbrt((cbrt((Float64(1.0 + Float64(Float64(alpha + beta) * Float64(Float64(Float64(beta - alpha) / Float64(beta + fma(2.0, i, alpha))) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))))) ^ 3.0)) ^ 3.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], -0.9995], N[(N[(N[(N[(N[(beta * 0.0), $MachinePrecision] + N[(N[Power[beta, 2.0], $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * t$95$3 + N[(-1.0 * N[(N[(2.0 + t$95$0), $MachinePrecision] * N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(N[(N[(beta * 0.0), $MachinePrecision] + N[(t$95$1 - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Power[N[Power[N[Power[N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] * N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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))) < -0.99950000000000006

   1. Initial program 3.8%

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

   3. Simplified 18.1%

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

   6. Applied egg-rr 17.8%

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

   8. Simplified 17.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \sqrt[3]{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right)}}{2} \]

   9. Taylor expanded in alpha around inf 76.9%

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

   11. Simplified 89.6%

       \[\leadsto \frac{\color{blue}{\frac{\left(0 \cdot \beta + \frac{{\beta}^{2}}{\alpha}\right) - \mathsf{fma}\left(-1, 2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right), \mathsf{fma}\left(-1, \left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \frac{\beta + 2 \cdot i}{\alpha}, \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right) \cdot \frac{0 \cdot \beta - \left(-2 + \left(-\mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)\right)}{\alpha}\right)\right)}{\alpha}}}{2} \]

   if -0.99950000000000006 < (/.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 80.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 99.9%

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

   8. Simplified 99.9%

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

      1. add-cbrt-cube 99.9%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \sqrt[3]{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right) \cdot \mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \sqrt[3]{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right)\right) \cdot \mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \sqrt[3]{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right)}}}{2} \]

      2. pow3 99.9%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \sqrt[3]{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right)\right)}^{3}}}}{2} \]

   10. Applied egg-rr 99.9%

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

       1. rem-cbrt-cube 99.9%

          \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} + 1\right)}^{3}}\right)}}^{3}}}{2} \]

       2. pow1/3 99.9%

          \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left({\left({\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} + 1\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}}}{2} \]

       3. pow-to-exp 99.9%

          \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(e^{\log \left({\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} + 1\right)}^{3}\right) \cdot 0.3333333333333333}\right)}}^{3}}}{2} \]

   12. Applied egg-rr 99.9%

       \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(e^{\left(3 \cdot \mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}\right)\right) \cdot 0.3333333333333333}\right)}}^{3}}}{2} \]
   13. Step-by-step derivation

       1. exp-prod 99.9%

          \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left({\left(e^{3 \cdot \mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}\right)}^{0.3333333333333333}\right)}}^{3}}}{2} \]

       2. unpow1/3 99.9%

          \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(\sqrt[3]{e^{3 \cdot \mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}}\right)}}^{3}}}{2} \]

       3. *-commutative 99.9%

          \[\leadsto \frac{\sqrt[3]{{\left(\sqrt[3]{e^{\color{blue}{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}\right) \cdot 3}}}\right)}^{3}}}{2} \]

       4. log1p-undefine 99.9%

          \[\leadsto \frac{\sqrt[3]{{\left(\sqrt[3]{e^{\color{blue}{\log \left(1 + \frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}\right)} \cdot 3}}\right)}^{3}}}{2} \]

       5. exp-to-pow 99.9%

          \[\leadsto \frac{\sqrt[3]{{\left(\sqrt[3]{\color{blue}{{\left(1 + \frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{3}}}\right)}^{3}}}{2} \]

   14. Simplified 99.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9995:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 0 + \frac{{\beta}^{2}}{\alpha}\right) - \mathsf{fma}\left(-1, 2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right), \mathsf{fma}\left(-1, \left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \frac{\beta + 2 \cdot i}{\alpha}, \left(2 + \mathsf{fma}\left(2, \beta, i \cdot 4\right)\right) \cdot \frac{\beta \cdot 0 + \left(\mathsf{fma}\left(2, \beta, i \cdot 4\right) - -2\right)}{\alpha}\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\sqrt[3]{{\left(1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}\right)}^{3}}\right)}^{3}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}\right)}^{3}}}{2}\\ \end{array} \end{array} \]

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))
        -0.99999999)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* i 4.0) (* beta 2.0)))) alpha) 2.0)
     (/
      (cbrt
       (pow
        (+
         1.0
         (*
          (+ alpha beta)
          (/
           (/ (- beta alpha) (+ beta (fma 2.0 i alpha)))
           (+ (+ alpha beta) (fma 2.0 i 2.0)))))
        3.0))
      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)) <= -0.99999999) {
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	} else {
		tmp = cbrt(pow((1.0 + ((alpha + beta) * (((beta - alpha) / (beta + fma(2.0, i, alpha))) / ((alpha + beta) + fma(2.0, i, 2.0))))), 3.0)) / 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)) <= -0.99999999)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(i * 4.0) + Float64(beta * 2.0)))) / alpha) / 2.0);
	else
		tmp = Float64(cbrt((Float64(1.0 + Float64(Float64(alpha + beta) * Float64(Float64(Float64(beta - alpha) / Float64(beta + fma(2.0, i, alpha))) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))))) ^ 3.0)) / 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], -0.99999999], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(i * 4.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Power[N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] * N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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))) < -0.99999998999999995

   1. Initial program 2.7%

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

   3. Simplified 17.3%

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

   5. Taylor expanded in alpha around inf 88.9%

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

   if -0.99999998999999995 < (/.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 80.8%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 99.7%

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

   8. Simplified 99.7%

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

      1. add-cbrt-cube 99.7%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \sqrt[3]{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right) \cdot \mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \sqrt[3]{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right)\right) \cdot \mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \sqrt[3]{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right)}}}{2} \]

      2. pow3 99.7%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}\right)}^{2}}, \sqrt[3]{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha\right) + \beta}}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}}, 1\right)\right)}^{3}}}}{2} \]

   10. Applied egg-rr 99.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}\right)}^{3}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.9995:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \end{array} \]

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)) -0.9995)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* i 4.0) (* beta 2.0)))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        (+ 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)) <= -0.9995) {
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / (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)) <= -0.9995)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(i * 4.0) + Float64(beta * 2.0)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + 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))))) / 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], -0.9995], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(i * 4.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + 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]), $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))) < -0.99950000000000006

   1. Initial program 3.8%

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

   3. Simplified 18.1%

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

   5. Taylor expanded in alpha around inf 88.3%

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

   if -0.99950000000000006 < (/.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 80.9%

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

   3. Simplified 99.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9995:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.9995:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \end{array} \]

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)) -0.9995)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* i 4.0) (* beta 2.0)))) 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)) <= -0.9995) {
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / 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)) <= -0.9995)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(i * 4.0) + Float64(beta * 2.0)))) / 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], -0.9995], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(i * 4.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $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))) < -0.99950000000000006

   1. Initial program 3.8%

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

   3. Simplified 18.1%

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

   5. Taylor expanded in alpha around inf 88.3%

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

   if -0.99950000000000006 < (/.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 80.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in alpha around 0 99.7%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9995:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{t\_0}{t\_1}}{2 + t\_1}\\ \mathbf{if}\;t\_2 \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\frac{1 + \frac{t\_0}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* (+ alpha beta) (- beta alpha)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (/ (/ t_0 t_1) (+ 2.0 t_1))))
   (if (<= t_2 -0.99999999)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* i 4.0) (* beta 2.0)))) alpha) 2.0)
     (if (<= t_2 0.2)
       (/
        (+
         1.0
         (/
          t_0
          (*
           (+ (+ alpha beta) (+ 2.0 (* 2.0 i)))
           (+ beta (+ alpha (* 2.0 i))))))
        2.0)
       (/ (- 2.0 (/ 2.0 beta)) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) * (beta - alpha);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (t_0 / t_1) / (2.0 + t_1);
	double tmp;
	if (t_2 <= -0.99999999) {
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	} else if (t_2 <= 0.2) {
		tmp = (1.0 + (t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i)))))) / 2.0;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 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) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) * (beta - alpha)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = (t_0 / t_1) / (2.0d0 + t_1)
    if (t_2 <= (-0.99999999d0)) then
        tmp = (((beta - beta) + (2.0d0 + ((i * 4.0d0) + (beta * 2.0d0)))) / alpha) / 2.0d0
    else if (t_2 <= 0.2d0) then
        tmp = (1.0d0 + (t_0 / (((alpha + beta) + (2.0d0 + (2.0d0 * i))) * (beta + (alpha + (2.0d0 * i)))))) / 2.0d0
    else
        tmp = (2.0d0 - (2.0d0 / beta)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) * (beta - alpha);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (t_0 / t_1) / (2.0 + t_1);
	double tmp;
	if (t_2 <= -0.99999999) {
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	} else if (t_2 <= 0.2) {
		tmp = (1.0 + (t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i)))))) / 2.0;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) * (beta - alpha)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = (t_0 / t_1) / (2.0 + t_1)
	tmp = 0
	if t_2 <= -0.99999999:
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0
	elif t_2 <= 0.2:
		tmp = (1.0 + (t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i)))))) / 2.0
	else:
		tmp = (2.0 - (2.0 / beta)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) * Float64(beta - alpha))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(Float64(t_0 / t_1) / Float64(2.0 + t_1))
	tmp = 0.0
	if (t_2 <= -0.99999999)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(i * 4.0) + Float64(beta * 2.0)))) / alpha) / 2.0);
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(1.0 + Float64(t_0 / Float64(Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))) * Float64(beta + Float64(alpha + Float64(2.0 * i)))))) / 2.0);
	else
		tmp = Float64(Float64(2.0 - Float64(2.0 / beta)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) * (beta - alpha);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = (t_0 / t_1) / (2.0 + t_1);
	tmp = 0.0;
	if (t_2 <= -0.99999999)
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	elseif (t_2 <= 0.2)
		tmp = (1.0 + (t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i)))))) / 2.0;
	else
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 / t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.99999999], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(i * 4.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$2, 0.2], N[(N[(1.0 + N[(t$95$0 / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 - N[(2.0 / beta), $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))) < -0.99999998999999995

   1. Initial program 2.7%

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

   3. Simplified 17.3%

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

   5. Taylor expanded in alpha around inf 88.9%

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

   if -0.99999998999999995 < (/.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))) < 0.20000000000000001

   1. Initial program 99.6%

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

      1. associate-/l/ 99.6%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   if 0.20000000000000001 < (/.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 40.4%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in i around 0 95.3%

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

      1. associate-+r+ 95.3%

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

      2. +-commutative 95.3%

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

   7. Simplified 95.3%

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

   8. Taylor expanded in beta around inf 95.3%

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

      1. mul-1-neg 95.3%

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

   10. Simplified 95.3%

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

   11. Taylor expanded in alpha around 0 95.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.2:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq 6 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.7 \cdot 10^{+123}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))
   (if (<= alpha 1.1e+19)
     t_0
     (if (<= alpha 6e+46)
       (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)
       (if (<= alpha 3.7e+123)
         t_0
         (/
          (/ (+ (- beta beta) (+ 2.0 (+ (* i 4.0) (* beta 2.0)))) alpha)
          2.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 1.1e+19) {
		tmp = t_0;
	} else if (alpha <= 6e+46) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	} else if (alpha <= 3.7e+123) {
		tmp = t_0;
	} else {
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    if (alpha <= 1.1d+19) then
        tmp = t_0
    else if (alpha <= 6d+46) then
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    else if (alpha <= 3.7d+123) then
        tmp = t_0
    else
        tmp = (((beta - beta) + (2.0d0 + ((i * 4.0d0) + (beta * 2.0d0)))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 1.1e+19) {
		tmp = t_0;
	} else if (alpha <= 6e+46) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	} else if (alpha <= 3.7e+123) {
		tmp = t_0;
	} else {
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0
	tmp = 0
	if alpha <= 1.1e+19:
		tmp = t_0
	elif alpha <= 6e+46:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	elif alpha <= 3.7e+123:
		tmp = t_0
	else:
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0)
	tmp = 0.0
	if (alpha <= 1.1e+19)
		tmp = t_0;
	elseif (alpha <= 6e+46)
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	elseif (alpha <= 3.7e+123)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(i * 4.0) + Float64(beta * 2.0)))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	tmp = 0.0;
	if (alpha <= 1.1e+19)
		tmp = t_0;
	elseif (alpha <= 6e+46)
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	elseif (alpha <= 3.7e+123)
		tmp = t_0;
	else
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 1.1e+19], t$95$0, If[LessEqual[alpha, 6e+46], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 3.7e+123], t$95$0, N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(i * 4.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.1e19 or 6.00000000000000047e46 < alpha < 3.69999999999999996e123

   1. Initial program 81.5%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in i around 0 86.6%

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

      1. associate-+r+ 86.6%

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

      2. +-commutative 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in alpha around 0 90.7%

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

      1. +-commutative 90.7%

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

   10. Simplified 90.7%

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

   if 1.1e19 < alpha < 6.00000000000000047e46

   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 24.8%

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

   5. Taylor expanded in alpha around inf 78.8%

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

   6. Taylor expanded in beta around 0 79.2%

      \[\leadsto \frac{\color{blue}{\frac{2 + 4 \cdot i}{\alpha}}}{2} \]

   if 3.69999999999999996e123 < alpha

   1. Initial program 1.5%

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

   3. Simplified 27.8%

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

   5. Taylor expanded in alpha around inf 78.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 6 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.7 \cdot 10^{+123}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+18} \lor \neg \left(\alpha \leq 2.05 \cdot 10^{+46}\right) \land \alpha \leq 3.4 \cdot 10^{+117}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (or (<= alpha 4.3e+18)
         (and (not (<= alpha 2.05e+46)) (<= alpha 3.4e+117)))
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((alpha <= 4.3e+18) || (!(alpha <= 2.05e+46) && (alpha <= 3.4e+117))) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((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 <= 4.3d+18) .or. (.not. (alpha <= 2.05d+46)) .and. (alpha <= 3.4d+117)) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((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 <= 4.3e+18) || (!(alpha <= 2.05e+46) && (alpha <= 3.4e+117))) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (alpha <= 4.3e+18) or (not (alpha <= 2.05e+46) and (alpha <= 3.4e+117)):
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if ((alpha <= 4.3e+18) || (!(alpha <= 2.05e+46) && (alpha <= 3.4e+117)))
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(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 <= 4.3e+18) || (~((alpha <= 2.05e+46)) && (alpha <= 3.4e+117)))
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[Or[LessEqual[alpha, 4.3e+18], And[N[Not[LessEqual[alpha, 2.05e+46]], $MachinePrecision], LessEqual[alpha, 3.4e+117]]], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.3e18 or 2.05e46 < alpha < 3.4000000000000001e117

   1. Initial program 81.5%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in i around 0 86.6%

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

      1. associate-+r+ 86.6%

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

      2. +-commutative 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in alpha around 0 90.7%

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

      1. +-commutative 90.7%

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

   10. Simplified 90.7%

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

   if 4.3e18 < alpha < 2.05e46 or 3.4000000000000001e117 < alpha

   1. Initial program 1.8%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in alpha around inf 78.6%

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

   6. Taylor expanded in beta around 0 68.1%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+18} \lor \neg \left(\alpha \leq 2.05 \cdot 10^{+46}\right) \land \alpha \leq 3.4 \cdot 10^{+117}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.06 \cdot 10^{+122}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.6 \cdot 10^{+253}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.06e+122)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (<= alpha 2.6e+253)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (* 4.0 (/ i alpha)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.06e+122) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 2.6e+253) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (4.0 * (i / 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 <= 1.06d+122) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if (alpha <= 2.6d+253) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (4.0d0 * (i / alpha)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.06e+122) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 2.6e+253) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (4.0 * (i / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.06e+122:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif alpha <= 2.6e+253:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (4.0 * (i / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.06e+122)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif (alpha <= 2.6e+253)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(4.0 * Float64(i / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.06e+122)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif (alpha <= 2.6e+253)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = (4.0 * (i / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.06e+122], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 2.6e+253], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.06000000000000002e122

   1. Initial program 78.2%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in i around 0 83.6%

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

      1. associate-+r+ 83.6%

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

      2. +-commutative 83.6%

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

   7. Simplified 83.6%

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

   8. Taylor expanded in alpha around 0 87.7%

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

      1. +-commutative 87.7%

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

   10. Simplified 87.7%

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

   if 1.06000000000000002e122 < alpha < 2.6e253

   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. Step-by-step derivation

   3. Simplified 34.8%

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

   5. Taylor expanded in i around 0 11.8%

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

      1. associate-+r+ 11.8%

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

      2. +-commutative 11.8%

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

   7. Simplified 11.8%

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

   8. Taylor expanded in alpha around inf 49.6%

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

   if 2.6e253 < alpha

   1. Initial program 1.2%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in alpha around inf 92.1%

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

   6. Taylor expanded in i around inf 73.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.06 \cdot 10^{+122}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.6 \cdot 10^{+253}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.92 \cdot 10^{+199}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.92e+199)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (* 4.0 (/ i alpha)) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.92e+199) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (4.0 * (i / 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 <= 1.92d+199) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (4.0d0 * (i / alpha)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.92e+199) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (4.0 * (i / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.92e+199:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (4.0 * (i / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.92e+199)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(4.0 * Float64(i / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.92e+199)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (4.0 * (i / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.92e+199], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.9200000000000001e199

   1. Initial program 72.8%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in i around 0 78.9%

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

      1. associate-+r+ 78.9%

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

      2. +-commutative 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in alpha around 0 83.5%

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

      1. +-commutative 83.5%

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

   10. Simplified 83.5%

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

   if 1.9200000000000001e199 < alpha

   1. Initial program 1.2%

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

   3. Simplified 17.5%

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

   5. Taylor expanded in alpha around inf 88.8%

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

   6. Taylor expanded in i around inf 53.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.92 \cdot 10^{+199}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+16}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 7e+16) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7e+16) {
		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 <= 7d+16) 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 <= 7e+16) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 7e+16:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7e+16)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7e+16)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 7e+16], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 7e16

   1. Initial program 75.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in i around inf 75.5%

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

   if 7e16 < beta

   1. Initial program 43.1%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in beta around inf 77.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+16}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.4% 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 65.3%

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

3. Simplified 83.3%

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

5. Taylor expanded in i around inf 61.6%

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

6. Final simplification 61.6%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024087 
(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))