Octave 3.8, jcobi/4

Percentage Accurate: 16.5% → 84.4%

Time: 10.3s

Alternatives: 9

Speedup: 115.0×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

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
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

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

Initial Program: 16.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

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
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

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

Alternative 1: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \left(\beta + \alpha\right) + i\\ t_4 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_5 := -t\_4\\ t_6 := t\_4 - 1\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_3, i, \beta \cdot \alpha\right)}{t\_5}}{t\_6} \cdot \frac{\frac{t\_3 \cdot i}{t\_5}}{1 + t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{t\_4}}{t\_6} \cdot \left(i \cdot \left(\frac{0.25 \cdot \beta}{i} + 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i)))
        (t_3 (+ (+ beta alpha) i))
        (t_4 (fma 2.0 i (+ beta alpha)))
        (t_5 (- t_4))
        (t_6 (- t_4 1.0)))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) INFINITY)
     (*
      (/ (/ (fma t_3 i (* beta alpha)) t_5) t_6)
      (/ (/ (* t_3 i) t_5) (+ 1.0 t_4)))
     (* (/ (/ i t_4) t_6) (* i (+ (/ (* 0.25 beta) i) 0.25))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double t_3 = (beta + alpha) + i;
	double t_4 = fma(2.0, i, (beta + alpha));
	double t_5 = -t_4;
	double t_6 = t_4 - 1.0;
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= ((double) INFINITY)) {
		tmp = ((fma(t_3, i, (beta * alpha)) / t_5) / t_6) * (((t_3 * i) / t_5) / (1.0 + t_4));
	} else {
		tmp = ((i / t_4) / t_6) * (i * (((0.25 * beta) / i) + 0.25));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_3 = Float64(Float64(beta + alpha) + i)
	t_4 = fma(2.0, i, Float64(beta + alpha))
	t_5 = Float64(-t_4)
	t_6 = Float64(t_4 - 1.0)
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= Inf)
		tmp = Float64(Float64(Float64(fma(t_3, i, Float64(beta * alpha)) / t_5) / t_6) * Float64(Float64(Float64(t_3 * i) / t_5) / Float64(1.0 + t_4)));
	else
		tmp = Float64(Float64(Float64(i / t_4) / t_6) * Float64(i * Float64(Float64(Float64(0.25 * beta) / i) + 0.25)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-t$95$4)}, Block[{t$95$6 = N[(t$95$4 - 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(t$95$3 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] / t$95$6), $MachinePrecision] * N[(N[(N[(t$95$3 * i), $MachinePrecision] / t$95$5), $MachinePrecision] / N[(1.0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i / t$95$4), $MachinePrecision] / t$95$6), $MachinePrecision] * N[(i * N[(N[(N[(0.25 * beta), $MachinePrecision] / i), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 48.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-neg-rev N/A

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

      7. times-frac N/A

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

      8. lift--.f64 N/A

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

   4. Applied rewrites 99.6%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift--.f64 N/A

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

   4. Applied rewrites 2.5%

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

   5. Taylor expanded in i around -inf

      \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \color{blue}{\left(-1 \cdot \left(i \cdot \left(-1 \cdot \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{-1}{16} \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right)}{i} - \frac{1}{4}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \color{blue}{\left(\left(-1 \cdot i\right) \cdot \left(-1 \cdot \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{-1}{16} \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right)}{i} - \frac{1}{4}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \color{blue}{\left(\left(-1 \cdot i\right) \cdot \left(-1 \cdot \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{-1}{16} \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right)}{i} - \frac{1}{4}\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right)} \cdot \left(-1 \cdot \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{-1}{16} \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right)}{i} - \frac{1}{4}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \left(\color{blue}{\left(-i\right)} \cdot \left(-1 \cdot \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{-1}{16} \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right)}{i} - \frac{1}{4}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \left(\left(-i\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{-1}{16} \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right)}{i} - \frac{1}{4}\right)}\right) \]

   7. Applied rewrites 81.1%

      \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \color{blue}{\left(\left(-i\right) \cdot \left(\left(-\frac{\mathsf{fma}\left(0.25, 2 \cdot \left(\alpha + \beta\right), 0.0625 \cdot \left(-2 \cdot \left(\left(1 + \left(\alpha + \beta\right)\right) + \left(\alpha + \beta\right)\right)\right)\right)}{i}\right) - 0.25\right)\right)} \]

   8. Taylor expanded in beta around inf

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

   10. Applied rewrites 81.3%

       \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \left(\left(-i\right) \cdot \left(\left(-\frac{0.25 \cdot \beta}{i}\right) - 0.25\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\beta + \alpha\right) + i, i, \beta \cdot \alpha\right)}{-\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \frac{\frac{\left(\left(\beta + \alpha\right) + i\right) \cdot i}{-\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \left(i \cdot \left(\frac{0.25 \cdot \beta}{i} + 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+123}:\\ \;\;\;\;\frac{i}{t\_0} \cdot \frac{\left(\frac{\mathsf{fma}\left(0.5, \beta + \alpha, -0.125 \cdot \mathsf{fma}\left(2, \beta + \alpha, 1\right)\right)}{i} + 0.25\right) \cdot i}{t\_0 - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ beta alpha))))
   (if (<= beta 1.56e+123)
     (*
      (/ i t_0)
      (/
       (*
        (+
         (/ (fma 0.5 (+ beta alpha) (* -0.125 (fma 2.0 (+ beta alpha) 1.0))) i)
         0.25)
        i)
       (- t_0 1.0)))
     (* (/ (+ alpha i) beta) (/ i beta)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if (beta <= 1.56e+123) {
		tmp = (i / t_0) * ((((fma(0.5, (beta + alpha), (-0.125 * fma(2.0, (beta + alpha), 1.0))) / i) + 0.25) * i) / (t_0 - 1.0));
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.56e+123)
		tmp = Float64(Float64(i / t_0) * Float64(Float64(Float64(Float64(fma(0.5, Float64(beta + alpha), Float64(-0.125 * fma(2.0, Float64(beta + alpha), 1.0))) / i) + 0.25) * i) / Float64(t_0 - 1.0)));
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(i / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.56e+123], N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(N[(N[(N[(0.5 * N[(beta + alpha), $MachinePrecision] + N[(-0.125 * N[(2.0 * N[(beta + alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + 0.25), $MachinePrecision] * i), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.56e123

   1. Initial program 20.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift--.f64 N/A

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

   4. Applied rewrites 30.0%

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

   5. Taylor expanded in i around -inf

      \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \color{blue}{\left(-1 \cdot \left(i \cdot \left(-1 \cdot \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{-1}{16} \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right)}{i} - \frac{1}{4}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \color{blue}{\left(\left(-1 \cdot i\right) \cdot \left(-1 \cdot \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{-1}{16} \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right)}{i} - \frac{1}{4}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \color{blue}{\left(\left(-1 \cdot i\right) \cdot \left(-1 \cdot \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{-1}{16} \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right)}{i} - \frac{1}{4}\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right)} \cdot \left(-1 \cdot \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{-1}{16} \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right)}{i} - \frac{1}{4}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \left(\color{blue}{\left(-i\right)} \cdot \left(-1 \cdot \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{-1}{16} \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right)}{i} - \frac{1}{4}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \left(\left(-i\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{-1}{16} \cdot \left(-2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + -2 \cdot \left(\alpha + \beta\right)\right)}{i} - \frac{1}{4}\right)}\right) \]

   7. Applied rewrites 86.1%

      \[\leadsto \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \color{blue}{\left(\left(-i\right) \cdot \left(\left(-\frac{\mathsf{fma}\left(0.25, 2 \cdot \left(\alpha + \beta\right), 0.0625 \cdot \left(-2 \cdot \left(\left(1 + \left(\alpha + \beta\right)\right) + \left(\alpha + \beta\right)\right)\right)\right)}{i}\right) - 0.25\right)\right)} \]

   9. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\left(\frac{\mathsf{fma}\left(0.5, \beta + \alpha, -0.125 \cdot \mathsf{fma}\left(2, \beta + \alpha, 1\right)\right)}{-i} - 0.25\right) \cdot \left(-i\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) - 1}} \]

   if 1.56e123 < beta

   1. Initial program 2.2%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+123}:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\left(\frac{\mathsf{fma}\left(0.5, \beta + \alpha, -0.125 \cdot \mathsf{fma}\left(2, \beta + \alpha, 1\right)\right)}{i} + 0.25\right) \cdot i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+123}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.56e+123) 0.0625 (* (/ (+ alpha i) beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.56e+123) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	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 <= 1.56d+123) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + i) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.56e+123) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.56e+123:
		tmp = 0.0625
	else:
		tmp = ((alpha + i) / beta) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.56e+123)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.56e+123)
		tmp = 0.0625;
	else
		tmp = ((alpha + i) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.56e+123], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.56e123

   1. Initial program 20.9%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 80.9%

      \[\leadsto \color{blue}{0.0625} \]

   if 1.56e123 < beta

   1. Initial program 2.2%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

Alternative 4: 76.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+123}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.56e+123) 0.0625 (/ (* (/ i beta) i) beta)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.56e+123) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * i) / beta;
	}
	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 <= 1.56d+123) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) * i) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.56e+123) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * i) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.56e+123:
		tmp = 0.0625
	else:
		tmp = ((i / beta) * i) / beta
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.56e+123)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) * i) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.56e+123)
		tmp = 0.0625;
	else
		tmp = ((i / beta) * i) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.56e+123], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] * i), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.56e123

   1. Initial program 20.9%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 80.9%

      \[\leadsto \color{blue}{0.0625} \]

   if 1.56e123 < beta

   1. Initial program 2.2%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{i}{\beta} \cdot \frac{\color{blue}{i}}{\beta} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.1%

      \[\leadsto \frac{i}{\beta} \cdot \frac{\color{blue}{i}}{\beta} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.2%

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

Alternative 5: 76.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+123}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.56e+123) 0.0625 (* (/ i beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.56e+123) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	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 <= 1.56d+123) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.56e+123) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.56e+123:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.56e+123)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.56e+123)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.56e+123], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.56e123

   1. Initial program 20.9%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 80.9%

      \[\leadsto \color{blue}{0.0625} \]

   if 1.56e123 < beta

   1. Initial program 2.2%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{i}{\beta} \cdot \frac{\color{blue}{i}}{\beta} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.1%

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

Alternative 6: 73.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+123}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\beta} \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.56e+123) 0.0625 (* (/ (/ i beta) beta) i)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.56e+123) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) / beta) * i;
	}
	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 <= 1.56d+123) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) / beta) * i
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.56e+123) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) / beta) * i;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.56e+123:
		tmp = 0.0625
	else:
		tmp = ((i / beta) / beta) * i
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.56e+123)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) / beta) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.56e+123)
		tmp = 0.0625;
	else
		tmp = ((i / beta) / beta) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.56e+123], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision] * i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.56e123

   1. Initial program 20.9%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 80.9%

      \[\leadsto \color{blue}{0.0625} \]

   if 1.56e123 < beta

   1. Initial program 2.2%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{{i}^{2}}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.3%

      \[\leadsto \frac{i \cdot i}{\color{blue}{\beta \cdot \beta}} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.8%

       \[\leadsto \frac{\frac{i}{\beta}}{\beta} \cdot i \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 72.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+244}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha + i\right) \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8.5e+244) 0.0625 (* (+ alpha i) (/ i (* beta beta)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.5e+244) {
		tmp = 0.0625;
	} else {
		tmp = (alpha + i) * (i / (beta * beta));
	}
	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 <= 8.5d+244) then
        tmp = 0.0625d0
    else
        tmp = (alpha + i) * (i / (beta * beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.5e+244) {
		tmp = 0.0625;
	} else {
		tmp = (alpha + i) * (i / (beta * beta));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 8.5e+244:
		tmp = 0.0625
	else:
		tmp = (alpha + i) * (i / (beta * beta))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8.5e+244)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(alpha + i) * Float64(i / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8.5e+244)
		tmp = 0.0625;
	else
		tmp = (alpha + i) * (i / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 8.5e+244], 0.0625, N[(N[(alpha + i), $MachinePrecision] * N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.49999999999999995e244

   1. Initial program 18.3%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 74.8%

      \[\leadsto \color{blue}{0.0625} \]

   if 8.49999999999999995e244 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 90.7

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 52.2%

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

Alternative 8: 72.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+244}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha \cdot i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8.5e+244) 0.0625 (/ (* alpha i) (* beta beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.5e+244) {
		tmp = 0.0625;
	} else {
		tmp = (alpha * i) / (beta * beta);
	}
	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 <= 8.5d+244) then
        tmp = 0.0625d0
    else
        tmp = (alpha * i) / (beta * beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.5e+244) {
		tmp = 0.0625;
	} else {
		tmp = (alpha * i) / (beta * beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 8.5e+244:
		tmp = 0.0625
	else:
		tmp = (alpha * i) / (beta * beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8.5e+244)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(alpha * i) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8.5e+244)
		tmp = 0.0625;
	else
		tmp = (alpha * i) / (beta * beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 8.5e+244], 0.0625, N[(N[(alpha * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.49999999999999995e244

   1. Initial program 18.3%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 74.8%

      \[\leadsto \color{blue}{0.0625} \]

   if 8.49999999999999995e244 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 90.7

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in alpha around inf

      \[\leadsto \frac{\alpha \cdot i}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.0%

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

Alternative 9: 70.7% accurate, 115.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.0625)

C

double code(double alpha, double beta, double i) {
	return 0.0625;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

def code(alpha, beta, i):
	return 0.0625

Julia

function code(alpha, beta, i)
	return 0.0625
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 17.3%

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

3. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\frac{1}{16}} \]
4. Step-by-step derivation

5. Applied rewrites 71.3%

   \[\leadsto \color{blue}{0.0625} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024343 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))