Octave 3.8, jcobi/4

Percentage Accurate: 16.1% → 83.4%

Time: 5.2s

Alternatives: 11

Speedup: 75.4×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    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: 16.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    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: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \left(\beta + \alpha\right) + i\\ t_4 := \mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)\\ t_5 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq \infty:\\ \;\;\;\;\frac{t\_3}{t\_5 - -1} \cdot \frac{\frac{\mathsf{fma}\left(t\_3, i, \beta \cdot \alpha\right)}{t\_5 \cdot t\_5} \cdot i}{t\_5 - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\left(\beta + \alpha\right) \cdot \frac{0.125}{i}}{t\_4}\right) \cdot t\_4\\ \end{array} \end{array} \]

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 (+ beta alpha) (/ -0.125 i) 0.0625))
        (t_5 (fma 2.0 i (+ beta alpha))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) INFINITY)
     (*
      (/ t_3 (- t_5 -1.0))
      (/ (* (/ (fma t_3 i (* beta alpha)) (* t_5 t_5)) i) (- t_5 1.0)))
     (* (+ 1.0 (/ (* (+ beta alpha) (/ 0.125 i)) t_4)) t_4))))

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((beta + alpha), (-0.125 / i), 0.0625);
	double t_5 = fma(2.0, i, (beta + alpha));
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= ((double) INFINITY)) {
		tmp = (t_3 / (t_5 - -1.0)) * (((fma(t_3, i, (beta * alpha)) / (t_5 * t_5)) * i) / (t_5 - 1.0));
	} else {
		tmp = (1.0 + (((beta + alpha) * (0.125 / i)) / t_4)) * t_4;
	}
	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(Float64(beta + alpha), Float64(-0.125 / i), 0.0625)
	t_5 = fma(2.0, i, Float64(beta + alpha))
	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(t_3 / Float64(t_5 - -1.0)) * Float64(Float64(Float64(fma(t_3, i, Float64(beta * alpha)) / Float64(t_5 * t_5)) * i) / Float64(t_5 - 1.0)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta + alpha) * Float64(0.125 / i)) / t_4)) * t_4);
	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[(N[(beta + alpha), $MachinePrecision] * N[(-0.125 / i), $MachinePrecision] + 0.0625), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $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[(t$95$3 / N[(t$95$5 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(t$95$3 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision] / N[(t$95$5 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta + alpha), $MachinePrecision] * N[(0.125 / i), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] * t$95$4), $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 16.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

   3. Applied rewrites 24.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-fma.f64 N/A

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

      5. difference-of-sqr--1 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 37.4%

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

   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 16.1%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 76.8

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. sum-to-mult N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      4. lower-unsound-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      5. lower-unsound-/.f64 70.4

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

   6. Applied rewrites 70.4%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

   8. Applied rewrites 69.3%

      \[\leadsto \mathsf{fma}\left(\frac{\beta}{\alpha} - -1, \color{blue}{\frac{\alpha}{i} \cdot 0.125}, 0.0625 - \frac{\beta + \alpha}{i} \cdot 0.125\right) \]

   10. Applied rewrites 76.8%

       \[\leadsto \left(1 + \frac{\left(\beta + \alpha\right) \cdot \frac{0.125}{i}}{\mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \left(\beta + \alpha\right) + i\\ t_3 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_4 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_5 := \mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)\\ \mathbf{if}\;\frac{\frac{t\_4 \cdot \left(\beta \cdot \alpha + t\_4\right)}{t\_1}}{t\_1 - 1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(t\_3, t\_3, -1\right)} \cdot \frac{t\_2 \cdot i}{t\_3 \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\left(\beta + \alpha\right) \cdot \frac{0.125}{i}}{t\_5}\right) \cdot t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (+ (+ beta alpha) i))
        (t_3 (fma 2.0 i (+ beta alpha)))
        (t_4 (* i (+ (+ alpha beta) i)))
        (t_5 (fma (+ beta alpha) (/ -0.125 i) 0.0625)))
   (if (<= (/ (/ (* t_4 (+ (* beta alpha) t_4)) t_1) (- t_1 1.0)) INFINITY)
     (*
      (/ (fma t_2 i (* beta alpha)) (fma t_3 t_3 -1.0))
      (/ (* t_2 i) (* t_3 t_3)))
     (* (+ 1.0 (/ (* (+ beta alpha) (/ 0.125 i)) t_5)) t_5))))

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 = (beta + alpha) + i;
	double t_3 = fma(2.0, i, (beta + alpha));
	double t_4 = i * ((alpha + beta) + i);
	double t_5 = fma((beta + alpha), (-0.125 / i), 0.0625);
	double tmp;
	if ((((t_4 * ((beta * alpha) + t_4)) / t_1) / (t_1 - 1.0)) <= ((double) INFINITY)) {
		tmp = (fma(t_2, i, (beta * alpha)) / fma(t_3, t_3, -1.0)) * ((t_2 * i) / (t_3 * t_3));
	} else {
		tmp = (1.0 + (((beta + alpha) * (0.125 / i)) / t_5)) * t_5;
	}
	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(Float64(beta + alpha) + i)
	t_3 = fma(2.0, i, Float64(beta + alpha))
	t_4 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_5 = fma(Float64(beta + alpha), Float64(-0.125 / i), 0.0625)
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(Float64(beta * alpha) + t_4)) / t_1) / Float64(t_1 - 1.0)) <= Inf)
		tmp = Float64(Float64(fma(t_2, i, Float64(beta * alpha)) / fma(t_3, t_3, -1.0)) * Float64(Float64(t_2 * i) / Float64(t_3 * t_3)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta + alpha) * Float64(0.125 / i)) / t_5)) * t_5);
	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[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(beta + alpha), $MachinePrecision] * N[(-0.125 / i), $MachinePrecision] + 0.0625), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(N[(beta * alpha), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(t$95$2 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * i), $MachinePrecision] / N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta + alpha), $MachinePrecision] * N[(0.125 / i), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision] * t$95$5), $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 16.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

   3. Applied rewrites 37.4%

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

   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 16.1%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 76.8

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. sum-to-mult N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      4. lower-unsound-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      5. lower-unsound-/.f64 70.4

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

   6. Applied rewrites 70.4%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

   8. Applied rewrites 69.3%

      \[\leadsto \mathsf{fma}\left(\frac{\beta}{\alpha} - -1, \color{blue}{\frac{\alpha}{i} \cdot 0.125}, 0.0625 - \frac{\beta + \alpha}{i} \cdot 0.125\right) \]

   10. Applied rewrites 76.8%

       \[\leadsto \left(1 + \frac{\left(\beta + \alpha\right) \cdot \frac{0.125}{i}}{\mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta + i, i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(2, i, \beta\right) \cdot \mathsf{fma}\left(2, i, \beta\right)} \cdot \frac{\left(\beta + i\right) \cdot i}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta\right), \mathsf{fma}\left(2, i, \beta\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\left(\beta + \alpha\right) \cdot \frac{0.125}{i}}{t\_3}\right) \cdot t\_3\\ \end{array} \end{array} \]

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 (fma (+ beta alpha) (/ -0.125 i) 0.0625)))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) INFINITY)
     (*
      (/
       (fma (+ beta i) i (* beta alpha))
       (* (fma 2.0 i beta) (fma 2.0 i beta)))
      (/ (* (+ beta i) i) (fma (fma 2.0 i beta) (fma 2.0 i beta) -1.0)))
     (* (+ 1.0 (/ (* (+ beta alpha) (/ 0.125 i)) t_3)) t_3))))

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 = fma((beta + alpha), (-0.125 / i), 0.0625);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= ((double) INFINITY)) {
		tmp = (fma((beta + i), i, (beta * alpha)) / (fma(2.0, i, beta) * fma(2.0, i, beta))) * (((beta + i) * i) / fma(fma(2.0, i, beta), fma(2.0, i, beta), -1.0));
	} else {
		tmp = (1.0 + (((beta + alpha) * (0.125 / i)) / t_3)) * t_3;
	}
	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 = fma(Float64(beta + alpha), Float64(-0.125 / i), 0.0625)
	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(fma(Float64(beta + i), i, Float64(beta * alpha)) / Float64(fma(2.0, i, beta) * fma(2.0, i, beta))) * Float64(Float64(Float64(beta + i) * i) / fma(fma(2.0, i, beta), fma(2.0, i, beta), -1.0)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta + alpha) * Float64(0.125 / i)) / t_3)) * t_3);
	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] * N[(-0.125 / i), $MachinePrecision] + 0.0625), $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[(beta + i), $MachinePrecision] * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(beta + i), $MachinePrecision] * i), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(2.0 * i + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta + alpha), $MachinePrecision] * N[(0.125 / i), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$3), $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 16.1%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 15.4%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 16.4%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 16.5%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 16.6%

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

   14. Taylor expanded in alpha around 0

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

   16. Applied rewrites 15.1%

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

   17. Taylor expanded in alpha around 0

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

   19. Applied rewrites 14.7%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

   21. Applied rewrites 33.0%

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

   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 16.1%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 76.8

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. sum-to-mult N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      4. lower-unsound-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      5. lower-unsound-/.f64 70.4

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

   6. Applied rewrites 70.4%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

   8. Applied rewrites 69.3%

      \[\leadsto \mathsf{fma}\left(\frac{\beta}{\alpha} - -1, \color{blue}{\frac{\alpha}{i} \cdot 0.125}, 0.0625 - \frac{\beta + \alpha}{i} \cdot 0.125\right) \]

   10. Applied rewrites 76.8%

       \[\leadsto \left(1 + \frac{\left(\beta + \alpha\right) \cdot \frac{0.125}{i}}{\mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \beta + 2 \cdot i\\ t_3 := t\_2 \cdot t\_2\\ t_4 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_5 := \mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)\\ \mathbf{if}\;\frac{\frac{t\_4 \cdot \left(\beta \cdot \alpha + t\_4\right)}{t\_1}}{t\_1 - 1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(\beta + i\right)\right) \cdot \left(\beta \cdot \left(\alpha + i\right)\right)}{t\_3}}{t\_3 - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\left(\beta + \alpha\right) \cdot \frac{0.125}{i}}{t\_5}\right) \cdot t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (+ beta (* 2.0 i)))
        (t_3 (* t_2 t_2))
        (t_4 (* i (+ (+ alpha beta) i)))
        (t_5 (fma (+ beta alpha) (/ -0.125 i) 0.0625)))
   (if (<= (/ (/ (* t_4 (+ (* beta alpha) t_4)) t_1) (- t_1 1.0)) 5e-12)
     (/ (/ (* (* i (+ beta i)) (* beta (+ alpha i))) t_3) (- t_3 1.0))
     (* (+ 1.0 (/ (* (+ beta alpha) (/ 0.125 i)) t_5)) t_5))))

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 = beta + (2.0 * i);
	double t_3 = t_2 * t_2;
	double t_4 = i * ((alpha + beta) + i);
	double t_5 = fma((beta + alpha), (-0.125 / i), 0.0625);
	double tmp;
	if ((((t_4 * ((beta * alpha) + t_4)) / t_1) / (t_1 - 1.0)) <= 5e-12) {
		tmp = (((i * (beta + i)) * (beta * (alpha + i))) / t_3) / (t_3 - 1.0);
	} else {
		tmp = (1.0 + (((beta + alpha) * (0.125 / i)) / t_5)) * t_5;
	}
	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(beta + Float64(2.0 * i))
	t_3 = Float64(t_2 * t_2)
	t_4 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_5 = fma(Float64(beta + alpha), Float64(-0.125 / i), 0.0625)
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(Float64(beta * alpha) + t_4)) / t_1) / Float64(t_1 - 1.0)) <= 5e-12)
		tmp = Float64(Float64(Float64(Float64(i * Float64(beta + i)) * Float64(beta * Float64(alpha + i))) / t_3) / Float64(t_3 - 1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta + alpha) * Float64(0.125 / i)) / t_5)) * t_5);
	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[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(beta + alpha), $MachinePrecision] * N[(-0.125 / i), $MachinePrecision] + 0.0625), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(N[(beta * alpha), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], 5e-12], N[(N[(N[(N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision] * N[(beta * N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(t$95$3 - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta + alpha), $MachinePrecision] * N[(0.125 / i), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision] * t$95$5), $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))) < 4.9999999999999997e-12

   1. Initial program 16.1%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 15.4%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 16.4%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 16.5%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 16.6%

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

   14. Taylor expanded in alpha around 0

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

   16. Applied rewrites 15.1%

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

   17. Taylor expanded in alpha around 0

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

   19. Applied rewrites 14.7%

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

   20. Taylor expanded in beta around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 2.9

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

   22. Applied rewrites 2.9%

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

   if 4.9999999999999997e-12 < (/.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 16.1%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 76.8

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. sum-to-mult N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      4. lower-unsound-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      5. lower-unsound-/.f64 70.4

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

   6. Applied rewrites 70.4%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

   8. Applied rewrites 69.3%

      \[\leadsto \mathsf{fma}\left(\frac{\beta}{\alpha} - -1, \color{blue}{\frac{\alpha}{i} \cdot 0.125}, 0.0625 - \frac{\beta + \alpha}{i} \cdot 0.125\right) \]

   10. Applied rewrites 76.8%

       \[\leadsto \left(1 + \frac{\left(\beta + \alpha\right) \cdot \frac{0.125}{i}}{\mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := t\_1 - 1\\ t_4 := \mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_3} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1 \cdot \left(i \cdot \mathsf{fma}\left(-1, \alpha, -1 \cdot i\right)\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\left(\beta + \alpha\right) \cdot \frac{0.125}{i}}{t\_4}\right) \cdot t\_4\\ \end{array} \end{array} \]

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 (- t_1 1.0))
        (t_4 (fma (+ beta alpha) (/ -0.125 i) 0.0625)))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) t_3) 5e-12)
     (/ (* -1.0 (* i (fma -1.0 alpha (* -1.0 i)))) t_3)
     (* (+ 1.0 (/ (* (+ beta alpha) (/ 0.125 i)) t_4)) t_4))))

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 = t_1 - 1.0;
	double t_4 = fma((beta + alpha), (-0.125 / i), 0.0625);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / t_3) <= 5e-12) {
		tmp = (-1.0 * (i * fma(-1.0, alpha, (-1.0 * i)))) / t_3;
	} else {
		tmp = (1.0 + (((beta + alpha) * (0.125 / i)) / t_4)) * t_4;
	}
	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(t_1 - 1.0)
	t_4 = fma(Float64(beta + alpha), Float64(-0.125 / i), 0.0625)
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / t_3) <= 5e-12)
		tmp = Float64(Float64(-1.0 * Float64(i * fma(-1.0, alpha, Float64(-1.0 * i)))) / t_3);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta + alpha) * Float64(0.125 / i)) / t_4)) * t_4);
	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[(t$95$1 - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(beta + alpha), $MachinePrecision] * N[(-0.125 / i), $MachinePrecision] + 0.0625), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision], 5e-12], N[(N[(-1.0 * N[(i * N[(-1.0 * alpha + N[(-1.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta + alpha), $MachinePrecision] * N[(0.125 / i), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] * t$95$4), $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))) < 4.9999999999999997e-12

   1. Initial program 16.1%

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

   2. Taylor expanded in beta around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 13.5

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

   4. Applied rewrites 13.5%

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

   if 4.9999999999999997e-12 < (/.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 16.1%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 76.8

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. sum-to-mult N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      4. lower-unsound-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

      5. lower-unsound-/.f64 70.4

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

   6. Applied rewrites 70.4%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\left(1 + \frac{\beta}{\alpha}\right) \cdot \alpha}{i} \]

   8. Applied rewrites 69.3%

      \[\leadsto \mathsf{fma}\left(\frac{\beta}{\alpha} - -1, \color{blue}{\frac{\alpha}{i} \cdot 0.125}, 0.0625 - \frac{\beta + \alpha}{i} \cdot 0.125\right) \]

   10. Applied rewrites 76.8%

       \[\leadsto \left(1 + \frac{\left(\beta + \alpha\right) \cdot \frac{0.125}{i}}{\mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\beta + \alpha, \frac{-0.125}{i}, 0.0625\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 - 1\\ t_3 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(\beta \cdot \alpha + t\_3\right)}{t\_1}}{t\_2} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1 \cdot \left(i \cdot \mathsf{fma}\left(-1, \alpha, -1 \cdot i\right)\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{i}, -0.125, \frac{\beta}{i} \cdot 0.125 - -0.0625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (- t_1 1.0))
        (t_3 (* i (+ (+ alpha beta) i))))
   (if (<= (/ (/ (* t_3 (+ (* beta alpha) t_3)) t_1) t_2) 5e-12)
     (/ (* -1.0 (* i (fma -1.0 alpha (* -1.0 i)))) t_2)
     (fma (/ beta i) -0.125 (- (* (/ beta i) 0.125) -0.0625)))))

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 = t_1 - 1.0;
	double t_3 = i * ((alpha + beta) + i);
	double tmp;
	if ((((t_3 * ((beta * alpha) + t_3)) / t_1) / t_2) <= 5e-12) {
		tmp = (-1.0 * (i * fma(-1.0, alpha, (-1.0 * i)))) / t_2;
	} else {
		tmp = fma((beta / i), -0.125, (((beta / i) * 0.125) - -0.0625));
	}
	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(t_1 - 1.0)
	t_3 = Float64(i * Float64(Float64(alpha + beta) + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(Float64(beta * alpha) + t_3)) / t_1) / t_2) <= 5e-12)
		tmp = Float64(Float64(-1.0 * Float64(i * fma(-1.0, alpha, Float64(-1.0 * i)))) / t_2);
	else
		tmp = fma(Float64(beta / i), -0.125, Float64(Float64(Float64(beta / i) * 0.125) - -0.0625));
	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[(t$95$1 - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(N[(beta * alpha), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], 5e-12], N[(N[(-1.0 * N[(i * N[(-1.0 * alpha + N[(-1.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(beta / i), $MachinePrecision] * -0.125 + N[(N[(N[(beta / i), $MachinePrecision] * 0.125), $MachinePrecision] - -0.0625), $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))) < 4.9999999999999997e-12

   1. Initial program 16.1%

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

   2. Taylor expanded in beta around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 13.5

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

   4. Applied rewrites 13.5%

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

   if 4.9999999999999997e-12 < (/.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 16.1%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 76.8

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. lower-/.f64 73.0

         \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied rewrites 73.0%

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\beta}{\color{blue}{i}} \]
   9. Step-by-step derivation

      1. lower-/.f64 74.1

         \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i} \]

   10. Applied rewrites 74.1%

       \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{\color{blue}{i}} \]
   11. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\beta}{i}} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\beta}{i}} \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\beta}{i}} \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\beta}{i} + \color{blue}{\left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right)} \]

       5. metadata-eval N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\beta}{i} + \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) \]

       6. *-commutative N/A

          \[\leadsto \frac{\beta}{i} \cdot \frac{-1}{8} + \left(\color{blue}{\frac{1}{16}} + \frac{1}{8} \cdot \frac{\beta}{i}\right) \]

       7. lower-fma.f64 74.1

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \color{blue}{-0.125}, 0.0625 + 0.125 \cdot \frac{\beta}{i}\right) \]

       8. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{-1}{8}, \frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{-1}{8}, \frac{1}{8} \cdot \frac{\beta}{i} + \frac{1}{16}\right) \]

   12. Applied rewrites 74.1%

       \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \color{blue}{-0.125}, \frac{\beta}{i} \cdot 0.125 - -0.0625\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \beta + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1 \cdot \left(i \cdot \mathsf{fma}\left(-1, \alpha, -1 \cdot i\right)\right)}{t\_3 \cdot t\_3 - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{i}, -0.125, \frac{\beta}{i} \cdot 0.125 - -0.0625\right)\\ \end{array} \end{array} \]

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 (* 2.0 i))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) 5e-12)
     (/ (* -1.0 (* i (fma -1.0 alpha (* -1.0 i)))) (- (* t_3 t_3) 1.0))
     (fma (/ beta i) -0.125 (- (* (/ beta i) 0.125) -0.0625)))))

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 + (2.0 * i);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 5e-12) {
		tmp = (-1.0 * (i * fma(-1.0, alpha, (-1.0 * i)))) / ((t_3 * t_3) - 1.0);
	} else {
		tmp = fma((beta / i), -0.125, (((beta / i) * 0.125) - -0.0625));
	}
	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(beta + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= 5e-12)
		tmp = Float64(Float64(-1.0 * Float64(i * fma(-1.0, alpha, Float64(-1.0 * i)))) / Float64(Float64(t_3 * t_3) - 1.0));
	else
		tmp = fma(Float64(beta / i), -0.125, Float64(Float64(Float64(beta / i) * 0.125) - -0.0625));
	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[(beta + N[(2.0 * i), $MachinePrecision]), $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], 5e-12], N[(N[(-1.0 * N[(i * N[(-1.0 * alpha + N[(-1.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$3 * t$95$3), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(beta / i), $MachinePrecision] * -0.125 + N[(N[(N[(beta / i), $MachinePrecision] * 0.125), $MachinePrecision] - -0.0625), $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))) < 4.9999999999999997e-12

   1. Initial program 16.1%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 15.4%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 16.4%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 16.5%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 16.6%

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

   14. Taylor expanded in alpha around 0

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

   16. Applied rewrites 15.1%

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

   17. Taylor expanded in alpha around 0

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

   19. Applied rewrites 14.7%

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

   20. Taylor expanded in beta around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-*.f64 11.8

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

   22. Applied rewrites 11.8%

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

   if 4.9999999999999997e-12 < (/.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 16.1%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 76.8

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. lower-/.f64 73.0

         \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied rewrites 73.0%

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\beta}{\color{blue}{i}} \]
   9. Step-by-step derivation

      1. lower-/.f64 74.1

         \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i} \]

   10. Applied rewrites 74.1%

       \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{\color{blue}{i}} \]
   11. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\beta}{i}} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\beta}{i}} \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\beta}{i}} \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\beta}{i} + \color{blue}{\left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right)} \]

       5. metadata-eval N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\beta}{i} + \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) \]

       6. *-commutative N/A

          \[\leadsto \frac{\beta}{i} \cdot \frac{-1}{8} + \left(\color{blue}{\frac{1}{16}} + \frac{1}{8} \cdot \frac{\beta}{i}\right) \]

       7. lower-fma.f64 74.1

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \color{blue}{-0.125}, 0.0625 + 0.125 \cdot \frac{\beta}{i}\right) \]

       8. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{-1}{8}, \frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{-1}{8}, \frac{1}{8} \cdot \frac{\beta}{i} + \frac{1}{16}\right) \]

   12. Applied rewrites 74.1%

       \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \color{blue}{-0.125}, \frac{\beta}{i} \cdot 0.125 - -0.0625\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{i}, -0.125, \frac{\beta}{i} \cdot 0.125 - -0.0625\right)\\ \end{array} \end{array} \]

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))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) 5e-12)
     (/ (* i (+ alpha i)) (pow beta 2.0))
     (fma (/ beta i) -0.125 (- (* (/ beta i) 0.125) -0.0625)))))

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 tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 5e-12) {
		tmp = (i * (alpha + i)) / pow(beta, 2.0);
	} else {
		tmp = fma((beta / i), -0.125, (((beta / i) * 0.125) - -0.0625));
	}
	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))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= 5e-12)
		tmp = Float64(Float64(i * Float64(alpha + i)) / (beta ^ 2.0));
	else
		tmp = fma(Float64(beta / i), -0.125, Float64(Float64(Float64(beta / i) * 0.125) - -0.0625));
	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]}, 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], 5e-12], N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[Power[beta, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(beta / i), $MachinePrecision] * -0.125 + N[(N[(N[(beta / i), $MachinePrecision] * 0.125), $MachinePrecision] - -0.0625), $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))) < 4.9999999999999997e-12

   1. Initial program 16.1%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-pow.f64 9.3

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

   4. Applied rewrites 9.3%

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

   if 4.9999999999999997e-12 < (/.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 16.1%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 76.8

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. lower-/.f64 73.0

         \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied rewrites 73.0%

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\beta}{\color{blue}{i}} \]
   9. Step-by-step derivation

      1. lower-/.f64 74.1

         \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i} \]

   10. Applied rewrites 74.1%

       \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{\color{blue}{i}} \]
   11. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\beta}{i}} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\beta}{i}} \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\beta}{i}} \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\beta}{i} + \color{blue}{\left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right)} \]

       5. metadata-eval N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\beta}{i} + \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) \]

       6. *-commutative N/A

          \[\leadsto \frac{\beta}{i} \cdot \frac{-1}{8} + \left(\color{blue}{\frac{1}{16}} + \frac{1}{8} \cdot \frac{\beta}{i}\right) \]

       7. lower-fma.f64 74.1

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \color{blue}{-0.125}, 0.0625 + 0.125 \cdot \frac{\beta}{i}\right) \]

       8. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{-1}{8}, \frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{-1}{8}, \frac{1}{8} \cdot \frac{\beta}{i} + \frac{1}{16}\right) \]

   12. Applied rewrites 74.1%

       \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \color{blue}{-0.125}, \frac{\beta}{i} \cdot 0.125 - -0.0625\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (- (+ 0.0625 (* 0.125 (/ beta i))) (* 0.125 (/ (+ alpha beta) i))))

C

double code(double alpha, double beta, double i) {
	return (0.0625 + (0.125 * (beta / i))) - (0.125 * ((alpha + beta) / i));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (0.0625d0 + (0.125d0 * (beta / i))) - (0.125d0 * ((alpha + beta) / i))
end function

Java

public static double code(double alpha, double beta, double i) {
	return (0.0625 + (0.125 * (beta / i))) - (0.125 * ((alpha + beta) / i));
}

Python

def code(alpha, beta, i):
	return (0.0625 + (0.125 * (beta / i))) - (0.125 * ((alpha + beta) / i))

Julia

function code(alpha, beta, i)
	return Float64(Float64(0.0625 + Float64(0.125 * Float64(beta / i))) - Float64(0.125 * Float64(Float64(alpha + beta) / i)))
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (0.0625 + (0.125 * (beta / i))) - (0.125 * ((alpha + beta) / i));
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(0.0625 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.1%

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

2. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   4. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

   8. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

   9. lower-+.f64 76.8

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

4. Applied rewrites 76.8%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

5. Taylor expanded in alpha around 0

   \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   2. lower-/.f64 73.0

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

7. Applied rewrites 73.0%

   \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]
8. Add Preprocessing

Alternative 10: 73.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\beta}{i}, -0.125, \frac{\beta}{i} \cdot 0.125 - -0.0625\right) \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (fma (/ beta i) -0.125 (- (* (/ beta i) 0.125) -0.0625)))

C

double code(double alpha, double beta, double i) {
	return fma((beta / i), -0.125, (((beta / i) * 0.125) - -0.0625));
}

Julia

function code(alpha, beta, i)
	return fma(Float64(beta / i), -0.125, Float64(Float64(Float64(beta / i) * 0.125) - -0.0625))
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(beta / i), $MachinePrecision] * -0.125 + N[(N[(N[(beta / i), $MachinePrecision] * 0.125), $MachinePrecision] - -0.0625), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.1%

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

2. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   4. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

   8. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

   9. lower-+.f64 76.8

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

4. Applied rewrites 76.8%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

5. Taylor expanded in alpha around 0

   \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   2. lower-/.f64 73.0

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

7. Applied rewrites 73.0%

   \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

8. Taylor expanded in alpha around 0

   \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\beta}{\color{blue}{i}} \]
9. Step-by-step derivation

   1. lower-/.f64 74.1

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i} \]

10. Applied rewrites 74.1%

    \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{\color{blue}{i}} \]
11. Step-by-step derivation

    1. lift--.f64 N/A

       \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\beta}{i}} \]

    2. lift-*.f64 N/A

       \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\beta}{i}} \]

    3. fp-cancel-sub-sign-inv N/A

       \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\beta}{i}} \]

    4. +-commutative N/A

       \[\leadsto \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{\beta}{i} + \color{blue}{\left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right)} \]

    5. metadata-eval N/A

       \[\leadsto \frac{-1}{8} \cdot \frac{\beta}{i} + \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) \]

    6. *-commutative N/A

       \[\leadsto \frac{\beta}{i} \cdot \frac{-1}{8} + \left(\color{blue}{\frac{1}{16}} + \frac{1}{8} \cdot \frac{\beta}{i}\right) \]

    7. lower-fma.f64 74.1

       \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \color{blue}{-0.125}, 0.0625 + 0.125 \cdot \frac{\beta}{i}\right) \]

    8. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{-1}{8}, \frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) \]

    9. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \frac{-1}{8}, \frac{1}{8} \cdot \frac{\beta}{i} + \frac{1}{16}\right) \]

12. Applied rewrites 74.1%

    \[\leadsto \mathsf{fma}\left(\frac{\beta}{i}, \color{blue}{-0.125}, \frac{\beta}{i} \cdot 0.125 - -0.0625\right) \]
13. Add Preprocessing

Alternative 11: 70.5% accurate, 75.4× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    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 16.1%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 70.5%

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

Reproduce

herbie shell --seed 2025154 
(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)))